{"content": "Equation: What is the solution to $\\frac { 2 } { x - 4 } = \\frac { 3 } { x - 8 }$?", "answer": "x = - 4", "steps": "Going to the denominator, we get: $2 x - 16 = 3 x - 12$. Solving for $x$, we get $x = - 4$. After checking, we find that $x = - 4$ is a solution to the fractional equation.", "expr_cands": ["\\frac { 2 } { x - 4 } = \\frac { 3 } { x - 8 }", "x", "2 x - 16 = 3 x - 12", "x = - 4"], "exprs": ["x = - 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 2 } { x - 4 } = \\frac { 3 } { x - 8 }"}, {"id": "x = - 4"}], "links": [{"rel": "等式方程求解", "source": "\\frac { 2 } { x - 4 } = \\frac { 3 } { x - 8 }", "target": "x = - 4"}]}}
{"content": "If the value of $3 a - 4$ is the opposite of the value of $2 a + 9$, then the value of $a$ is ____?", "answer": "- 1", "steps": "$\\because$ The value of $3 a - 4$ is the opposite of the value of $2 a + 9$, $\\therefore$ $3 a - 4 + 2 a + 9 = 0$, solving for $a$ yields $a = - 1$.", "expr_cands": ["3 a - 4", "a", "2 a + 9", "3 a - 4 + 2 a + 9 = 0", "a = - 1"], "exprs": ["3 a - 4 + 2 a + 9 = 0", "a = - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 a - 4"}, {"id": "3 a - 4 + 2 a + 9 = 0"}, {"id": "2 a + 9"}, {"id": "$3 a - 4$ 的值与 $2 a + 9$ 的值互为相反数"}, {"id": "a = - 1"}], "links": [{"rel": "被描述", "source": "3 a - 4", "target": "3 a - 4 + 2 a + 9 = 0"}, {"rel": "等式方程求解", "source": "3 a - 4 + 2 a + 9 = 0", "target": "a = - 1"}, {"rel": "被描述", "source": "2 a + 9", "target": "3 a - 4 + 2 a + 9 = 0"}, {"rel": "限制性描述", "source": "$3 a - 4$ 的值与 $2 a + 9$ 的值互为相反数", "target": "3 a - 4 + 2 a + 9 = 0"}]}}
{"content": "Calculate: $a ^ { - 1 } \\times a ^ { - 2 } \\times a ^ { - 3 }$ = ____ ?", "answer": "a ^ { - 6 }", "steps": "Because $a ^ { m } \\times a ^ { n } = a ^ { m + n }$, therefore $a ^ { - 1 } \\times a ^ { - 2 } \\times a ^ { - 3 } = a ^ { - 6 }$.", "expr_cands": ["a ^ { - 1 } \\times a ^ { - 2 } \\times a ^ { - 3 }", "a", "a ^ { m } \\times a ^ { n }", "a ^ { m + n }", "m", "n", "a ^ { - 6 }"], "exprs": ["a ^ { - 6 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a ^ { - 1 } \\times a ^ { - 2 } \\times a ^ { - 3 }"}, {"id": "a ^ { - 6 }"}], "links": [{"rel": "计算", "source": "a ^ { - 1 } \\times a ^ { - 2 } \\times a ^ { - 3 }", "target": "a ^ { - 6 }"}]}}
{"content": "If $x _ 1$, $x _ 2$ are the two real roots of the quadratic equation $x ^ 2 - 2 x - 4 = 0$, then $x _ 1 + x _ 2 - x _ 1 x _ 2$ = ____ ?", "answer": "6", "steps": "From the relationship between the root and the coefficient, we know that $x _ 1 + x _ 2 = 2$ and $x _ 1 x _ 2 = - 4$. Therefore, the original expression is equal to $2 - ( - 4 ) = 6$.", "expr_cands": ["x _ { 1 }", "x _ { 2 }", "x ^ { 2 } - 2 x - 4 = 0", "x", "x _ { 1 } + x _ { 2 } - x _ { 1 } x _ { 2 }", "x _ { 1 } + x _ { 2 } = 2", "x _ { 1 } x _ { 2 } = - 4", "2 - ( - 4 )", "6"], "exprs": ["x _ { 1 } + x _ { 2 } = 2", "x _ { 1 } x _ { 2 } = - 4", "2 - ( - 4 )", "6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x _ { 1 }"}, {"id": "x _ { 1 } + x _ { 2 } = 2"}, {"id": "x _ { 2 }"}, {"id": "x ^ { 2 } - 2 x - 4 = 0"}, {"id": "$x _ { 1 }$ , $x _ { 2 }$ 是一元二次方程 $x ^ { 2 } - 2 x - 4 = 0$ 的两个实数根"}, {"id": "一元二次方程根与系数关系，两根之和"}, {"id": "x _ { 1 } x _ { 2 } = - 4"}, {"id": "一元二次方程根与系数关系，两根之积"}, {"id": "x _ { 1 } + x _ { 2 } - x _ { 1 } x _ { 2 }"}, {"id": "2 - ( - 4 )"}, {"id": "6"}], "links": [{"rel": "被描述", "source": "x _ { 1 }", "target": "x _ { 1 } + x _ { 2 } = 2"}, {"rel": "被描述", "source": "x _ { 1 }", "target": "x _ { 1 } x _ { 2 } = - 4"}, {"rel": "代入", "source": "x _ { 1 } + x _ { 2 } = 2", "target": "2 - ( - 4 )"}, {"rel": "被描述", "source": "x _ { 2 }", "target": "x _ { 1 } + x _ { 2 } = 2"}, {"rel": "被描述", "source": "x _ { 2 }", "target": "x _ { 1 } x _ { 2 } = - 4"}, {"rel": "被描述", "source": "x ^ { 2 } - 2 x - 4 = 0", "target": "x _ { 1 } + x _ { 2 } = 2"}, {"rel": "被描述", "source": "x ^ { 2 } - 2 x - 4 = 0", "target": "x _ { 1 } x _ { 2 } = - 4"}, {"rel": "限制性描述", "source": "$x _ { 1 }$ , $x _ { 2 }$ 是一元二次方程 $x ^ { 2 } - 2 x - 4 = 0$ 的两个实数根", "target": "x _ { 1 } + x _ { 2 } = 2"}, {"rel": "限制性描述", "source": "$x _ { 1 }$ , $x _ { 2 }$ 是一元二次方程 $x ^ { 2 } - 2 x - 4 = 0$ 的两个实数根", "target": "x _ { 1 } x _ { 2 } = - 4"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "x _ { 1 } + x _ { 2 } = 2"}, {"rel": "代入", "source": "x _ { 1 } x _ { 2 } = - 4", "target": "2 - ( - 4 )"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之积", "target": "x _ { 1 } x _ { 2 } = - 4"}, {"rel": "被代入", "source": "x _ { 1 } + x _ { 2 } - x _ { 1 } x _ { 2 }", "target": "2 - ( - 4 )"}, {"rel": "计算", "source": "2 - ( - 4 )", "target": "6"}]}}
{"content": "Find the value of $a$ that makes the expression $\\frac { 2 a - 1 } { a + 2 }$ equal to $0$.", "answer": "\\frac { 1 } { 2 }", "steps": "$\\because$ The value of the expression $\\frac { 2 a - 1 } { a + 2 }$ is $0$, $\\therefore$ $2 a - 1 = 0$, solving for $a$ gives $a = \\frac { 1 } { 2 }$.", "expr_cands": ["\\frac { 2 a - 1 } { a + 2 }", "a", "0", "2 a - 1 = 0", "a = \\frac { 1 } { 2 }"], "exprs": ["2 a - 1 = 0", "a = \\frac { 1 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 2 a - 1 } { a + 2 }"}, {"id": "2 a - 1 = 0"}, {"id": "0"}, {"id": "使式子 $\\frac { 2 a - 1 } { a + 2 }$ 的值为 $0$"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "a = \\frac { 1 } { 2 }"}], "links": [{"rel": "被描述", "source": "\\frac { 2 a - 1 } { a + 2 }", "target": "2 a - 1 = 0"}, {"rel": "等式方程求解", "source": "2 a - 1 = 0", "target": "a = \\frac { 1 } { 2 }"}, {"rel": "被描述", "source": "0", "target": "2 a - 1 = 0"}, {"rel": "限制性描述", "source": "使式子 $\\frac { 2 a - 1 } { a + 2 }$ 的值为 $0$", "target": "2 a - 1 = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "2 a - 1 = 0"}]}}
{"content": "The equation in terms of $x$ is $4 m - 3 x = 1$. If the solution to this equation is $x = - 1$, then the value of $m$ is...", "answer": "- \\frac { 1 } { 2 }", "steps": "$\\because$ The solution to the one-variable linear equation $4 m - 3 x = 1$ with respect to $x$ is $x = - 1$, $\\therefore$ $4 m + 3 = - 1$, which gives the solution $m = - \\frac { 1 } { 2 }$.", "expr_cands": ["x", "4 m - 3 x = 1", "m", "x = - 1", "4 m + 3 = 1", "m = - \\frac { 1 } { 2 }"], "exprs": ["4 m + 3 = 1", "m = - \\frac { 1 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 m - 3 x = 1"}, {"id": "4 m + 3 = 1"}, {"id": "x = - 1"}, {"id": "m = - \\frac { 1 } { 2 }"}], "links": [{"rel": "被代入", "source": "4 m - 3 x = 1", "target": "4 m + 3 = 1"}, {"rel": "等式方程求解", "source": "4 m + 3 = 1", "target": "m = - \\frac { 1 } { 2 }"}, {"rel": "代入", "source": "x = - 1", "target": "4 m + 3 = 1"}]}}
{"content": "Given $a = 20$, $b = - 30$, $c = - 36$, what is $- a - b - c$?", "answer": "46", "steps": "$a = 20$ , $b = - 30$ , $c = - 36$ , $- a - b - c = - 20 - ( - 30 ) - ( - 36 ) = - 20 + 30 + 36 = 46$ This is a mathematical expression that defines the values of variables $a$, $b$, and $c$. It then calculates the value of $- a - b - c$ by substituting the given values and simplifying the expression. The final result is $46$.", "expr_cands": ["a = 20", "a", "b = - 30", "b", "c = - 36", "c", "- a - b - c", "46"], "exprs": ["46"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- a - b - c"}, {"id": "46"}, {"id": "a = 20"}, {"id": "b = - 30"}, {"id": "c = - 36"}], "links": [{"rel": "被代入", "source": "- a - b - c", "target": "46"}, {"rel": "代入", "source": "a = 20", "target": "46"}, {"rel": "代入", "source": "b = - 30", "target": "46"}, {"rel": "代入", "source": "c = - 36", "target": "46"}]}}
{"content": "If $a - 3$ is the arithmetic square root of a number, then ____?", "answer": "a \\ge 3", "steps": "A non-negative number has an arithmetic square root, and the arithmetic square root of the number is greater than or equal to $0$. Therefore, $a - 3 \\geq 0$, which means $a \\geq 3$.", "expr_cands": ["a - 3", "a", "0", "a - 3 \\ge 0", "3 \\le a", "a \\ge 3"], "exprs": ["a - 3 \\ge 0", "a \\ge 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a - 3"}, {"id": "a - 3 \\ge 0"}, {"id": "$a - 3$ 是一个数的算术平方根"}, {"id": "一个非负数才有算术平方根"}, {"id": "且该数的算术平方根大于或等于 $0$"}, {"id": "a \\ge 3"}], "links": [{"rel": "被描述", "source": "a - 3", "target": "a - 3 \\ge 0"}, {"rel": "不等式方程求解", "source": "a - 3 \\ge 0", "target": "a \\ge 3"}, {"rel": "限制性描述", "source": "$a - 3$ 是一个数的算术平方根", "target": "a - 3 \\ge 0"}, {"rel": "限制性描述", "source": "一个非负数才有算术平方根", "target": "a - 3 \\ge 0"}, {"rel": "限制性描述", "source": "且该数的算术平方根大于或等于 $0$", "target": "a - 3 \\ge 0"}]}}
{"content": "Given: $\\frac { x } { y } = \\frac { 2 } { 3 }$, then $\\frac { 2 x - y } { x + y } =$ ____?", "answer": "\\frac { 1 } { 5 }", "steps": "Because $\\frac { x } { y } = \\frac { 2 } { 3 }$, therefore let $x = 2 a$, $y = 3 a$, thus $\\frac { 2 x - y } { x + y } = \\frac { 4 a - 3 a } { 2 a + 3 a } = \\frac { 1 } { 5 }$.", "expr_cands": ["\\frac { x } { y } = \\frac { 2 } { 3 }", "y", "x", "\\frac { 2 x - y } { x + y }", "x = 2 a", "a", "y = 3 a", "\\frac { 1 } { 5 }"], "exprs": ["x = 2 a", "y = 3 a", "\\frac { 1 } { 5 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "设 $x = 2 a$ , $y = 3 a$"}, {"id": "x = 2 a"}, {"id": "y = 3 a"}, {"id": "\\frac { 2 x - y } { x + y }"}, {"id": "\\frac { 1 } { 5 }"}], "links": [{"rel": "假设描述", "source": "设 $x = 2 a$ , $y = 3 a$", "target": "x = 2 a"}, {"rel": "假设描述", "source": "设 $x = 2 a$ , $y = 3 a$", "target": "y = 3 a"}, {"rel": "代入", "source": "x = 2 a", "target": "\\frac { 1 } { 5 }"}, {"rel": "代入", "source": "y = 3 a", "target": "\\frac { 1 } { 5 }"}, {"rel": "被代入", "source": "\\frac { 2 x - y } { x + y }", "target": "\\frac { 1 } { 5 }"}]}}
{"content": "Given that $- 5$ is a solution to the equation $ax + b = 0$ in terms of $x$, what is the solution to the equation $a ( x + 3 ) + b = 0$ in terms of $x$?", "answer": "x = - 8", "steps": "Since $- 5$ is a solution of the equation $ax + b = 0$ with respect to $x$, we have $- 5 a + b = 0$. Therefore, $b = 5 a$. Since $a ( x + 3 ) + b = 0$, we can solve for $x$ to get $x = - \\frac { 3 a + b } { a }$. Substituting $b = 5 a$ into $x = - \\frac { 3 a + b } { a }$, we get $x = - \\frac { 3 a + 5 a } { a } = - 8$.", "expr_cands": ["- 5", "x", "ax + b = 0", "a", "b", "a ( x + 3 ) + b = 0", "- 5 a + b = 0", "b = 5 a", "a ( x + 3 ) + 5 a = 0", "x = \\frac { - 3 a - b } { a }", "x = - 8", "5 a", "\\frac { - 3 a - b } { a } = - 8", "\\frac { - 3 a - b } { a }"], "exprs": ["- 5 a + b = 0", "x = \\frac { - 3 a - b } { a }", "b = 5 a", "x = - 8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 5"}, {"id": "- 5 a + b = 0"}, {"id": "x"}, {"id": "ax + b = 0"}, {"id": "$- 5$ 是关于 $x$ 的方程 $ax + b = 0$ 的解"}, {"id": "b = 5 a"}, {"id": "a ( x + 3 ) + b = 0"}, {"id": "x = \\frac { - 3 a - b } { a }"}, {"id": "x = - 8"}], "links": [{"rel": "被描述", "source": "- 5", "target": "- 5 a + b = 0"}, {"rel": "移项", "source": "- 5 a + b = 0", "target": "b = 5 a"}, {"rel": "被描述", "source": "x", "target": "- 5 a + b = 0"}, {"rel": "被描述", "source": "ax + b = 0", "target": "- 5 a + b = 0"}, {"rel": "限制性描述", "source": "$- 5$ 是关于 $x$ 的方程 $ax + b = 0$ 的解", "target": "- 5 a + b = 0"}, {"rel": "代入", "source": "b = 5 a", "target": "x = - 8"}, {"rel": "等式方程部分求解", "source": "a ( x + 3 ) + b = 0", "target": "x = \\frac { - 3 a - b } { a }"}, {"rel": "被代入", "source": "x = \\frac { - 3 a - b } { a }", "target": "x = - 8"}]}}
{"content": "The line $y = 2 ( x - 1 )$ translated down by $3$ units becomes ____ ?", "answer": "y = 2 x - 5", "steps": "The line $y = 2 ( x - 1 )$ is translated down by $3$ units to obtain the equation $y = 2 ( x - 1 ) - 3$, which simplifies to $y = 2 x - 5$.", "expr_cands": ["y = 2 ( x - 1 )", "x", "y", "3", "y = 2 ( x - 1 ) - 3", "2 x - 2 = 2 ( x - 1 ) - 3", "2 x - 5"], "exprs": ["2 x - 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = 2 ( x - 1 )"}, {"id": "2 x - 5"}, {"id": "3"}, {"id": "直线 $y = 2 ( x - 1 )$ 向下平移 $3$ 个单位长度得到的直线"}], "links": [{"rel": "被描述", "source": "y = 2 ( x - 1 )", "target": "2 x - 5"}, {"rel": "被描述", "source": "3", "target": "2 x - 5"}, {"rel": "限制性描述", "source": "直线 $y = 2 ( x - 1 )$ 向下平移 $3$ 个单位长度得到的直线", "target": "2 x - 5"}]}}
{"content": "If the equation $6 x + 3 a = 22$ and the equation $5 ( x + 1 ) = 4 x + 7$ have the same solution for $x$, then the value of $a$ is ____?", "answer": "\\frac { 10 } { 3 }", "steps": "$5 ( x + 1 ) = 4 x + 7$, remove parentheses, $5 x + 5 = 4 x + 7$, move terms, we get $5 x - 4 x = 7 - 5$, combine like terms, we get $x = 2$. Substitute $x = 2$ into $6 x + 3 a = 22$, we get $6 * 2 + 3 a = 22$, $\\therefore a = \\frac { 10 } { 3 }$.", "expr_cands": ["x", "6 x + 3 a = 22", "a", "5 ( x + 1 ) = 4 x + 7", "x = 2", "5 x + 5 = 4 x + 7", "5 x - 4 x = 7 - 5", "3 a + 12 = 22", "6 * 2 + 3 a = 22", "a = \\frac { 10 } { 3 }"], "exprs": ["x = 2", "6 * 2 + 3 a = 22", "a = \\frac { 10 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5 ( x + 1 ) = 4 x + 7"}, {"id": "x = 2"}, {"id": "6 x + 3 a = 22"}, {"id": "6 * 2 + 3 a = 22"}, {"id": "a = \\frac { 10 } { 3 }"}], "links": [{"rel": "等式方程求解", "source": "5 ( x + 1 ) = 4 x + 7", "target": "x = 2"}, {"rel": "代入", "source": "x = 2", "target": "6 * 2 + 3 a = 22"}, {"rel": "被代入", "source": "6 x + 3 a = 22", "target": "6 * 2 + 3 a = 22"}, {"rel": "等式方程求解", "source": "6 * 2 + 3 a = 22", "target": "a = \\frac { 10 } { 3 }"}]}}
{"content": "Given that $x = - 3$ is a solution of the linear equation $6 - ax = x$, then $a$ = ____ ?", "answer": "- 3", "steps": "Substituting $x = - 3$ into $6 - ax = x$, we get: $6 + 3 a = - 3$. Solving for $a$, we get: $a = - 3$.", "expr_cands": ["x = - 3", "x", "6 - ax = x", "a", "3 a + 6 = - 3", "6 + 3 a = - 3", "a = - 3"], "exprs": ["6 + 3 a = - 3", "a = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "6 - ax = x"}, {"id": "6 + 3 a = - 3"}, {"id": "x = - 3"}, {"id": "a = - 3"}], "links": [{"rel": "被代入", "source": "6 - ax = x", "target": "6 + 3 a = - 3"}, {"rel": "等式方程求解", "source": "6 + 3 a = - 3", "target": "a = - 3"}, {"rel": "代入", "source": "x = - 3", "target": "6 + 3 a = - 3"}]}}
{"content": "Given $| a + 2019 | = - | b - 2020 |$, what is $a + b$?", "answer": "1", "steps": "According to the problem, we know that $| a + 2019 | + | b - 2020 | = 0$. Since $| a + 2019 |$ and $| b - 2020 |$ are both non-negative, we have $a + 2019 = 0$ and $b - 2020 = 0$. Solving for $a$ and $b$, we get $a = - 2019$ and $b = 2020$. Therefore, $a + b = - 2019 + 2020 = 1$.", "expr_cands": ["| a + 2019 | = - | b - 2020 |", "b", "a", "a + b", "| a + 2019 | + | b - 2020 | = 0", "| a + 2019 |", "| b - 2020 |", "a + 2019 = 0", "a = - 2019", "b - 2020 = 0", "b = 2020", "1"], "exprs": ["| a + 2019 | + | b - 2020 | = 0", "a + 2019 = 0", "b - 2020 = 0", "a = - 2019", "b = 2020", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| a + 2019 | = - | b - 2020 |"}, {"id": "| a + 2019 | + | b - 2020 | = 0"}, {"id": "a + 2019 = 0"}, {"id": "绝对值恒大于等于0"}, {"id": "b - 2020 = 0"}, {"id": "a = - 2019"}, {"id": "b = 2020"}, {"id": "a + b"}, {"id": "1"}], "links": [{"rel": "移项", "source": "| a + 2019 | = - | b - 2020 |", "target": "| a + 2019 | + | b - 2020 | = 0"}, {"rel": "被描述", "source": "| a + 2019 | + | b - 2020 | = 0", "target": "a + 2019 = 0"}, {"rel": "被描述", "source": "| a + 2019 | + | b - 2020 | = 0", "target": "b - 2020 = 0"}, {"rel": "等式方程求解", "source": "a + 2019 = 0", "target": "a = - 2019"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "a + 2019 = 0"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "b - 2020 = 0"}, {"rel": "等式方程求解", "source": "b - 2020 = 0", "target": "b = 2020"}, {"rel": "代入", "source": "a = - 2019", "target": "1"}, {"rel": "代入", "source": "b = 2020", "target": "1"}, {"rel": "被代入", "source": "a + b", "target": "1"}]}}
{"content": "When $y$ = ____ ?, the value of the algebraic expression $\\frac { 2 y + 5 } { 3 }$ is the opposite of $2$.", "answer": "- \\frac { 11 } { 2 }", "steps": "$\\because$ The value of the algebraic expression $\\frac { 2 y + 5 } { 3 }$ is the opposite of $2$, $\\therefore$ $\\frac { 2 y + 5 } { 3 } + 2 = 0$, $\\therefore$ $2 y + 5 + 6 = 0$, $\\therefore$ $y = - \\frac { 11 } { 2 }$.", "expr_cands": ["y", "\\frac { 2 y + 5 } { 3 }", "2", "\\frac { 2 y + 5 } { 3 } + 2 = 0", "y = - \\frac { 11 } { 2 }", "2 y + 5 + 6 = 0"], "exprs": ["\\frac { 2 y + 5 } { 3 } + 2 = 0", "y = - \\frac { 11 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 2 y + 5 } { 3 }"}, {"id": "\\frac { 2 y + 5 } { 3 } + 2 = 0"}, {"id": "2"}, {"id": "代数式 $\\frac { 2 y + 5 } { 3 }$ 的值与 $2$ 互为相反数"}, {"id": "y = - \\frac { 11 } { 2 }"}], "links": [{"rel": "被描述", "source": "\\frac { 2 y + 5 } { 3 }", "target": "\\frac { 2 y + 5 } { 3 } + 2 = 0"}, {"rel": "等式方程求解", "source": "\\frac { 2 y + 5 } { 3 } + 2 = 0", "target": "y = - \\frac { 11 } { 2 }"}, {"rel": "被描述", "source": "2", "target": "\\frac { 2 y + 5 } { 3 } + 2 = 0"}, {"rel": "限制性描述", "source": "代数式 $\\frac { 2 y + 5 } { 3 }$ 的值与 $2$ 互为相反数", "target": "\\frac { 2 y + 5 } { 3 } + 2 = 0"}]}}
{"content": "The degree of the polynomial $3 mn ^ { 2 } - 2 mn + 5$ is ____ ?", "answer": "3", "steps": "The degree of the polynomial $3 mn ^ { 2 } - 2 mn + 5$ is $1 + 2 = 3$.", "expr_cands": ["3 mn ^ { 2 } - 2 mn + 5", "n", "m", "1 + 2", "3"], "exprs": ["3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 mn ^ { 2 } - 2 mn + 5"}, {"id": "3"}, {"id": "多项式 $3 mn ^ { 2 } - 2 mn + 5$ 的次数"}], "links": [{"rel": "被描述", "source": "3 mn ^ { 2 } - 2 mn + 5", "target": "3"}, {"rel": "限制性描述", "source": "多项式 $3 mn ^ { 2 } - 2 mn + 5$ 的次数", "target": "3"}]}}
{"content": "The minimum value of the expression $\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }$ is _____.", "answer": "26", "steps": "From the given conditions, we have $2 x + 90 \\geq 0$, $x + 59 \\geq 0$, and $x - 5 \\geq 0$. Solving these inequalities, we get $x \\geq 5$. Therefore, when $x = 5$, the expression $\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }$ attains its minimum value, which is $26$.", "expr_cands": ["\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }", "x", "2 x + 90 \\ge 0", "- 45 \\le x", "x + 59 \\ge 0", "- 59 \\le x", "x - 5 \\ge 0", "5 \\le x", "x \\ge 5", "x = 5", "26"], "exprs": ["2 x + 90 \\ge 0", "x + 59 \\ge 0", "x - 5 \\ge 0", "x = 5", "26"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }"}, {"id": "2 x + 90 \\ge 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "式子 $\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }$ 的最小值"}, {"id": "x + 59 \\ge 0"}, {"id": "x - 5 \\ge 0"}, {"id": "x = 5"}, {"id": "式子 $\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }$ 的值最小"}, {"id": "26"}], "links": [{"rel": "被描述", "source": "\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }", "target": "2 x + 90 \\ge 0"}, {"rel": "被描述", "source": "\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }", "target": "x + 59 \\ge 0"}, {"rel": "被描述", "source": "\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }", "target": "x - 5 \\ge 0"}, {"rel": "被代入", "source": "\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }", "target": "26"}, {"rel": "被描述", "source": "2 x + 90 \\ge 0", "target": "x = 5"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "2 x + 90 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x + 59 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 5 \\ge 0"}, {"rel": "限制性描述", "source": "式子 $\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }$ 的最小值", "target": "2 x + 90 \\ge 0"}, {"rel": "限制性描述", "source": "式子 $\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }$ 的最小值", "target": "x + 59 \\ge 0"}, {"rel": "限制性描述", "source": "式子 $\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }$ 的最小值", "target": "x - 5 \\ge 0"}, {"rel": "限制性描述", "source": "式子 $\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }$ 的最小值", "target": "x = 5"}, {"rel": "被描述", "source": "x + 59 \\ge 0", "target": "x = 5"}, {"rel": "被描述", "source": "x - 5 \\ge 0", "target": "x = 5"}, {"rel": "代入", "source": "x = 5", "target": "26"}, {"rel": "限制性描述", "source": "式子 $\\sqrt { 2 x + 90 } + 2 \\sqrt { x + 59 } + 3 \\sqrt { x - 5 }$ 的值最小", "target": "x = 5"}]}}
{"content": "Given $\\frac { c } { 3 } = \\frac { b } { 9 } = \\frac { a } { 2 } \\neq 0$, find $\\frac { b + c } { a }$ in the form of $a / b$.", "answer": "6", "steps": "Let $\\frac { c } { 3 } = \\frac { b } { 9 } = \\frac { a } { 2 } = k$, then we have $c = 3 k$, $b = 9 k$, and $a = 2 k$. Substituting $c = 3 k$, $b = 9 k$, and $a = 2 k$ into $\\frac { b + c } { a }$, we get $\\frac { 3 k + 9 k } { 2 k } = 6$.", "expr_cands": ["\\frac { c } { 3 } = \\frac { b } { 9 } = \\frac { a } { 2 } \\neq 0", "\\frac { b + c } { a }", "b", "a", "c", "a / b", "\\frac { c } { 3 } = k", "k", "c = 3 k", "b = 9 k", "a = 2 k", "3 k", "9 k", "2 k", "\\frac { 3 k + 9 k } { 2 k }", "6"], "exprs": ["c = 3 k", "b = 9 k", "a = 2 k", "6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "设 $\\frac { c } { 3 } = \\frac { b } { 9 } = \\frac { a } { 2 } = k$"}, {"id": "c = 3 k"}, {"id": "b = 9 k"}, {"id": "a = 2 k"}, {"id": "\\frac { 3 k + 9 k } { 2 k }"}, {"id": "6"}], "links": [{"rel": "假设描述", "source": "设 $\\frac { c } { 3 } = \\frac { b } { 9 } = \\frac { a } { 2 } = k$", "target": "c = 3 k"}, {"rel": "假设描述", "source": "设 $\\frac { c } { 3 } = \\frac { b } { 9 } = \\frac { a } { 2 } = k$", "target": "b = 9 k"}, {"rel": "假设描述", "source": "设 $\\frac { c } { 3 } = \\frac { b } { 9 } = \\frac { a } { 2 } = k$", "target": "a = 2 k"}, {"rel": "代入", "source": "c = 3 k", "target": "6"}, {"rel": "代入", "source": "b = 9 k", "target": "6"}, {"rel": "代入", "source": "a = 2 k", "target": "6"}, {"rel": "被代入", "source": "\\frac { 3 k + 9 k } { 2 k }", "target": "6"}]}}
{"content": "Given the inequality $\\frac { 1 + x } { 2 } < \\frac { 1 + 2 x } { 3 } + 1$ with solution set $x > - 5$, find the solution set of the inequality $\\frac { 1 + ( 3 x - 1 )} { 2 } < \\frac { 1 + 2 ( 3 x - 1 )} { 3 } + 1$.", "answer": "x > - \\frac { 4 } { 3 }", "steps": "$\\because$ The solution to the inequality $\\frac { 1 + x } { 2 } < \\frac { 1 + 2 x } { 3 } + 1$ is $x > - 5$, $\\therefore$ in the inequality $\\frac { 1 + ( 3 x - 1 )} { 2 } < \\frac { 1 + 2 ( 3 x - 1 )} { 3 } + 1$, we have $3 x - 1 > - 5$, and solving for $x$ gives $x > - \\frac { 4 } { 3 }$.", "expr_cands": ["\\frac { 1 + x } { 2 } < \\frac { 1 + 2 x } { 3 } + 1", "x", "x > - 5", "\\frac { 1 + ( 3 x - 1 ) } { 2 } < \\frac { 1 + 2 ( 3 x - 1 ) } { 3 } + 1", "- 5 < x", "- \\frac { 4 } { 3 } < x", "3 x - 1 > - 5", "x > - \\frac { 4 } { 3 }"], "exprs": ["x > - \\frac { 4 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 1 + ( 3 x - 1 ) } { 2 } < \\frac { 1 + 2 ( 3 x - 1 ) } { 3 } + 1"}, {"id": "x > - \\frac { 4 } { 3 }"}], "links": [{"rel": "不等式方程求解", "source": "\\frac { 1 + ( 3 x - 1 ) } { 2 } < \\frac { 1 + 2 ( 3 x - 1 ) } { 3 } + 1", "target": "x > - \\frac { 4 } { 3 }"}]}}
{"content": "If the function $y = kx ^ { k - 2 }$ is an inverse proportion function, then $k$ = ____?", "answer": "1", "steps": "According to the problem, we have $k - 2 = - 1$, and $k \\neq 0$. Solving for $k$, we get $k = 1$.", "expr_cands": ["y = kx ^ { k - 2 }", "k", "y", "x", "k - 2 = - 1", "k = 1", "k \\neq 0"], "exprs": ["k - 2 = - 1", "k = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = kx ^ { k - 2 }"}, {"id": "k - 2 = - 1"}, {"id": "函数 $y = kx ^ { k - 2 }$ 是反比例函数"}, {"id": "且 $k \\neq 0$"}, {"id": "k = 1"}], "links": [{"rel": "被描述", "source": "y = kx ^ { k - 2 }", "target": "k - 2 = - 1"}, {"rel": "等式方程求解", "source": "k - 2 = - 1", "target": "k = 1"}, {"rel": "限制性描述", "source": "函数 $y = kx ^ { k - 2 }$ 是反比例函数", "target": "k - 2 = - 1"}, {"rel": "限制性描述", "source": "且 $k \\neq 0$", "target": "k - 2 = - 1"}]}}
{"content": "Given $m = \\sqrt { 2 } + 2$, find $( 1 - \\frac { 1 } { m - 1 }) ( 1 - m )$.", "answer": "- \\sqrt { 2 }", "steps": "Original expression = $( \\frac { m - 1 } { m - 1 } - \\frac { 1 } { m - 1 } ) * ( 1 - m ) = \\frac { m - 2 } { m - 1 } * ( 1 - m ) = 2 - m$. When $m = \\sqrt { 2 } + 2$, the original expression equals $2 - \\sqrt { 2 } - 2 = - \\sqrt { 2 }$.", "expr_cands": ["m = \\sqrt { 2 } + 2", "m", "( 1 - \\frac { 1 } { m - 1 } ) ( 1 - m )", "( \\frac { m - 1 } { m - 1 } - \\frac { 1 } { m - 1 } ) * ( 1 - m )", "2 - m", "2 - \\sqrt { 2 } - 2", "- \\sqrt { 2 }"], "exprs": ["- \\sqrt { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "m = \\sqrt { 2 } + 2"}, {"id": "- \\sqrt { 2 }"}, {"id": "( 1 - \\frac { 1 } { m - 1 } ) ( 1 - m )"}], "links": [{"rel": "代入", "source": "m = \\sqrt { 2 } + 2", "target": "- \\sqrt { 2 }"}, {"rel": "被代入", "source": "( 1 - \\frac { 1 } { m - 1 } ) ( 1 - m )", "target": "- \\sqrt { 2 }"}]}}
{"content": "Given that $x$ and $y$ are real numbers and $y = \\sqrt { x - 6 } + \\sqrt { 6 - x } + 25$, what is the arithmetic square root of $x + 3 y$?", "answer": "9", "steps": "$\\because y = \\sqrt { x - 6 } + \\sqrt { 6 - x } + 25$, by the definition of square roots, we have $x - 6 \\ge 0$, $6 - x \\ge 0$, $\\therefore x - 6 = 6 - x = 0$, $\\therefore x = 6$, then $y = 25$, so the arithmetic square root of $x + 3 y = 81$ is $9$.", "expr_cands": ["x", "y", "y = \\sqrt { x - 6 } + \\sqrt { 6 - x } + 25", "x + 3 y", "x - 6 \\ge 0", "6 \\le x", "6 - x \\ge 0", "x \\le 6", "x - 6 = 0", "x = 6", "y = 25", "81", "9"], "exprs": ["x - 6 \\ge 0", "6 - x \\ge 0", "x = 6", "y = 25", "9"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = \\sqrt { x - 6 } + \\sqrt { 6 - x } + 25"}, {"id": "x - 6 \\ge 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "6 - x \\ge 0"}, {"id": "x = 6"}, {"id": "y = 25"}, {"id": "x + 3 y"}, {"id": "9"}, {"id": "$x + 3 y$ 的算术平方根"}], "links": [{"rel": "被描述", "source": "y = \\sqrt { x - 6 } + \\sqrt { 6 - x } + 25", "target": "x - 6 \\ge 0"}, {"rel": "被描述", "source": "y = \\sqrt { x - 6 } + \\sqrt { 6 - x } + 25", "target": "6 - x \\ge 0"}, {"rel": "被代入", "source": "y = \\sqrt { x - 6 } + \\sqrt { 6 - x } + 25", "target": "y = 25"}, {"rel": "联立", "source": "x - 6 \\ge 0", "target": "x = 6"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 6 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "6 - x \\ge 0"}, {"rel": "联立", "source": "6 - x \\ge 0", "target": "x = 6"}, {"rel": "代入", "source": "x = 6", "target": "y = 25"}, {"rel": "被描述", "source": "x = 6", "target": "9"}, {"rel": "被描述", "source": "y = 25", "target": "9"}, {"rel": "被描述", "source": "x + 3 y", "target": "9"}, {"rel": "限制性描述", "source": "$x + 3 y$ 的算术平方根", "target": "9"}]}}
{"content": "If a real number $a$ satisfies $| a - 8 | + \\sqrt { a - 9 } = a$, then $a$ = ____?", "answer": "73", "steps": "According to the problem, we have $a - 9 \\ge 0$, which implies $a \\ge 9$. Therefore, the original equation can be rewritten as $a - 8 + \\sqrt { a - 9 } = a$. This simplifies to $\\sqrt { a - 9 } = 8$, which implies $a - 9 = 64$. Solving for $a$, we get $a = 73$.", "expr_cands": ["a", "| a - 8 | + \\sqrt { a - 9 } = a", "a - 9 \\ge 0", "9 \\le a", "a \\ge 9", "a - 8 + \\sqrt { a - 9 } = a", "a = 73", "\\sqrt { a - 9 } = 8", "a - 9 = 64"], "exprs": ["a - 9 \\ge 0", "a \\ge 9", "a - 8 + \\sqrt { a - 9 } = a", "a = 73"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| a - 8 | + \\sqrt { a - 9 } = a"}, {"id": "a - 9 \\ge 0"}, {"id": "实数 $a$ 满足 $| a - 8 | + \\sqrt { a - 9 } = a$"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "a \\ge 9"}, {"id": "a - 8 + \\sqrt { a - 9 } = a"}, {"id": "a = 73"}], "links": [{"rel": "被描述", "source": "| a - 8 | + \\sqrt { a - 9 } = a", "target": "a - 9 \\ge 0"}, {"rel": "联立", "source": "| a - 8 | + \\sqrt { a - 9 } = a", "target": "a - 8 + \\sqrt { a - 9 } = a"}, {"rel": "不等式方程求解", "source": "a - 9 \\ge 0", "target": "a \\ge 9"}, {"rel": "限制性描述", "source": "实数 $a$ 满足 $| a - 8 | + \\sqrt { a - 9 } = a$", "target": "a - 9 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "a - 9 \\ge 0"}, {"rel": "联立", "source": "a \\ge 9", "target": "a - 8 + \\sqrt { a - 9 } = a"}, {"rel": "等式方程求解", "source": "a - 8 + \\sqrt { a - 9 } = a", "target": "a = 73"}]}}
{"content": "The solution set of the inequality $4 x > 8$ is ____ ?", "answer": "x > 2", "steps": "$4 x > 8$ , divide both sides by $4$ we get: $x > 2$.", "expr_cands": ["4 x > 8", "x", "2 < x", "4", "x > 2"], "exprs": ["x > 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 x > 8"}, {"id": "x > 2"}], "links": [{"rel": "不等式方程求解", "source": "4 x > 8", "target": "x > 2"}]}}
{"content": "Given that $x$ is an integer and $\\frac { 1 } { x + 2 } + \\frac { 1 } { x - 2 } + \\frac { x + 6 } { x ^ 2 - 4 }$ is an integer, what is the sum of all possible values of $x$ that satisfy this condition?", "answer": "8", "steps": "$\\frac { 1 } { x + 2 } + \\frac { 1 } { x - 2 } + \\frac { x + 6 } { x ^ { 2 } - 4 } = \\frac { x - 2 } { x ^ { 2 } - 4 } + \\frac { x + 2 } { x ^ { 2 } - 4 } + \\frac { x + 6 } { x ^ { 2 } - 4 } = \\frac { 3 x + 6 } { x ^ { 2 } - 4 } = \\frac { 3 } { x - 2 }$, because $x$ is an integer and $\\frac { 1 } { x + 2 } + \\frac { 1 } { x - 2 } + \\frac { x + 6 } { x ^ { 2 } - 4 }$ is an integer, therefore the possible values of $x$ are $- 1$, $1$, $3$, $5$; hence the sum of all possible values of $x$ is $8$.", "expr_cands": ["x", "\\frac { 1 } { x + 2 } + \\frac { 1 } { x - 2 } + \\frac { x + 6 } { x ^ { 2 } - 4 }", "\\frac { 3 } { x - 2 }", "- 1", "1", "3", "5", "8"], "exprs": ["\\frac { 3 } { x - 2 }", "- 1", "1", "3", "5", "8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 1 } { x + 2 } + \\frac { 1 } { x - 2 } + \\frac { x + 6 } { x ^ { 2 } - 4 }"}, {"id": "\\frac { 3 } { x - 2 }"}, {"id": "- 1"}, {"id": "$x$ 为整数"}, {"id": "且 $\\frac { 1 } { x + 2 } + \\frac { 1 } { x - 2 } + \\frac { x + 6 } { x ^ { 2 } - 4 }$ 为整数"}, {"id": "符合条件的x的值"}, {"id": "1"}, {"id": "3"}, {"id": "5"}, {"id": "8"}, {"id": "所有符合条件的 $x$ 值的和"}], "links": [{"rel": "计算", "source": "\\frac { 1 } { x + 2 } + \\frac { 1 } { x - 2 } + \\frac { x + 6 } { x ^ { 2 } - 4 }", "target": "\\frac { 3 } { x - 2 }"}, {"rel": "被描述", "source": "\\frac { 3 } { x - 2 }", "target": "- 1"}, {"rel": "被描述", "source": "\\frac { 3 } { x - 2 }", "target": "1"}, {"rel": "被描述", "source": "\\frac { 3 } { x - 2 }", "target": "3"}, {"rel": "被描述", "source": "\\frac { 3 } { x - 2 }", "target": "5"}, {"rel": "被描述", "source": "- 1", "target": "8"}, {"rel": "限制性描述", "source": "$x$ 为整数", "target": "- 1"}, {"rel": "限制性描述", "source": "$x$ 为整数", "target": "1"}, {"rel": "限制性描述", "source": "$x$ 为整数", "target": "3"}, {"rel": "限制性描述", "source": "$x$ 为整数", "target": "5"}, {"rel": "限制性描述", "source": "且 $\\frac { 1 } { x + 2 } + \\frac { 1 } { x - 2 } + \\frac { x + 6 } { x ^ { 2 } - 4 }$ 为整数", "target": "- 1"}, {"rel": "限制性描述", "source": "且 $\\frac { 1 } { x + 2 } + \\frac { 1 } { x - 2 } + \\frac { x + 6 } { x ^ { 2 } - 4 }$ 为整数", "target": "1"}, {"rel": "限制性描述", "source": "且 $\\frac { 1 } { x + 2 } + \\frac { 1 } { x - 2 } + \\frac { x + 6 } { x ^ { 2 } - 4 }$ 为整数", "target": "3"}, {"rel": "限制性描述", "source": "且 $\\frac { 1 } { x + 2 } + \\frac { 1 } { x - 2 } + \\frac { x + 6 } { x ^ { 2 } - 4 }$ 为整数", "target": "5"}, {"rel": "限制性描述", "source": "符合条件的x的值", "target": "- 1"}, {"rel": "限制性描述", "source": "符合条件的x的值", "target": "1"}, {"rel": "限制性描述", "source": "符合条件的x的值", "target": "3"}, {"rel": "限制性描述", "source": "符合条件的x的值", "target": "5"}, {"rel": "被描述", "source": "1", "target": "8"}, {"rel": "被描述", "source": "3", "target": "8"}, {"rel": "被描述", "source": "5", "target": "8"}, {"rel": "限制性描述", "source": "所有符合条件的 $x$ 值的和", "target": "8"}]}}
{"content": "If $2 a = 3 b ( a \\neq 0 )$, then the value of $\\frac { b } { a - b }$ is ____?", "answer": "2", "steps": "Because $2 a = 3 b$, therefore $a = \\frac { 3 } { 2 } b$, therefore $\\frac { b } { a - b } = \\frac { b } { \\frac { 3 } { 2 } b - b } = 2$.", "expr_cands": ["2 a = 3 b ( a \\neq 0 )", "a", "b", "\\frac { b } { a - b }", "2 a = 3 b", "a = \\frac { 3 } { 2 } b", "2"], "exprs": ["a = \\frac { 3 } { 2 } b", "2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 a = 3 b ( a \\neq 0 )"}, {"id": "a = \\frac { 3 } { 2 } b"}, {"id": "\\frac { b } { a - b }"}, {"id": "2"}], "links": [{"rel": "等式方程部分求解", "source": "2 a = 3 b ( a \\neq 0 )", "target": "a = \\frac { 3 } { 2 } b"}, {"rel": "代入", "source": "a = \\frac { 3 } { 2 } b", "target": "2"}, {"rel": "被代入", "source": "\\frac { b } { a - b }", "target": "2"}]}}
{"content": "Given that a positive number $m$ has two different square roots of $3 a + 1$ and $a + 11$, then $m$ = ____?", "answer": "64", "steps": "Because $3 a + 1$ and $a + 11$ are two different square roots of a positive number $m$, therefore $3 a + 1 + a + 11 = 0$, $a = - 3$, $a + 11 = 8$, and $m = 8 ^ { 2 } = 64$.", "expr_cands": ["m", "3 a + 1", "a", "a + 11", "3 a + 1 + a + 11 = 0", "a = - 3", "8", "m = 64"], "exprs": ["3 a + 1 + a + 11 = 0", "a = - 3", "8", "m = 64"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 a + 1"}, {"id": "3 a + 1 + a + 11 = 0"}, {"id": "m"}, {"id": "a + 11"}, {"id": "一个正数 $m$ 的两个不同的平方根是 $3 a + 1$ 和 $a + 11$"}, {"id": "平方根互为相反数"}, {"id": "a = - 3"}, {"id": "8"}, {"id": "m = 64"}, {"id": "平方"}], "links": [{"rel": "被描述", "source": "3 a + 1", "target": "3 a + 1 + a + 11 = 0"}, {"rel": "等式方程求解", "source": "3 a + 1 + a + 11 = 0", "target": "a = - 3"}, {"rel": "被描述", "source": "m", "target": "3 a + 1 + a + 11 = 0"}, {"rel": "被描述", "source": "a + 11", "target": "3 a + 1 + a + 11 = 0"}, {"rel": "被代入", "source": "a + 11", "target": "8"}, {"rel": "限制性描述", "source": "一个正数 $m$ 的两个不同的平方根是 $3 a + 1$ 和 $a + 11$", "target": "3 a + 1 + a + 11 = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "3 a + 1 + a + 11 = 0"}, {"rel": "代入", "source": "a = - 3", "target": "8"}, {"rel": "被描述", "source": "8", "target": "m = 64"}, {"rel": "限制性描述", "source": "平方", "target": "m = 64"}]}}
{"content": "If $2 m - 3 n = 2$, then the algebraic expression $4 m ^ 2 - 12 mn + 9 n ^ 2$ = ____ ?", "answer": "4", "steps": "Because $2 m - 3 n = 2$, therefore $4 m ^ 2 - 12 mn + 9 n ^ 2 = ( 2 m - 3 n ) ^ 2 = 2 ^ 2 = 4$.", "expr_cands": ["2 m - 3 n = 2", "n", "m", "4 m ^ { 2 } - 12 mn + 9 n ^ { 2 }", "( 2 m - 3 n ) ^ { 2 }", "4"], "exprs": ["( 2 m - 3 n ) ^ { 2 }", "4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 m ^ { 2 } - 12 mn + 9 n ^ { 2 }"}, {"id": "( 2 m - 3 n ) ^ { 2 }"}, {"id": "2 m - 3 n = 2"}, {"id": "4"}], "links": [{"rel": "提取因式", "source": "4 m ^ { 2 } - 12 mn + 9 n ^ { 2 }", "target": "( 2 m - 3 n ) ^ { 2 }"}, {"rel": "被代入", "source": "( 2 m - 3 n ) ^ { 2 }", "target": "4"}, {"rel": "提取因式参考", "source": "2 m - 3 n = 2", "target": "( 2 m - 3 n ) ^ { 2 }"}, {"rel": "代入", "source": "2 m - 3 n = 2", "target": "4"}]}}
{"content": "If $a - 2 b = 3$, then the value of $9 - 2 a + 4 b$ is ____?", "answer": "3", "steps": "Since $a - 2 b = 3$, therefore $9 - 2 a + 4 b = 9 - 2 ( a - 2 b ) = 9 - 2 * 3 = 3$.", "expr_cands": ["a - 2 b = 3", "a", "b", "9 - 2 a + 4 b", "9 - 2 ( a - 2 b )", "3"], "exprs": ["9 - 2 ( a - 2 b )", "3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "9 - 2 a + 4 b"}, {"id": "9 - 2 ( a - 2 b )"}, {"id": "a - 2 b = 3"}, {"id": "3"}], "links": [{"rel": "提取因式", "source": "9 - 2 a + 4 b", "target": "9 - 2 ( a - 2 b )"}, {"rel": "被代入", "source": "9 - 2 ( a - 2 b )", "target": "3"}, {"rel": "提取因式参考", "source": "a - 2 b = 3", "target": "9 - 2 ( a - 2 b )"}, {"rel": "代入", "source": "a - 2 b = 3", "target": "3"}]}}
{"content": "If $x$ is a rational number, what is the minimum value of $| x - 2 | + | x + 1 | + | x + 3 |$?", "answer": "5", "steps": "\\because $| x - 2 | + | x + 1 | + | x + 3 |$ represents the sum of distances to $- 1$, $- 3$, and $2$, respectively. \\therefore When $x = - 1$, it is the minimum, and the minimum value is $2 - ( - 3 ) = 2 + 3 = 5$.", "expr_cands": ["x", "| x - 2 | + | x + 1 | + | x + 3 |", "- 1", "- 3", "2", "x = - 1", "2 - ( - 3 )", "5"], "exprs": ["5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| x - 2 | + | x + 1 | + | x + 3 |"}, {"id": "5"}, {"id": "x = - 1"}, {"id": ", $| x - 2 | + | x + 1 | + | x + 3 |$ 表示到 $- 1$ , $- 3$ , $2$ 三个数的距离的和"}, {"id": "当 $x = - 1$ 时"}], "links": [{"rel": "被描述", "source": "| x - 2 | + | x + 1 | + | x + 3 |", "target": "5"}, {"rel": "被描述", "source": "x = - 1", "target": "5"}, {"rel": "限制性描述", "source": ", $| x - 2 | + | x + 1 | + | x + 3 |$ 表示到 $- 1$ , $- 3$ , $2$ 三个数的距离的和", "target": "5"}, {"rel": "限制性描述", "source": "当 $x = - 1$ 时", "target": "5"}]}}
{"content": "Given the function $y = x ^ { n ^ { 2 } - 3 } - n + 2$, when $n$ = ____ ?, it is a proportional function.", "answer": "2", "steps": "$\\because$ The function $y = x ^ { n ^ 2 - 3 } - n + 2$ is a proportional function. $\\therefore$ $n ^ 2 - 3 = 1$, $- n + 2 = 0$, solving for $n$ gives $n = 2$.", "expr_cands": ["y = x ^ { n ^ { 2 } - 3 } - n + 2", "n", "y", "x", "n ^ { 2 } - 3 = 1", "n = - 2", "n = 2", "- n + 2 = 0"], "exprs": ["n ^ { 2 } - 3 = 1", "- n + 2 = 0", "n = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = x ^ { n ^ { 2 } - 3 } - n + 2"}, {"id": "n ^ { 2 } - 3 = 1"}, {"id": "函数 $y = x ^ { n ^ { 2 } - 3 } - n + 2$"}, {"id": "它是正比例函数"}, {"id": "- n + 2 = 0"}, {"id": "n = 2"}], "links": [{"rel": "被描述", "source": "y = x ^ { n ^ { 2 } - 3 } - n + 2", "target": "n ^ { 2 } - 3 = 1"}, {"rel": "被描述", "source": "y = x ^ { n ^ { 2 } - 3 } - n + 2", "target": "- n + 2 = 0"}, {"rel": "联立", "source": "n ^ { 2 } - 3 = 1", "target": "n = 2"}, {"rel": "限制性描述", "source": "函数 $y = x ^ { n ^ { 2 } - 3 } - n + 2$", "target": "n ^ { 2 } - 3 = 1"}, {"rel": "限制性描述", "source": "函数 $y = x ^ { n ^ { 2 } - 3 } - n + 2$", "target": "- n + 2 = 0"}, {"rel": "限制性描述", "source": "它是正比例函数", "target": "n ^ { 2 } - 3 = 1"}, {"rel": "限制性描述", "source": "它是正比例函数", "target": "- n + 2 = 0"}, {"rel": "联立", "source": "- n + 2 = 0", "target": "n = 2"}]}}
{"content": "If $1$ is a solution of the equation $a ( x + 1 ) = a - x$, then $a$ = ____ ?", "answer": "- 1", "steps": "According to the problem, substituting $x = 1$ into the equation yields $2 a = a - 1$. Solving for $a$, we get $a = - 1$.", "expr_cands": ["1", "a ( x + 1 ) = a - x", "x", "a", "x = 1", "2 a = a - 1", "a = - 1"], "exprs": ["x = 1", "2 a = a - 1", "a = - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "1"}, {"id": "x = 1"}, {"id": "a ( x + 1 ) = a - x"}, {"id": "x"}, {"id": "$1$ 是方程 $a ( x + 1 ) = a - x$ 的解"}, {"id": "2 a = a - 1"}, {"id": "a = - 1"}], "links": [{"rel": "被描述", "source": "1", "target": "x = 1"}, {"rel": "代入", "source": "x = 1", "target": "2 a = a - 1"}, {"rel": "被描述", "source": "a ( x + 1 ) = a - x", "target": "x = 1"}, {"rel": "被代入", "source": "a ( x + 1 ) = a - x", "target": "2 a = a - 1"}, {"rel": "被描述", "source": "x", "target": "x = 1"}, {"rel": "限制性描述", "source": "$1$ 是方程 $a ( x + 1 ) = a - x$ 的解", "target": "x = 1"}, {"rel": "等式方程求解", "source": "2 a = a - 1", "target": "a = - 1"}]}}
{"content": "If the coefficient of the linear term of $( x + a ) ( x - 3 )$ is $0$, then the value of $a$ is ____?", "answer": "3", "steps": "$\\because$ $( x + a ) ( x - 3 ) = x ^ { 2 } + ( a - 3 ) x - 3 a$ , and $\\because$ there is no linear term in the result, $\\therefore$ $a - 3 = 0$, which implies $a = 3$.", "expr_cands": ["( x + a ) ( x - 3 )", "x", "a", "0", "x ^ { 2 } + ( a - 3 ) x - 3 a", "a - 3 = 0", "a = 3"], "exprs": ["x ^ { 2 } + ( a - 3 ) x - 3 a", "a - 3 = 0", "a = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( x + a ) ( x - 3 )"}, {"id": "x ^ { 2 } + ( a - 3 ) x - 3 a"}, {"id": "x"}, {"id": "a - 3 = 0"}, {"id": "0"}, {"id": "$( x + a ) ( x - 3 )$ 的一次项系数为 $0$"}, {"id": "a = 3"}], "links": [{"rel": "提取因式", "source": "( x + a ) ( x - 3 )", "target": "x ^ { 2 } + ( a - 3 ) x - 3 a"}, {"rel": "被描述", "source": "x ^ { 2 } + ( a - 3 ) x - 3 a", "target": "a - 3 = 0"}, {"rel": "提取因式参考", "source": "x", "target": "x ^ { 2 } + ( a - 3 ) x - 3 a"}, {"rel": "等式方程求解", "source": "a - 3 = 0", "target": "a = 3"}, {"rel": "被描述", "source": "0", "target": "a - 3 = 0"}, {"rel": "限制性描述", "source": "$( x + a ) ( x - 3 )$ 的一次项系数为 $0$", "target": "a - 3 = 0"}]}}
{"content": "If $a$ is the opposite of $3$, then the value of $- 2 a$ is ____?", "answer": "6", "steps": "$\\because$ $a$ is the opposite of $3$, $\\therefore$ $a = - 3$, $\\therefore$ $- 2 a = - 2 * ( - 3 ) = 6$.", "expr_cands": ["a", "3", "- 2 a", "a = - 3", "6"], "exprs": ["a = - 3", "6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3"}, {"id": "a = - 3"}, {"id": "a"}, {"id": "$a$ 是 $3$ 的相反数"}, {"id": "- 2 a"}, {"id": "6"}], "links": [{"rel": "被描述", "source": "3", "target": "a = - 3"}, {"rel": "代入", "source": "a = - 3", "target": "6"}, {"rel": "被描述", "source": "a", "target": "a = - 3"}, {"rel": "限制性描述", "source": "$a$ 是 $3$ 的相反数", "target": "a = - 3"}, {"rel": "被代入", "source": "- 2 a", "target": "6"}]}}
{"content": "The equation $\\frac { 4 x - 1 } { 5 } - \\frac { x + 2 } { 2 } = 0$ has a solution of $x$ = ____ ?", "answer": "4", "steps": "$\\frac { 4 x - 1 } { 5 } - \\frac { x + 2 } { 2 } = 0$ is simplified by eliminating the denominators to get $2 ( 4 x - 1 ) - 5 ( x + 2 ) = 0$. Combining like terms and rearranging yields $3 x = 12$, and solving for $x$ gives $x = 4$.", "expr_cands": ["\\frac { 4 x - 1 } { 5 } - \\frac { x + 2 } { 2 } = 0", "x", "x = 4", "2 ( 4 x - 1 ) - 5 ( x + 2 ) = 0", "3 x = 12"], "exprs": ["x = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 4 x - 1 } { 5 } - \\frac { x + 2 } { 2 } = 0"}, {"id": "x = 4"}], "links": [{"rel": "等式方程求解", "source": "\\frac { 4 x - 1 } { 5 } - \\frac { x + 2 } { 2 } = 0", "target": "x = 4"}]}}
{"content": "If $y = ( k + 2 ) { x } ^ {{ k } ^ { 2 } - 3 } + 2$ is a linear function, then $k$ = ____?", "answer": "2", "steps": "According to the problem, we have $k ^ { 2 } - 3 = 1$, and $k + 2 \\neq 0$. Solving for $k$, we get $k = 2$.", "expr_cands": ["y = ( k + 2 ) { x } ^ { { k } ^ { 2 } - 3 } + 2", "x", "k", "y", "k ^ { 2 } - 3 = 1", "k = - 2", "k = 2", "k + 2 \\neq 0", "k \\neq - 2"], "exprs": ["k ^ { 2 } - 3 = 1", "k + 2 \\neq 0", "k = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = ( k + 2 ) { x } ^ { { k } ^ { 2 } - 3 } + 2"}, {"id": "k ^ { 2 } - 3 = 1"}, {"id": "$y = ( k + 2 ) { x } ^ { { k } ^ { 2 } - 3 } + 2$ 是一次函数"}, {"id": "k + 2 \\neq 0"}, {"id": "k = 2"}], "links": [{"rel": "被描述", "source": "y = ( k + 2 ) { x } ^ { { k } ^ { 2 } - 3 } + 2", "target": "k ^ { 2 } - 3 = 1"}, {"rel": "被描述", "source": "y = ( k + 2 ) { x } ^ { { k } ^ { 2 } - 3 } + 2", "target": "k + 2 \\neq 0"}, {"rel": "联立", "source": "k ^ { 2 } - 3 = 1", "target": "k = 2"}, {"rel": "限制性描述", "source": "$y = ( k + 2 ) { x } ^ { { k } ^ { 2 } - 3 } + 2$ 是一次函数", "target": "k ^ { 2 } - 3 = 1"}, {"rel": "限制性描述", "source": "$y = ( k + 2 ) { x } ^ { { k } ^ { 2 } - 3 } + 2$ 是一次函数", "target": "k + 2 \\neq 0"}, {"rel": "联立", "source": "k + 2 \\neq 0", "target": "k = 2"}]}}
{"content": "If the value of the algebraic expression $4 y ^ { 2 } - 2 y + 5$ is $7$, then the value of $2 y ^ { 2 } - y$ is ____?", "answer": "1", "steps": "Since $4 y ^ { 2 } - 2 y + 5 = 7$, it follows that $4 y ^ { 2 } - 2 y = 2$. Therefore, $2 y ^ { 2 } - y = 1$. Thus, the original expression equals $1$.", "expr_cands": ["4 y ^ { 2 } - 2 y + 5", "y", "7", "2 y ^ { 2 } - y", "4 y ^ { 2 } - 2 y + 5 = 7", "y = - \\frac { 1 } { 2 }", "y = 1", "4 y ^ { 2 } - 2 y = 2", "2 y ^ { 2 } - y = 1", "1"], "exprs": ["4 y ^ { 2 } - 2 y + 5 = 7", "4 y ^ { 2 } - 2 y = 2", "2 y ^ { 2 } - y = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 y ^ { 2 } - 2 y + 5"}, {"id": "4 y ^ { 2 } - 2 y + 5 = 7"}, {"id": "7"}, {"id": "代数式 $4 y ^ { 2 } - 2 y + 5$ 的值为 $7$"}, {"id": "4 y ^ { 2 } - 2 y = 2"}, {"id": "2 y ^ { 2 } - y = 1"}], "links": [{"rel": "被描述", "source": "4 y ^ { 2 } - 2 y + 5", "target": "4 y ^ { 2 } - 2 y + 5 = 7"}, {"rel": "移项", "source": "4 y ^ { 2 } - 2 y + 5 = 7", "target": "4 y ^ { 2 } - 2 y = 2"}, {"rel": "被描述", "source": "7", "target": "4 y ^ { 2 } - 2 y + 5 = 7"}, {"rel": "限制性描述", "source": "代数式 $4 y ^ { 2 } - 2 y + 5$ 的值为 $7$", "target": "4 y ^ { 2 } - 2 y + 5 = 7"}, {"rel": "同乘除", "source": "4 y ^ { 2 } - 2 y = 2", "target": "2 y ^ { 2 } - y = 1"}]}}
{"content": "If $\\frac { 1 } { 6 } a ^ 2 bc ^ 3$ and $- 0.5 a ^ 2 b ^ mc ^ n$ are like terms, what is the value of $m - n$?", "answer": "- 2", "steps": "Because $\\frac { 1 } { 6 } a ^ 2 bc ^ 3$ and $- 0.5 a ^ 2 b ^ mc ^ n$ are like terms, therefore $m = 1$, $n = 3$, and thus $m - n = 1 - 3 = - 2$.", "expr_cands": ["\\frac { 1 } { 6 } { a } ^ { 2 } b { c } ^ { 3 }", "c", "b", "a", "- 0.5 a ^ { 2 } b ^ { m } c ^ { n }", "n", "m", "m - n", "m = 1", "n = 3", "- 2"], "exprs": ["m = 1", "n = 3", "- 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 1 } { 6 } { a } ^ { 2 } b { c } ^ { 3 }"}, {"id": "m = 1"}, {"id": "- 0.5 a ^ { 2 } b ^ { m } c ^ { n }"}, {"id": "$\\frac { 1 } { 6 } { a } ^ { 2 } b { c } ^ { 3 }$ 与 $- 0.5 a ^ { 2 } b ^ { m } c ^ { n }$ : 是同类项"}, {"id": "n = 3"}, {"id": "m - n"}, {"id": "- 2"}], "links": [{"rel": "被描述", "source": "\\frac { 1 } { 6 } { a } ^ { 2 } b { c } ^ { 3 }", "target": "m = 1"}, {"rel": "被描述", "source": "\\frac { 1 } { 6 } { a } ^ { 2 } b { c } ^ { 3 }", "target": "n = 3"}, {"rel": "代入", "source": "m = 1", "target": "- 2"}, {"rel": "被描述", "source": "- 0.5 a ^ { 2 } b ^ { m } c ^ { n }", "target": "m = 1"}, {"rel": "被描述", "source": "- 0.5 a ^ { 2 } b ^ { m } c ^ { n }", "target": "n = 3"}, {"rel": "限制性描述", "source": "$\\frac { 1 } { 6 } { a } ^ { 2 } b { c } ^ { 3 }$ 与 $- 0.5 a ^ { 2 } b ^ { m } c ^ { n }$ : 是同类项", "target": "m = 1"}, {"rel": "限制性描述", "source": "$\\frac { 1 } { 6 } { a } ^ { 2 } b { c } ^ { 3 }$ 与 $- 0.5 a ^ { 2 } b ^ { m } c ^ { n }$ : 是同类项", "target": "n = 3"}, {"rel": "代入", "source": "n = 3", "target": "- 2"}, {"rel": "被代入", "source": "m - n", "target": "- 2"}]}}
{"content": "Given that $x = 3$ is a solution of the quadratic equation $2 ax ^ 2 - ax = 5$, then $a$ = ____ ?", "answer": "\\frac { 1 } { 3 }", "steps": "Since $x = 3$ is a solution of the quadratic equation $2 ax ^ 2 - ax = 5$, we have $18 a - 3 a = 5$. Solving for $a$, we get $a = \\frac { 1 } { 3 }$.", "expr_cands": ["x = 3", "x", "2 ax ^ { 2 } - ax = 5", "a", "15 a = 5", "18 a - 3 a = 5", "a = \\frac { 1 } { 3 }"], "exprs": ["18 a - 3 a = 5", "a = \\frac { 1 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 ax ^ { 2 } - ax = 5"}, {"id": "18 a - 3 a = 5"}, {"id": "x = 3"}, {"id": "a = \\frac { 1 } { 3 }"}], "links": [{"rel": "被代入", "source": "2 ax ^ { 2 } - ax = 5", "target": "18 a - 3 a = 5"}, {"rel": "等式方程求解", "source": "18 a - 3 a = 5", "target": "a = \\frac { 1 } { 3 }"}, {"rel": "代入", "source": "x = 3", "target": "18 a - 3 a = 5"}]}}
{"content": "When $x$ is what value, the values of the algebraic expressions $2 x + 1$ and $5 x - 8$ are opposite?", "answer": "x = 1", "steps": "According to the problem, we have $2 x + 1 + ( 5 x - 8 ) = 0$. Removing the parentheses, we get $2 x + 1 + 5 x - 8 = 0$. Solving for $x$, we get $x = 1$.", "expr_cands": ["x", "2 x + 1", "5 x - 8", "2 x + 1 + ( 5 x - 8 ) = 0", "x = 1", "2 x + 1 + 5 x - 8 = 0"], "exprs": ["2 x + 1 + ( 5 x - 8 ) = 0", "x = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x + 1"}, {"id": "2 x + 1 + ( 5 x - 8 ) = 0"}, {"id": "5 x - 8"}, {"id": "代数式 $2 x + 1$ 与 $5 x - 8$ 的值互为相反数"}, {"id": "x = 1"}], "links": [{"rel": "被描述", "source": "2 x + 1", "target": "2 x + 1 + ( 5 x - 8 ) = 0"}, {"rel": "等式方程求解", "source": "2 x + 1 + ( 5 x - 8 ) = 0", "target": "x = 1"}, {"rel": "被描述", "source": "5 x - 8", "target": "2 x + 1 + ( 5 x - 8 ) = 0"}, {"rel": "限制性描述", "source": "代数式 $2 x + 1$ 与 $5 x - 8$ 的值互为相反数", "target": "2 x + 1 + ( 5 x - 8 ) = 0"}]}}
{"content": "The larger root of the quadratic equation $x ^ 2 - 7 x = 0$ is _____.", "answer": "7", "steps": "$\\because$ The quadratic equation $x ^ 2 - 7 x = 0$ can be factored as $x ( x - 7 ) = 0$, $\\therefore$ the solutions are $x _ 1 = 0$ and $x _ 2 = 7$, $\\therefore$ the larger root of this equation is $7$.", "expr_cands": ["x ^ { 2 } - 7 x = 0", "x", "x = 0", "x = 7", "x ( x - 7 ) = 0", "x _ { 1 } = 0", "x _ { 1 }", "x _ { 2 } = 7", "x _ { 2 }", "7"], "exprs": ["7"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } - 7 x = 0"}, {"id": "7"}, {"id": "一元二次方程 $x ^ { 2 } - 7 x = 0$ 的较大根"}], "links": [{"rel": "被描述", "source": "x ^ { 2 } - 7 x = 0", "target": "7"}, {"rel": "限制性描述", "source": "一元二次方程 $x ^ { 2 } - 7 x = 0$ 的较大根", "target": "7"}]}}
{"content": "Given $x < - 2$, what is the value of $| x + 2 | - | 1 - x |$?", "answer": "- 3", "steps": "Since $x < - 2$, it follows that $x + 2 < 0$ and $1 - x > 0$. Therefore, $| x + 2 | - | 1 - x | = - x - 2 - ( 1 - x ) = - 3$.", "expr_cands": ["x < - 2", "x", "| x + 2 | - | 1 - x |", "x + 2 < 0", "1 - x > 0", "x < 1", "- x - 2 - ( 1 - x )", "- 3"], "exprs": ["x + 2 < 0", "1 - x > 0", "- x - 2 - ( 1 - x )", "- 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x < - 2"}, {"id": "x + 2 < 0"}, {"id": "1 - x > 0"}, {"id": "不等式性质"}, {"id": "| x + 2 | - | 1 - x |"}, {"id": "- x - 2 - ( 1 - x )"}, {"id": "绝对值恒大于等于0"}, {"id": "- 3"}], "links": [{"rel": "移项", "source": "x < - 2", "target": "x + 2 < 0"}, {"rel": "被描述", "source": "x < - 2", "target": "1 - x > 0"}, {"rel": "被描述", "source": "x + 2 < 0", "target": "- x - 2 - ( 1 - x )"}, {"rel": "被描述", "source": "1 - x > 0", "target": "- x - 2 - ( 1 - x )"}, {"rel": "属性描述", "source": "不等式性质", "target": "1 - x > 0"}, {"rel": "被描述", "source": "| x + 2 | - | 1 - x |", "target": "- x - 2 - ( 1 - x )"}, {"rel": "计算", "source": "- x - 2 - ( 1 - x )", "target": "- 3"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "- x - 2 - ( 1 - x )"}]}}
{"content": "If $\\sqrt { x - 2 }$ is a quadratic radical, then $x$ should satisfy ____?", "answer": "x \\ge 2", "steps": "From the given condition, it is known that $x - 2 \\ge 0$, so $x \\ge 2$.", "expr_cands": ["\\sqrt { x - 2 }", "x", "x - 2 \\ge 0", "2 \\le x", "x \\ge 2"], "exprs": ["x - 2 \\ge 0", "x \\ge 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x - 2 }"}, {"id": "x - 2 \\ge 0"}, {"id": "$\\sqrt { x - 2 }$ 是二次根式"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x \\ge 2"}], "links": [{"rel": "被描述", "source": "\\sqrt { x - 2 }", "target": "x - 2 \\ge 0"}, {"rel": "不等式方程求解", "source": "x - 2 \\ge 0", "target": "x \\ge 2"}, {"rel": "限制性描述", "source": "$\\sqrt { x - 2 }$ 是二次根式", "target": "x - 2 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 2 \\ge 0"}]}}
{"content": "Given $x$ and $y$ satisfy: $y = \\sqrt { x - 2 } + \\sqrt { 2 - x } - 3$, what is the value of $xy$?", "answer": "- 6", "steps": "From the given conditions, $x - 2 \\ge 0$ and $2 - x \\ge 0$, we can solve for $x$ and get $x = 2$. Therefore, $y = - 3$ and $xy = - 6$.", "expr_cands": ["x", "y", "y = \\sqrt { x - 2 } + \\sqrt { 2 - x } - 3", "xy", "x - 2 \\ge 0", "2 \\le x", "2 - x \\ge 0", "x \\le 2", "x = 2", "y = - 3", "- 6"], "exprs": ["x - 2 \\ge 0", "2 - x \\ge 0", "x = 2", "y = - 3", "- 6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = \\sqrt { x - 2 } + \\sqrt { 2 - x } - 3"}, {"id": "x - 2 \\ge 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "2 - x \\ge 0"}, {"id": "x = 2"}, {"id": "y = - 3"}, {"id": "xy"}, {"id": "- 6"}], "links": [{"rel": "被描述", "source": "y = \\sqrt { x - 2 } + \\sqrt { 2 - x } - 3", "target": "x - 2 \\ge 0"}, {"rel": "被描述", "source": "y = \\sqrt { x - 2 } + \\sqrt { 2 - x } - 3", "target": "2 - x \\ge 0"}, {"rel": "被代入", "source": "y = \\sqrt { x - 2 } + \\sqrt { 2 - x } - 3", "target": "y = - 3"}, {"rel": "联立", "source": "x - 2 \\ge 0", "target": "x = 2"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 2 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "2 - x \\ge 0"}, {"rel": "联立", "source": "2 - x \\ge 0", "target": "x = 2"}, {"rel": "代入", "source": "x = 2", "target": "y = - 3"}, {"rel": "代入", "source": "x = 2", "target": "- 6"}, {"rel": "代入", "source": "y = - 3", "target": "- 6"}, {"rel": "被代入", "source": "xy", "target": "- 6"}]}}
{"content": "If $2 x - y = 2$, then the value of $1 - 4 x + 2 y$ is ____?", "answer": "- 3", "steps": "Because $2 x - y = 2$, therefore the original expression is equal to $1 - 2 ( 2 x - y ) = 1 - 2 * 2 = 1 - 4 = - 3$.", "expr_cands": ["2 x - y = 2", "x", "y", "1 - 4 x + 2 y", "1 - 2 ( 2 x - y )", "- 3"], "exprs": ["1 - 2 ( 2 x - y )", "- 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "1 - 4 x + 2 y"}, {"id": "1 - 2 ( 2 x - y )"}, {"id": "2 x - y = 2"}, {"id": "- 3"}], "links": [{"rel": "提取因式", "source": "1 - 4 x + 2 y", "target": "1 - 2 ( 2 x - y )"}, {"rel": "被代入", "source": "1 - 2 ( 2 x - y )", "target": "- 3"}, {"rel": "提取因式参考", "source": "2 x - y = 2", "target": "1 - 2 ( 2 x - y )"}, {"rel": "代入", "source": "2 x - y = 2", "target": "- 3"}]}}
{"content": "If $a$ and $b$ are rational numbers, and $\\sqrt { 4 } - \\sqrt { 18 } + \\sqrt { \\frac { 1 } { 2 }} = a + b \\sqrt { 2 }$, then $a + b$ = ____?", "answer": "- \\frac { 1 } { 2 }", "steps": "Since $\\sqrt { 4 } - \\sqrt { 18 } + \\sqrt { \\frac { 1 } { 2 }} = a + b \\sqrt { 2 }$, therefore $2 - 3 \\sqrt { 2 } + \\frac { 1 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }$, therefore $2 - \\frac { 5 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }$, therefore $a = 2$, $b = - \\frac { 5 } { 2 }$, therefore $a + b = - \\frac { 1 } { 2 }$.", "expr_cands": ["a", "b", "\\sqrt { 4 } - \\sqrt { 18 } + \\sqrt { \\frac { 1 } { 2 } } = a + b \\sqrt { 2 }", "a + b", "2 - 3 \\sqrt { 2 } + \\frac { 1 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }", "2 - \\frac { 5 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }", "a = 2", "b = - \\frac { 5 } { 2 }", "- \\frac { 1 } { 2 }"], "exprs": ["2 - 3 \\sqrt { 2 } + \\frac { 1 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }", "2 - \\frac { 5 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }", "a = 2", "b = - \\frac { 5 } { 2 }", "- \\frac { 1 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { 4 } - \\sqrt { 18 } + \\sqrt { \\frac { 1 } { 2 } } = a + b \\sqrt { 2 }"}, {"id": "2 - 3 \\sqrt { 2 } + \\frac { 1 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }"}, {"id": "2 - \\frac { 5 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }"}, {"id": "a = 2"}, {"id": "b = - \\frac { 5 } { 2 }"}, {"id": "a + b"}, {"id": "- \\frac { 1 } { 2 }"}], "links": [{"rel": "计算", "source": "\\sqrt { 4 } - \\sqrt { 18 } + \\sqrt { \\frac { 1 } { 2 } } = a + b \\sqrt { 2 }", "target": "2 - 3 \\sqrt { 2 } + \\frac { 1 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }"}, {"rel": "计算", "source": "2 - 3 \\sqrt { 2 } + \\frac { 1 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }", "target": "2 - \\frac { 5 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }"}, {"rel": "移项", "source": "2 - \\frac { 5 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }", "target": "a = 2"}, {"rel": "移项", "source": "2 - \\frac { 5 } { 2 } \\sqrt { 2 } = a + b \\sqrt { 2 }", "target": "b = - \\frac { 5 } { 2 }"}, {"rel": "代入", "source": "a = 2", "target": "- \\frac { 1 } { 2 }"}, {"rel": "代入", "source": "b = - \\frac { 5 } { 2 }", "target": "- \\frac { 1 } { 2 }"}, {"rel": "被代入", "source": "a + b", "target": "- \\frac { 1 } { 2 }"}]}}
{"content": "Given the function $y = 2 x + 6$, what is the range of values for $x$ if $y > 0$?", "answer": "x > - 3", "steps": "The function $y = 2 x + 6$ will have $y > 0$ if $2 x + 6 > 0$, which can be solved to obtain $x > - 3$.", "expr_cands": ["y = 2 x + 6", "y", "x", "y > 0", "2 x + 6 > 0", "- 3 < x", "x > - 3"], "exprs": ["2 x + 6 > 0", "x > - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = 2 x + 6"}, {"id": "2 x + 6 > 0"}, {"id": "y > 0"}, {"id": "x > - 3"}], "links": [{"rel": "联立", "source": "y = 2 x + 6", "target": "2 x + 6 > 0"}, {"rel": "不等式方程求解", "source": "2 x + 6 > 0", "target": "x > - 3"}, {"rel": "联立", "source": "y > 0", "target": "2 x + 6 > 0"}]}}
{"content": "The degree of the monomial $2 ^ 5 \\pi a b ^ 2$ is ____ ?", "answer": "3", "steps": "The degree of the monomial ${ 2 } ^ { 5 } \\pi a { b } ^ { 2 }$ is: $1 + 2 = 3$, therefore the degree of the monomial ${ 2 } ^ { 5 } \\pi a { b } ^ { 2 }$ is 3.", "expr_cands": ["{ 2 } ^ { 5 } \\pi a { b } ^ { 2 }", "a", "b", "1 + 2", "3"], "exprs": ["1 + 2", "3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "{ 2 } ^ { 5 } \\pi a { b } ^ { 2 }"}, {"id": "1 + 2"}, {"id": "单项式 ${ 2 } ^ { 5 } \\pi a { b } ^ { 2 }$ 的次数是次"}, {"id": "3"}], "links": [{"rel": "被描述", "source": "{ 2 } ^ { 5 } \\pi a { b } ^ { 2 }", "target": "1 + 2"}, {"rel": "计算", "source": "1 + 2", "target": "3"}, {"rel": "限制性描述", "source": "单项式 ${ 2 } ^ { 5 } \\pi a { b } ^ { 2 }$ 的次数是次", "target": "1 + 2"}]}}
{"content": "If the equations $2 x - 3 = 1$ and $\\frac { x - k } { 2 } = k - 3 x$ have the same solution for $x$, find the value of $k$ = ____?", "answer": "\\frac { 14 } { 3 }", "steps": "Since $2 x - 3 = 1$, therefore $x = 2$. Also, since $\\frac { x - k } { 2 } = k - 3 x$, we have $\\frac { 2 - k } { 2 } = k - 3 * 2$, thus $k = \\frac { 14 } { 3 }$.", "expr_cands": ["x", "2 x - 3 = 1", "\\frac { x - k } { 2 } = k - 3 x", "k", "x = 2", "1 - \\frac { k } { 2 } = k - 6", "\\frac { 2 - k } { 2 } = k - 3 * 2", "k = \\frac { 14 } { 3 }"], "exprs": ["x = 2", "\\frac { 2 - k } { 2 } = k - 3 * 2", "k = \\frac { 14 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x - 3 = 1"}, {"id": "x = 2"}, {"id": "\\frac { x - k } { 2 } = k - 3 x"}, {"id": "\\frac { 2 - k } { 2 } = k - 3 * 2"}, {"id": "k = \\frac { 14 } { 3 }"}], "links": [{"rel": "等式方程求解", "source": "2 x - 3 = 1", "target": "x = 2"}, {"rel": "代入", "source": "x = 2", "target": "\\frac { 2 - k } { 2 } = k - 3 * 2"}, {"rel": "被代入", "source": "\\frac { x - k } { 2 } = k - 3 x", "target": "\\frac { 2 - k } { 2 } = k - 3 * 2"}, {"rel": "等式方程求解", "source": "\\frac { 2 - k } { 2 } = k - 3 * 2", "target": "k = \\frac { 14 } { 3 }"}]}}
{"content": "When is $\\sqrt { x + 1 }$ defined?", "answer": "x \\ge - 1", "steps": "According to the problem, we have: $x + 1 \\ge 0$. Solving for $x$, we get: $x \\ge - 1$.", "expr_cands": ["x", "\\sqrt { x + 1 }", "x + 1 \\ge 0", "- 1 \\le x", "x \\ge - 1"], "exprs": ["x + 1 \\ge 0", "x \\ge - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x + 1 }"}, {"id": "x + 1 \\ge 0"}, {"id": "二次根式 $\\sqrt { x + 1 }$ 有意义"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x \\ge - 1"}], "links": [{"rel": "被描述", "source": "\\sqrt { x + 1 }", "target": "x + 1 \\ge 0"}, {"rel": "不等式方程求解", "source": "x + 1 \\ge 0", "target": "x \\ge - 1"}, {"rel": "限制性描述", "source": "二次根式 $\\sqrt { x + 1 }$ 有意义", "target": "x + 1 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x + 1 \\ge 0"}]}}
{"content": "$5$, $12$, $m$ form a Pythagorean triple. What is the value of $m$?", "answer": "13", "steps": "When $12$ is the longest side, $5 ^ 2 + m ^ 2 = 12 ^ 2$, $m = \\sqrt { 119 }$ (rounded down). When $m$ is the longest side, $m ^ 2 = 5 ^ 2 + 12 ^ 2$, $m = 13$.", "expr_cands": ["5", "12", "m", "5 ^ { 2 } + m ^ { 2 } = 12 ^ { 2 }", "m = \\sqrt { 119 }", "m = - \\sqrt { 119 }", "m ^ { 2 } = 5 ^ { 2 } + 12 ^ { 2 }", "m = - 13", "m = 13"], "exprs": ["5 ^ { 2 } + m ^ { 2 } = 12 ^ { 2 }", "m ^ { 2 } = 5 ^ { 2 } + 12 ^ { 2 }", "m = 13"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5"}, {"id": "5 ^ { 2 } + m ^ { 2 } = 12 ^ { 2 }"}, {"id": "12"}, {"id": "m"}, {"id": "$5$ , $12$ , $m$ 是一组勾股数"}, {"id": "当 $12$ 是最长边时"}, {"id": "m ^ { 2 } = 5 ^ { 2 } + 12 ^ { 2 }"}, {"id": "当 $m$ 是最长边时"}, {"id": "m = 13"}], "links": [{"rel": "被描述", "source": "5", "target": "5 ^ { 2 } + m ^ { 2 } = 12 ^ { 2 }"}, {"rel": "被描述", "source": "5", "target": "m ^ { 2 } = 5 ^ { 2 } + 12 ^ { 2 }"}, {"rel": "被描述", "source": "5 ^ { 2 } + m ^ { 2 } = 12 ^ { 2 }", "target": "m = 13"}, {"rel": "被描述", "source": "12", "target": "5 ^ { 2 } + m ^ { 2 } = 12 ^ { 2 }"}, {"rel": "被描述", "source": "12", "target": "m ^ { 2 } = 5 ^ { 2 } + 12 ^ { 2 }"}, {"rel": "被描述", "source": "m", "target": "5 ^ { 2 } + m ^ { 2 } = 12 ^ { 2 }"}, {"rel": "被描述", "source": "m", "target": "m ^ { 2 } = 5 ^ { 2 } + 12 ^ { 2 }"}, {"rel": "限制性描述", "source": "$5$ , $12$ , $m$ 是一组勾股数", "target": "5 ^ { 2 } + m ^ { 2 } = 12 ^ { 2 }"}, {"rel": "限制性描述", "source": "$5$ , $12$ , $m$ 是一组勾股数", "target": "m ^ { 2 } = 5 ^ { 2 } + 12 ^ { 2 }"}, {"rel": "限制性描述", "source": "$5$ , $12$ , $m$ 是一组勾股数", "target": "m = 13"}, {"rel": "限制性描述", "source": "当 $12$ 是最长边时", "target": "5 ^ { 2 } + m ^ { 2 } = 12 ^ { 2 }"}, {"rel": "被描述", "source": "m ^ { 2 } = 5 ^ { 2 } + 12 ^ { 2 }", "target": "m = 13"}, {"rel": "限制性描述", "source": "当 $m$ 是最长边时", "target": "m ^ { 2 } = 5 ^ { 2 } + 12 ^ { 2 }"}]}}
{"content": "If $a = - 4$, then the absolute value of $a$ is ____?", "answer": "4", "steps": "\\because $a = - 4$ , \\therefore The absolute value of $a$ is $| - 4 | = 4$.", "expr_cands": ["a = - 4", "a", "| - 4 |", "4"], "exprs": ["| - 4 |", "4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a = - 4"}, {"id": "| - 4 |"}, {"id": "$a$ 的绝对值"}, {"id": "4"}], "links": [{"rel": "被描述", "source": "a = - 4", "target": "| - 4 |"}, {"rel": "计算", "source": "| - 4 |", "target": "4"}, {"rel": "限制性描述", "source": "$a$ 的绝对值", "target": "| - 4 |"}]}}
{"content": "The equation $2 x ^ 2 - mx + 6 m = 9$ has a root of $3$. Find the value of $m$.", "answer": "- 3", "steps": "Substituting $x = 3$ into the equation $2 x ^ 2 - mx + 6 m = 9$, we get $2 * 3 ^ 2 - 3 m + 6 m = 9$, $3 m = - 9$, and $m = - 3$.", "expr_cands": ["2 x ^ { 2 } - mx + 6 m = 9", "m", "x", "3", "x = 3", "3 m + 18 = 9", "2 * 3 ^ { 2 } - 3 m + 6 m = 9", "m = - 3", "3 m = - 9"], "exprs": ["x = 3", "2 * 3 ^ { 2 } - 3 m + 6 m = 9", "m = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3"}, {"id": "x = 3"}, {"id": "2 x ^ { 2 } - mx + 6 m = 9"}, {"id": "x"}, {"id": "方程 $2 x ^ { 2 } - mx + 6 m = 9$ 的一个根是 $3$"}, {"id": "2 * 3 ^ { 2 } - 3 m + 6 m = 9"}, {"id": "m = - 3"}], "links": [{"rel": "被描述", "source": "3", "target": "x = 3"}, {"rel": "代入", "source": "x = 3", "target": "2 * 3 ^ { 2 } - 3 m + 6 m = 9"}, {"rel": "被描述", "source": "2 x ^ { 2 } - mx + 6 m = 9", "target": "x = 3"}, {"rel": "被代入", "source": "2 x ^ { 2 } - mx + 6 m = 9", "target": "2 * 3 ^ { 2 } - 3 m + 6 m = 9"}, {"rel": "被描述", "source": "x", "target": "x = 3"}, {"rel": "限制性描述", "source": "方程 $2 x ^ { 2 } - mx + 6 m = 9$ 的一个根是 $3$", "target": "x = 3"}, {"rel": "等式方程求解", "source": "2 * 3 ^ { 2 } - 3 m + 6 m = 9", "target": "m = - 3"}]}}
{"content": "Given $\\sqrt { 24 m } + 4 \\sqrt { \\frac { 3 m } { 2 } } + m \\sqrt { \\frac { 6 } { m } } = 30$, find the value of $m$.", "answer": "6", "steps": "Because $\\sqrt { 24 m } + 4 \\sqrt { \\frac { 3 m } { 2 } } + m \\sqrt { \\frac { 6 } { m } } = 30$, therefore $\\sqrt { 24 m } + \\sqrt { 24 m } + \\sqrt { 6 m } = 30$, therefore $5 \\sqrt { 6 m } = 30$, therefore $\\sqrt { 6 m } = 6$, therefore $m = 6$.", "expr_cands": ["\\sqrt { 24 m } + 4 \\sqrt { \\frac { 3 m } { 2 } } + m \\sqrt { \\frac { 6 } { m } } = 30", "m", "m = 6", "\\sqrt { 24 m } + \\sqrt { 24 m } + \\sqrt { 6 m } = 30", "5 \\sqrt { 6 m } = 30", "\\sqrt { 6 m } = 6"], "exprs": ["m = 6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { 24 m } + 4 \\sqrt { \\frac { 3 m } { 2 } } + m \\sqrt { \\frac { 6 } { m } } = 30"}, {"id": "m = 6"}], "links": [{"rel": "等式方程求解", "source": "\\sqrt { 24 m } + 4 \\sqrt { \\frac { 3 m } { 2 } } + m \\sqrt { \\frac { 6 } { m } } = 30", "target": "m = 6"}]}}
{"content": "If the square root of a number is $x ^ { 2 } + x$ and $1 - x ^ { 2 }$, then the number is ____?", "answer": "0", "steps": "According to the problem, we have $x ^ 2 + x + 1 - x ^ 2 = 0$, which gives us $x = - 1$, $x ^ 2 + x = 0$, and $1 - x ^ 2 = 0$. Therefore, the number is $0$.", "expr_cands": ["x ^ { 2 } + x", "x", "1 - x ^ { 2 }", "x ^ { 2 } + x + 1 - x ^ { 2 } = 0", "x = - 1", "0"], "exprs": ["x ^ { 2 } + x + 1 - x ^ { 2 } = 0", "x = - 1", "0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } + x"}, {"id": "x ^ { 2 } + x + 1 - x ^ { 2 } = 0"}, {"id": "1 - x ^ { 2 }"}, {"id": "一个数的平方根为 $x ^ { 2 } + x$ 和 $1 - x ^ { 2 }$"}, {"id": "平方根互为相反数"}, {"id": "x = - 1"}, {"id": "0"}, {"id": "这个数为 $0$"}], "links": [{"rel": "被描述", "source": "x ^ { 2 } + x", "target": "x ^ { 2 } + x + 1 - x ^ { 2 } = 0"}, {"rel": "被描述", "source": "x ^ { 2 } + x", "target": "0"}, {"rel": "等式方程求解", "source": "x ^ { 2 } + x + 1 - x ^ { 2 } = 0", "target": "x = - 1"}, {"rel": "被描述", "source": "1 - x ^ { 2 }", "target": "x ^ { 2 } + x + 1 - x ^ { 2 } = 0"}, {"rel": "被描述", "source": "1 - x ^ { 2 }", "target": "0"}, {"rel": "限制性描述", "source": "一个数的平方根为 $x ^ { 2 } + x$ 和 $1 - x ^ { 2 }$", "target": "x ^ { 2 } + x + 1 - x ^ { 2 } = 0"}, {"rel": "限制性描述", "source": "一个数的平方根为 $x ^ { 2 } + x$ 和 $1 - x ^ { 2 }$", "target": "0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "x ^ { 2 } + x + 1 - x ^ { 2 } = 0"}, {"rel": "被描述", "source": "x = - 1", "target": "0"}, {"rel": "限制性描述", "source": "这个数为 $0$", "target": "0"}]}}
{"content": "If $x = 0$ is a solution to the equation $2 x - 3 n = 1$ in terms of $x$, then $n$ = ____ ?", "answer": "- \\frac { 1 } { 3 }", "steps": "Substituting $x = 0$ into $2 x - 3 n = 1$, we get $- 3 n = 1$. Solving for $n$, we get $n = - \\frac { 1 } { 3 }$.", "expr_cands": ["x = 0", "x", "2 x - 3 n = 1", "n", "- 3 n = 1", "n = - \\frac { 1 } { 3 }"], "exprs": ["- 3 n = 1", "n = - \\frac { 1 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x - 3 n = 1"}, {"id": "- 3 n = 1"}, {"id": "x = 0"}, {"id": "n = - \\frac { 1 } { 3 }"}], "links": [{"rel": "被代入", "source": "2 x - 3 n = 1", "target": "- 3 n = 1"}, {"rel": "等式方程求解", "source": "- 3 n = 1", "target": "n = - \\frac { 1 } { 3 }"}, {"rel": "代入", "source": "x = 0", "target": "- 3 n = 1"}]}}
{"content": "If $x + 2 y = - 6$, then $12 - 2 x - 4 y$ = ____ ?", "answer": "24", "steps": "Because $x + 2 y = - 6$, therefore $12 - 2 x - 4 y = 12 - 2 ( x + 2 y ) = 12 - 2 * ( - 6 ) = 12 + 12 = 24$.", "expr_cands": ["x + 2 y = - 6", "x", "y", "12 - 2 x - 4 y", "12 - 2 ( x + 2 y )", "24"], "exprs": ["12 - 2 ( x + 2 y )", "24"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "12 - 2 x - 4 y"}, {"id": "12 - 2 ( x + 2 y )"}, {"id": "x + 2 y = - 6"}, {"id": "24"}], "links": [{"rel": "提取因式", "source": "12 - 2 x - 4 y", "target": "12 - 2 ( x + 2 y )"}, {"rel": "被代入", "source": "12 - 2 ( x + 2 y )", "target": "24"}, {"rel": "提取因式参考", "source": "x + 2 y = - 6", "target": "12 - 2 ( x + 2 y )"}, {"rel": "代入", "source": "x + 2 y = - 6", "target": "24"}]}}
{"content": "$| - 2 | = x$, what is the value of $x$?", "answer": "2", "steps": "Because the absolute value of negative two is two, therefore x equals two.", "expr_cands": ["| - 2 | = x", "x", "| - 2 |", "2", "x = 2"], "exprs": ["x = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| - 2 | = x"}, {"id": "x = 2"}, {"id": "绝对值恒大于等于0"}], "links": [{"rel": "被描述", "source": "| - 2 | = x", "target": "x = 2"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "x = 2"}]}}
{"content": "The range of values for $x$ that make the expression $\\frac { 1 } { \\sqrt { 1 } - 2 x }$ meaningful is _____.", "answer": "x < \\frac { 1 } { 2 }", "steps": "From the given condition, we have $1 - 2 x > 0$, which implies that $x < \\frac { 1 } { 2 }$.", "expr_cands": ["\\frac { 1 } { \\sqrt { 1 } - { 2 } x }", "x", "1 - 2 x > 0", "x < \\frac { 1 } { 2 }"], "exprs": ["1 - 2 x > 0", "x < \\frac { 1 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 1 } { \\sqrt { 1 } - { 2 } x }"}, {"id": "1 - 2 x > 0"}, {"id": "使式子 $\\frac { 1 } { \\sqrt { 1 } - { 2 } x }$ 有意义的 $x$ 的取值范围"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "分式有意义，则分母不为0"}, {"id": "x < \\frac { 1 } { 2 }"}], "links": [{"rel": "被描述", "source": "\\frac { 1 } { \\sqrt { 1 } - { 2 } x }", "target": "1 - 2 x > 0"}, {"rel": "不等式方程求解", "source": "1 - 2 x > 0", "target": "x < \\frac { 1 } { 2 }"}, {"rel": "限制性描述", "source": "使式子 $\\frac { 1 } { \\sqrt { 1 } - { 2 } x }$ 有意义的 $x$ 的取值范围", "target": "1 - 2 x > 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "1 - 2 x > 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "1 - 2 x > 0"}]}}
{"content": "The sum of the two roots of the equation $x ^ 2 = 2020 x$ is ____ ?", "answer": "2020", "steps": "Convert the equation to general form: $x ^ { 2 } - 2020 x = 0$, and let the two roots of the equation be $x _ { 1 }$ and $x _ { 2 }$. Then $x _ { 1 } + x _ { 2 } = 2020$. ", "expr_cands": ["x ^ { 2 } = 2020 x", "x", "x ^ { 2 } - 2020 x = 0", "x = 0", "x = 2020", "x _ { 1 }", "x _ { 2 }", "x _ { 1 } + x _ { 2 } = 2020", "x _ { 1 } + x _ { 2 }", "2020"], "exprs": ["x ^ { 2 } - 2020 x = 0", "x _ { 1 }", "x _ { 2 }", "2020"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } = 2020 x"}, {"id": "x ^ { 2 } - 2020 x = 0"}, {"id": "设方程的两根为 $x _ { 1 }$ , $x _ { 2 }$"}, {"id": "x _ { 1 }"}, {"id": "x _ { 2 }"}, {"id": "2020"}, {"id": "x _ { 1 } + x _ { 2 }"}, {"id": "方程 $x ^ { 2 } = 2020 x$ 的两根之和"}, {"id": "一元二次方程根与系数关系，两根之和"}], "links": [{"rel": "移项", "source": "x ^ { 2 } = 2020 x", "target": "x ^ { 2 } - 2020 x = 0"}, {"rel": "被描述", "source": "x ^ { 2 } - 2020 x = 0", "target": "2020"}, {"rel": "假设描述", "source": "设方程的两根为 $x _ { 1 }$ , $x _ { 2 }$", "target": "x _ { 1 }"}, {"rel": "假设描述", "source": "设方程的两根为 $x _ { 1 }$ , $x _ { 2 }$", "target": "x _ { 2 }"}, {"rel": "被描述", "source": "x _ { 1 } + x _ { 2 }", "target": "2020"}, {"rel": "限制性描述", "source": "方程 $x ^ { 2 } = 2020 x$ 的两根之和", "target": "2020"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "2020"}]}}
{"content": "The equation about $x$, $mx + 4 = 3 x + 5$, has a solution of $x = - 1$. What is the value of $m$?", "answer": "2", "steps": "Substituting $x = - 1$, we get $- m + 4 = - 3 + 5$. Solving for $m$, we get $m = 2$.", "expr_cands": ["x", "mx + 4 = 3 x + 5", "m", "x = - 1", "- m + 4 = - 3 + 5", "m = 2"], "exprs": ["- m + 4 = - 3 + 5", "m = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "mx + 4 = 3 x + 5"}, {"id": "- m + 4 = - 3 + 5"}, {"id": "x = - 1"}, {"id": "m = 2"}], "links": [{"rel": "被代入", "source": "mx + 4 = 3 x + 5", "target": "- m + 4 = - 3 + 5"}, {"rel": "等式方程求解", "source": "- m + 4 = - 3 + 5", "target": "m = 2"}, {"rel": "代入", "source": "x = - 1", "target": "- m + 4 = - 3 + 5"}]}}
{"content": "Given $m = \\frac { 15 ^ { 4 } } { 3 ^ { 44 } }$ , $n = \\frac { 5 ^ { 4 } } { 3 ^ { 40 } }$ , what is $( - \\frac { 5 } { 3 } ) ^ { m - n }$ ?", "answer": "1", "steps": "Because $m = \\frac { 15 ^ { 4 } } { 3 ^ { 44 } } = \\frac { 3 ^ { 4 } \\cdot 5 ^ { 4 } } { 3 ^ { 4 } \\cdot 3 ^ { 40 } } = \\frac { 5 ^ { 4 } } { 3 ^ { 40 } } = n$, so $m = n$. Substituting $m = n$ into $( - \\frac { 5 } { 3 } ) ^ { m - n } = ( - \\frac { 5 } { 3 } ) ^ { 0 } = 1$.", "expr_cands": ["m = \\frac { 15 ^ { 4 } } { 3 ^ { 44 } }", "m", "n = \\frac { 5 ^ { 4 } } { 3 ^ { 40 } }", "n", "( - \\frac { 5 } { 3 } ) ^ { m - n }", "m = n", "1"], "exprs": ["m = n", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "m = \\frac { 15 ^ { 4 } } { 3 ^ { 44 } }"}, {"id": "m = n"}, {"id": "n = \\frac { 5 ^ { 4 } } { 3 ^ { 40 } }"}, {"id": "( - \\frac { 5 } { 3 } ) ^ { m - n }"}, {"id": "1"}], "links": [{"rel": "联立", "source": "m = \\frac { 15 ^ { 4 } } { 3 ^ { 44 } }", "target": "m = n"}, {"rel": "代入", "source": "m = n", "target": "1"}, {"rel": "联立", "source": "n = \\frac { 5 ^ { 4 } } { 3 ^ { 40 } }", "target": "m = n"}, {"rel": "被代入", "source": "( - \\frac { 5 } { 3 } ) ^ { m - n }", "target": "1"}]}}
{"content": "If the roots of the equation $x ^ 2 + 2 x - 3 = 0$ are $m$ and $n$, then $m + n$ = ____?", "answer": "- 2", "steps": "Since the equation $x ^ 2 + 2 x - 3 = 0$ has two roots $m$ and $n$, therefore $m + n = - \\frac { 2 } { 1 } = - 2$.", "expr_cands": ["x ^ { 2 } + 2 x - 3 = 0", "x", "m", "n", "m + n", "x = - 3", "x = 1", "m + n = - 2", "- 2"], "exprs": ["x = - 3", "x = 1", "- 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } + 2 x - 3 = 0"}, {"id": "x = - 3"}, {"id": "x = 1"}, {"id": "m + n"}, {"id": "- 2"}, {"id": "方程 $x ^ { 2 } + 2 x - 3 = 0$ 的两根分别为 $m$ , $n$"}], "links": [{"rel": "等式方程求解", "source": "x ^ { 2 } + 2 x - 3 = 0", "target": "x = - 3"}, {"rel": "等式方程求解", "source": "x ^ { 2 } + 2 x - 3 = 0", "target": "x = 1"}, {"rel": "被描述", "source": "x = - 3", "target": "- 2"}, {"rel": "被描述", "source": "x = 1", "target": "- 2"}, {"rel": "被描述", "source": "m + n", "target": "- 2"}, {"rel": "限制性描述", "source": "方程 $x ^ { 2 } + 2 x - 3 = 0$ 的两根分别为 $m$ , $n$", "target": "- 2"}]}}
{"content": "If the value of the polynomial $2 a - 4 b + 6$ is $10$, then the value of the polynomial $a - 2 b + 6$ is ____?", "answer": "8", "steps": "From the given information, we have $2 ( a - 2 b ) + 6 = 10$, which implies $a - 2 b = 2$. Therefore, the original expression is equal to $2 + 6 = 8$.", "expr_cands": ["2 a - 4 b + 6", "b", "a", "10", "a - 2 b + 6", "2 ( a - 2 b ) + 6 = 10", "a - 2 b = 2", "2 + 6", "8"], "exprs": ["2 ( a - 2 b ) + 6 = 10", "a - 2 b = 2", "8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 a - 4 b + 6"}, {"id": "2 ( a - 2 b ) + 6 = 10"}, {"id": "10"}, {"id": "多项式 $2 a - 4 b + 6$ 的值为 $10$"}, {"id": "a - 2 b = 2"}, {"id": "a - 2 b + 6"}, {"id": "8"}], "links": [{"rel": "被描述", "source": "2 a - 4 b + 6", "target": "2 ( a - 2 b ) + 6 = 10"}, {"rel": "等式方程部分求解", "source": "2 ( a - 2 b ) + 6 = 10", "target": "a - 2 b = 2"}, {"rel": "被描述", "source": "10", "target": "2 ( a - 2 b ) + 6 = 10"}, {"rel": "限制性描述", "source": "多项式 $2 a - 4 b + 6$ 的值为 $10$", "target": "2 ( a - 2 b ) + 6 = 10"}, {"rel": "代入", "source": "a - 2 b = 2", "target": "8"}, {"rel": "被代入", "source": "a - 2 b + 6", "target": "8"}]}}
{"content": "The intersection point of the line $y = x + 3 m$ and the line $y = 2 x - 6$ is on the $y$-axis. Find the value of $m$.", "answer": "- 2", "steps": "According to the problem, when $x = 0$, $y = 0 + 3 m = 3 m$, and $y = 2 * 0 - 6 = - 6$. Since the intersection point of the line $y = x + 3 m$ and the line $y = 2 x - 6$ is on the $y$-axis, we have $3 m = - 6$. Solving for $m$, we get $m = - 2$.", "expr_cands": ["y = x + 3 m", "y", "x", "m", "y = 2 x - 6", "x = 0", "y = 3 m", "y = - 6", "- 6 = 3 m", "3 m + x = - 6", "3 m = - 6", "m = - 2"], "exprs": ["x = 0", "y = 3 m", "y = - 6", "m = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = x + 3 m"}, {"id": "x = 0"}, {"id": "y = 2 x - 6"}, {"id": "y"}, {"id": "直线 $y = x + 3 m$ 与直线 $y = 2 x - 6$ 的交点在 $y$ 轴上"}, {"id": "y = 3 m"}, {"id": "y = - 6"}, {"id": "m = - 2"}], "links": [{"rel": "被描述", "source": "y = x + 3 m", "target": "x = 0"}, {"rel": "被代入", "source": "y = x + 3 m", "target": "y = 3 m"}, {"rel": "代入", "source": "x = 0", "target": "y = 3 m"}, {"rel": "代入", "source": "x = 0", "target": "y = - 6"}, {"rel": "被描述", "source": "y = 2 x - 6", "target": "x = 0"}, {"rel": "被代入", "source": "y = 2 x - 6", "target": "y = - 6"}, {"rel": "被描述", "source": "y", "target": "x = 0"}, {"rel": "限制性描述", "source": "直线 $y = x + 3 m$ 与直线 $y = 2 x - 6$ 的交点在 $y$ 轴上", "target": "x = 0"}, {"rel": "联立", "source": "y = 3 m", "target": "m = - 2"}, {"rel": "联立", "source": "y = - 6", "target": "m = - 2"}]}}
{"content": "If $\\sqrt { x } \\cdot \\sqrt { x - 6 } = \\sqrt { x ( x - 6 )}$, then ____?", "answer": "x \\ge 6", "steps": "Because $\\sqrt { x ( x - 6 ) } = \\sqrt { x } \\cdot \\sqrt { x - 6 }$ , therefore $x \\ge 0$ and $x - 6 \\ge 0$ , therefore $x \\ge 6$.", "expr_cands": ["\\sqrt { x } { \\cdot } \\sqrt { x - 6 } = \\sqrt { x ( x - 6 ) }", "x", "\\sqrt { x ( x - 6 ) } = \\sqrt { x } \\cdot \\sqrt { x - 6 }", "x \\ge 0", "x - 6 \\ge 0", "6 \\le x", "x \\ge 6"], "exprs": ["x \\ge 0", "x - 6 \\ge 0", "x \\ge 6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x } { \\cdot } \\sqrt { x - 6 } = \\sqrt { x ( x - 6 ) }"}, {"id": "x \\ge 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x - 6 \\ge 0"}, {"id": "x \\ge 6"}], "links": [{"rel": "被描述", "source": "\\sqrt { x } { \\cdot } \\sqrt { x - 6 } = \\sqrt { x ( x - 6 ) }", "target": "x \\ge 0"}, {"rel": "被描述", "source": "\\sqrt { x } { \\cdot } \\sqrt { x - 6 } = \\sqrt { x ( x - 6 ) }", "target": "x - 6 \\ge 0"}, {"rel": "联立", "source": "x \\ge 0", "target": "x \\ge 6"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 6 \\ge 0"}, {"rel": "联立", "source": "x - 6 \\ge 0", "target": "x \\ge 6"}]}}
{"content": "If the square root of $a - 2$ is meaningful in the real number range, then the range of values for $a$ is _____.", "answer": "a \\ge 2", "steps": "From the given condition, we have $a - 2 \\ge 0$, which implies $a \\ge 2$.", "expr_cands": ["\\sqrt { a - 2 }", "a", "a - 2 \\ge 0", "2 \\le a", "a \\ge 2"], "exprs": ["a - 2 \\ge 0", "a \\ge 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { a - 2 }"}, {"id": "a - 2 \\ge 0"}, {"id": "二次根式 $\\sqrt { a - 2 }$ 在实数范围内有意义"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "a \\ge 2"}], "links": [{"rel": "被描述", "source": "\\sqrt { a - 2 }", "target": "a - 2 \\ge 0"}, {"rel": "不等式方程求解", "source": "a - 2 \\ge 0", "target": "a \\ge 2"}, {"rel": "限制性描述", "source": "二次根式 $\\sqrt { a - 2 }$ 在实数范围内有意义", "target": "a - 2 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "a - 2 \\ge 0"}]}}
{"content": "Given $a ^ { 3 } b ^ { 6 } \\div ( a ^ { 2 } b ^ { 2 } ) = 3$, what is the value of $a ^ { 2 } b ^ { 8 }$?", "answer": "9", "steps": "\\because $( a ^ { 3 } b ^ { 6 } ) \\div ( a ^ { 2 } b ^ { 2 } ) = 3$ , which means $ab ^ { 4 } = 3$ , \\therefore $a ^ { 2 } b ^ { 8 } = ab ^ { 4 } \\times ab ^ { 4 } = 3 ^ { 2 } = 9$ .", "expr_cands": ["a ^ { 3 } b ^ { 6 } \\div ( a ^ { 2 } b ^ { 2 } ) = 3", "a", "b", "a ^ { 2 } b ^ { 8 }", "( a ^ { 3 } b ^ { 6 } ) \\div ( a ^ { 2 } b ^ { 2 } ) = 3", "ab ^ { 4 } = 3", "9"], "exprs": ["ab ^ { 4 } = 3", "9"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a ^ { 3 } b ^ { 6 } \\div ( a ^ { 2 } b ^ { 2 } ) = 3"}, {"id": "ab ^ { 4 } = 3"}, {"id": "a ^ { 2 } b ^ { 8 }"}, {"id": "9"}], "links": [{"rel": "计算", "source": "a ^ { 3 } b ^ { 6 } \\div ( a ^ { 2 } b ^ { 2 } ) = 3", "target": "ab ^ { 4 } = 3"}, {"rel": "代入", "source": "ab ^ { 4 } = 3", "target": "9"}, {"rel": "被代入", "source": "a ^ { 2 } b ^ { 8 }", "target": "9"}]}}
{"content": "If $3 x + 2$ and $- 2 x + 1$ are opposite numbers, then the value of $x - 2$ is ____?", "answer": "- 5", "steps": "$\\because$ $3 x + 2$ is the opposite of $- 2 x + 1$, $\\therefore$ $3 x + 2 - 2 x + 1 = 0$, solving for $x$ gives $x = - 3$, so $x - 2 = - 3 - 2 = - 5$.", "expr_cands": ["3 x + 2", "x", "- 2 x + 1", "x - 2", "3 x + 2 - 2 x + 1 = 0", "x = - 3", "- 5"], "exprs": ["3 x + 2 - 2 x + 1 = 0", "x = - 3", "- 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 x + 2"}, {"id": "3 x + 2 - 2 x + 1 = 0"}, {"id": "- 2 x + 1"}, {"id": "$3 x + 2$ 与 $- 2 x + 1$ 互为相反数"}, {"id": "x = - 3"}, {"id": "x - 2"}, {"id": "- 5"}], "links": [{"rel": "被描述", "source": "3 x + 2", "target": "3 x + 2 - 2 x + 1 = 0"}, {"rel": "等式方程求解", "source": "3 x + 2 - 2 x + 1 = 0", "target": "x = - 3"}, {"rel": "被描述", "source": "- 2 x + 1", "target": "3 x + 2 - 2 x + 1 = 0"}, {"rel": "限制性描述", "source": "$3 x + 2$ 与 $- 2 x + 1$ 互为相反数", "target": "3 x + 2 - 2 x + 1 = 0"}, {"rel": "代入", "source": "x = - 3", "target": "- 5"}, {"rel": "被代入", "source": "x - 2", "target": "- 5"}]}}
{"content": "Given rational numbers $x$ and $y$ satisfying the equation: $| x - 2 | + | y - 1 | = 0$, then $x$ = ____ ?", "answer": "2", "steps": "Since $| x - 2 | + | y - 1 | = 0$, it follows that $x - 2 = 0$, and solving for $x$ yields $x = 2$.", "expr_cands": ["x", "y", "| x - 2 | + | y - 1 | = 0", "x - 2 = 0", "x = 2"], "exprs": ["x - 2 = 0", "x = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| x - 2 | + | y - 1 | = 0"}, {"id": "x - 2 = 0"}, {"id": "有理数 $x$ 与 $y$ 满足关系式 : $| x - 2 | + | y - 1 | = 0$"}, {"id": "绝对值恒大于等于0"}, {"id": "x = 2"}], "links": [{"rel": "被描述", "source": "| x - 2 | + | y - 1 | = 0", "target": "x - 2 = 0"}, {"rel": "等式方程求解", "source": "x - 2 = 0", "target": "x = 2"}, {"rel": "限制性描述", "source": "有理数 $x$ 与 $y$ 满足关系式 : $| x - 2 | + | y - 1 | = 0$", "target": "x - 2 = 0"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "x - 2 = 0"}]}}
{"content": "Given that the value of $a - 2 b$ is $1008$, what is the value of $1 - 2 a + 4 b$?", "answer": "- 2015", "steps": "Since $a - 2 b = 1008$, therefore $1 - 2 a + 4 b = 1 - 2 ( a - 2 b ) = 1 - 2 * 1008 = 1 - 2016 = - 2015$.", "expr_cands": ["a - 2 b", "b", "a", "1008", "1 - 2 a + 4 b", "a - 2 b = 1008", "1 - 2 ( a - 2 b )", "- 2015"], "exprs": ["a - 2 b = 1008", "1 - 2 ( a - 2 b )", "- 2015"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a - 2 b"}, {"id": "a - 2 b = 1008"}, {"id": "1008"}, {"id": "$a - 2 b$ 的值是 $1008$"}, {"id": "1 - 2 a + 4 b"}, {"id": "1 - 2 ( a - 2 b )"}, {"id": "- 2015"}], "links": [{"rel": "被描述", "source": "a - 2 b", "target": "a - 2 b = 1008"}, {"rel": "提取因式参考", "source": "a - 2 b = 1008", "target": "1 - 2 ( a - 2 b )"}, {"rel": "代入", "source": "a - 2 b = 1008", "target": "- 2015"}, {"rel": "被描述", "source": "1008", "target": "a - 2 b = 1008"}, {"rel": "限制性描述", "source": "$a - 2 b$ 的值是 $1008$", "target": "a - 2 b = 1008"}, {"rel": "提取因式", "source": "1 - 2 a + 4 b", "target": "1 - 2 ( a - 2 b )"}, {"rel": "被代入", "source": "1 - 2 ( a - 2 b )", "target": "- 2015"}]}}
{"content": "When $a = - 2$, what is the value of the quadratic radical $\\sqrt { 2 - a }$?", "answer": "2", "steps": "When $a = - 2$, the quadratic radical $\\sqrt { 2 - a } = \\sqrt { 2 + 2 } = 2$.", "expr_cands": ["a = - 2", "a", "\\sqrt { 2 - a }", "2"], "exprs": ["2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { 2 - a }"}, {"id": "2"}, {"id": "a = - 2"}], "links": [{"rel": "被代入", "source": "\\sqrt { 2 - a }", "target": "2"}, {"rel": "代入", "source": "a = - 2", "target": "2"}]}}
{"content": "If $a ^ { x } = 3$, $a ^ { y } = 9$, then $a ^ { 2 x - y }$ = ____?", "answer": "1", "steps": "Because $a ^ { x } = 3$, $a ^ { y } = 9$, we can obtain $a ^ { 2 x - y } = ( a ^ { x } ) ^ { 2 } \\div a ^ { y } = 9 \\div 9 = 1$.", "expr_cands": ["a ^ { x } = 3", "a", "x", "a ^ { y } = 9", "y", "a ^ { 2 x - y }", "1"], "exprs": ["1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a ^ { 2 x - y }"}, {"id": "1"}, {"id": "a ^ { x } = 3"}, {"id": "a ^ { y } = 9"}], "links": [{"rel": "被代入", "source": "a ^ { 2 x - y }", "target": "1"}, {"rel": "代入", "source": "a ^ { x } = 3", "target": "1"}, {"rel": "代入", "source": "a ^ { y } = 9", "target": "1"}]}}
{"content": "The equation of the line $y = - 3 x - 2$ after being translated $3$ units upward is _____.", "answer": "y = - 3 x + 1", "steps": "The equation of the line $y = - 3 x - 2$ after being translated 3 units upwards is $y = - 3 x - 2 + 3$, which simplifies to $y = - 3 x + 1$.", "expr_cands": ["y = - 3 x - 2", "y", "x", "3", "y = - 3 x - 2 + 3", "- 3 x - 2 = - 3 x - 2 + 3", "- 3 x + 1"], "exprs": ["y = - 3 x - 2 + 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = - 3 x - 2"}, {"id": "y = - 3 x - 2 + 3"}, {"id": "3"}, {"id": "直线 $y = - 3 x - 2$ 向上平移 $3$ 个单位后"}, {"id": "所得直线的表达式"}], "links": [{"rel": "被描述", "source": "y = - 3 x - 2", "target": "y = - 3 x - 2 + 3"}, {"rel": "被描述", "source": "3", "target": "y = - 3 x - 2 + 3"}, {"rel": "限制性描述", "source": "直线 $y = - 3 x - 2$ 向上平移 $3$ 个单位后", "target": "y = - 3 x - 2 + 3"}, {"rel": "限制性描述", "source": "所得直线的表达式", "target": "y = - 3 x - 2 + 3"}]}}
{"content": "If ${ a } ^ { 2 } + 2 ma + 25 = { ( a + 5 ) } ^ { 2 }$, then the value of $m$ is ____?", "answer": "5", "steps": "Because $a ^ 2 + 2 ma + 25 = ( a + 5 ) ^ 2 = a ^ 2 + 10 a + 25$, therefore $2 m = 10$, therefore $m = 5$.", "expr_cands": ["{ a } ^ { 2 } + 2 ma + 25 = { ( a + 5 ) } ^ { 2 }", "a", "m", "a ^ { 2 } + 2 ma + 25 = a ^ { 2 } + 10 a + 25", "2 m = 10", "m = 5"], "exprs": ["a ^ { 2 } + 2 ma + 25 = a ^ { 2 } + 10 a + 25", "2 m = 10", "m = 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "{ a } ^ { 2 } + 2 ma + 25 = { ( a + 5 ) } ^ { 2 }"}, {"id": "a ^ { 2 } + 2 ma + 25 = a ^ { 2 } + 10 a + 25"}, {"id": "2 m = 10"}, {"id": "m = 5"}], "links": [{"rel": "展开", "source": "{ a } ^ { 2 } + 2 ma + 25 = { ( a + 5 ) } ^ { 2 }", "target": "a ^ { 2 } + 2 ma + 25 = a ^ { 2 } + 10 a + 25"}, {"rel": "移项", "source": "a ^ { 2 } + 2 ma + 25 = a ^ { 2 } + 10 a + 25", "target": "2 m = 10"}, {"rel": "等式方程求解", "source": "2 m = 10", "target": "m = 5"}]}}
{"content": "Given that $x = 2$ is a solution to the equation $ax - 1 = x + 3$, what is the value of $a$?", "answer": "3", "steps": "Substituting $x = 2$ into the equation yields: $2 a - 1 = 2 + 3$, solving for $a$ gives $a = 3$.", "expr_cands": ["x = 2", "x", "ax - 1 = x + 3", "a", "2 a - 1 = 2 + 3", "a = 3"], "exprs": ["2 a - 1 = 2 + 3", "a = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x = 2"}, {"id": "2 a - 1 = 2 + 3"}, {"id": "ax - 1 = x + 3"}, {"id": "a = 3"}], "links": [{"rel": "代入", "source": "x = 2", "target": "2 a - 1 = 2 + 3"}, {"rel": "等式方程求解", "source": "2 a - 1 = 2 + 3", "target": "a = 3"}, {"rel": "被代入", "source": "ax - 1 = x + 3", "target": "2 a - 1 = 2 + 3"}]}}
{"content": "The solution set of the inequality $- 2 x + a \\ge 5$ is $x \\le - 1$. What is the value of $a$?", "answer": "3", "steps": "$- 2 x + a \\ge 5$, moving the terms gives $- 2 x \\ge 5 - a$, dividing by $- 2$ gives $x \\le \\frac { a - 5 } { 2 }$. Since the solution set of the equation $- 2 x + a \\ge 5$ is $x \\le - 1$, we have $\\frac { a - 5 } { 2 } = - 1$, solving for $a$ gives $a = 3$.", "expr_cands": ["- 2 x + a \\ge 5", "a", "x", "x \\le - 1", "- 2 x \\ge 5 - a", "1", "x \\le \\frac { a - 5 } { 2 }", "\\frac { a - 5 } { 2 } = - 1", "a = 3"], "exprs": ["x \\le \\frac { a - 5 } { 2 }", "\\frac { a - 5 } { 2 } = - 1", "a = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 2 x + a \\ge 5"}, {"id": "x \\le \\frac { a - 5 } { 2 }"}, {"id": "x \\le - 1"}, {"id": "\\frac { a - 5 } { 2 } = - 1"}, {"id": "a = 3"}], "links": [{"rel": "不等式方程部分求解", "source": "- 2 x + a \\ge 5", "target": "x \\le \\frac { a - 5 } { 2 }"}, {"rel": "联立", "source": "x \\le \\frac { a - 5 } { 2 }", "target": "\\frac { a - 5 } { 2 } = - 1"}, {"rel": "联立", "source": "x \\le - 1", "target": "\\frac { a - 5 } { 2 } = - 1"}, {"rel": "等式方程求解", "source": "\\frac { a - 5 } { 2 } = - 1", "target": "a = 3"}]}}
{"content": "If the solution to the equation $mx + 3 = x + 2$ is $x = 1$, then the value of $m$ is ____?", "answer": "0", "steps": "Substituting $x = 1$ into the equation, we get $m + 3 = 1 + 2$, which yields $m = 0$ upon solving.", "expr_cands": ["mx + 3 = x + 2", "m", "x", "x = 1", "m + 3 = 1 + 2", "m = 0"], "exprs": ["m + 3 = 1 + 2", "m = 0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "mx + 3 = x + 2"}, {"id": "m + 3 = 1 + 2"}, {"id": "x = 1"}, {"id": "m = 0"}], "links": [{"rel": "被代入", "source": "mx + 3 = x + 2", "target": "m + 3 = 1 + 2"}, {"rel": "等式方程求解", "source": "m + 3 = 1 + 2", "target": "m = 0"}, {"rel": "代入", "source": "x = 1", "target": "m + 3 = 1 + 2"}]}}
{"content": "If $- \\frac { 2 } { 7 } x ^ { m } y ^ { m - 2 } z ^ { m }$ is a monomial of degree seven, then $m$ = ____ ?", "answer": "3", "steps": "From the given information, we have $m + m - 2 + m = 7$. Solving for $m$, we get $m = 3$.", "expr_cands": ["- \\frac { 2 } { 7 } x ^ { m } y ^ { m - 2 } z ^ { m }", "y", "z", "x", "m", "m + m - 2 + m = 7", "m = 3"], "exprs": ["m + m - 2 + m = 7", "m = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- \\frac { 2 } { 7 } x ^ { m } y ^ { m - 2 } z ^ { m }"}, {"id": "m + m - 2 + m = 7"}, {"id": "$- \\frac { 2 } { 7 } x ^ { m } y ^ { m - 2 } z ^ { m }$ 是一个七次单项式"}, {"id": "m = 3"}], "links": [{"rel": "被描述", "source": "- \\frac { 2 } { 7 } x ^ { m } y ^ { m - 2 } z ^ { m }", "target": "m + m - 2 + m = 7"}, {"rel": "等式方程求解", "source": "m + m - 2 + m = 7", "target": "m = 3"}, {"rel": "限制性描述", "source": "$- \\frac { 2 } { 7 } x ^ { m } y ^ { m - 2 } z ^ { m }$ 是一个七次单项式", "target": "m + m - 2 + m = 7"}]}}
{"content": "Given that $x = - 2$ is a solution of the quadratic equation $x ^ 2 - mx + 5 = 0$, then $m$ = ____ ?", "answer": "- \\frac { 9 } { 2 }", "steps": "Substituting $x = - 2$ into $x ^ 2 - mx + 5 = 0$ yields $4 + 2 m + 5 = 0$, solving for $m$ gives $m = - \\frac { 9 } { 2 }$.", "expr_cands": ["x = - 2", "x", "x ^ { 2 } - mx + 5 = 0", "m", "2 m + 9 = 0", "4 + 2 m + 5 = 0", "m = - \\frac { 9 } { 2 }"], "exprs": ["4 + 2 m + 5 = 0", "m = - \\frac { 9 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } - mx + 5 = 0"}, {"id": "4 + 2 m + 5 = 0"}, {"id": "x = - 2"}, {"id": "m = - \\frac { 9 } { 2 }"}], "links": [{"rel": "被代入", "source": "x ^ { 2 } - mx + 5 = 0", "target": "4 + 2 m + 5 = 0"}, {"rel": "等式方程求解", "source": "4 + 2 m + 5 = 0", "target": "m = - \\frac { 9 } { 2 }"}, {"rel": "代入", "source": "x = - 2", "target": "4 + 2 m + 5 = 0"}]}}
{"content": "If the solution to the equation $ax = 3 x + 5$ is $x = 5$, then $a$ = ____?", "answer": "4", "steps": "$\\because$ The equation $ax = 3 x + 5$ has a solution of $x = 5$, $\\therefore$ $5 a = 15 + 5$, which solves for $a = 4$.", "expr_cands": ["ax = 3 x + 5", "a", "x", "x = 5", "5 a = 15 + 5", "a = 4"], "exprs": ["5 a = 15 + 5", "a = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "ax = 3 x + 5"}, {"id": "5 a = 15 + 5"}, {"id": "x = 5"}, {"id": "a = 4"}], "links": [{"rel": "被代入", "source": "ax = 3 x + 5", "target": "5 a = 15 + 5"}, {"rel": "等式方程求解", "source": "5 a = 15 + 5", "target": "a = 4"}, {"rel": "代入", "source": "x = 5", "target": "5 a = 15 + 5"}]}}
{"content": "Given a quadratic function $y = x ^ 2 - 4 x + c$ that intersects the $x$-axis at only one point, what is the value of $c$?", "answer": "4", "steps": "Because the quadratic function $y = x ^ 2 - 4 x + c$ has only one intersection with the $x$-axis, i.e. the quadratic equation $x ^ 2 - 4 x + c = 0$ has one root, we have $4 ^ 2 - 4 \\cdot 1 \\cdot c = 0$, which gives $c = 4$.", "expr_cands": ["y = x ^ { 2 } - 4 x + c", "c", "y", "x", "{ x } ^ { 2 } - 4 x + c = 0", "{ 4 } ^ { 2 } - 4 * 1 * c = 0", "c = 4"], "exprs": ["{ x } ^ { 2 } - 4 x + c = 0", "{ 4 } ^ { 2 } - 4 * 1 * c = 0", "c = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = x ^ { 2 } - 4 x + c"}, {"id": "{ x } ^ { 2 } - 4 x + c = 0"}, {"id": "x"}, {"id": "二次函数 $y = x ^ { 2 } - 4 x + c$ 与 $x$ 轴只有一个交点"}, {"id": "{ 4 } ^ { 2 } - 4 * 1 * c = 0"}, {"id": "即一元二次方程 ${ x } ^ { 2 } - 4 x + c = 0$ 有一根"}, {"id": "一元二次方程根与系数关系，方程有解，则根的判别式大于等于0"}, {"id": "c = 4"}], "links": [{"rel": "被描述", "source": "y = x ^ { 2 } - 4 x + c", "target": "{ x } ^ { 2 } - 4 x + c = 0"}, {"rel": "被描述", "source": "{ x } ^ { 2 } - 4 x + c = 0", "target": "{ 4 } ^ { 2 } - 4 * 1 * c = 0"}, {"rel": "被描述", "source": "x", "target": "{ x } ^ { 2 } - 4 x + c = 0"}, {"rel": "限制性描述", "source": "二次函数 $y = x ^ { 2 } - 4 x + c$ 与 $x$ 轴只有一个交点", "target": "{ x } ^ { 2 } - 4 x + c = 0"}, {"rel": "限制性描述", "source": "二次函数 $y = x ^ { 2 } - 4 x + c$ 与 $x$ 轴只有一个交点", "target": "{ 4 } ^ { 2 } - 4 * 1 * c = 0"}, {"rel": "等式方程求解", "source": "{ 4 } ^ { 2 } - 4 * 1 * c = 0", "target": "c = 4"}, {"rel": "限制性描述", "source": "即一元二次方程 ${ x } ^ { 2 } - 4 x + c = 0$ 有一根", "target": "{ 4 } ^ { 2 } - 4 * 1 * c = 0"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，方程有解，则根的判别式大于等于0", "target": "{ 4 } ^ { 2 } - 4 * 1 * c = 0"}]}}
{"content": "Given $( m ^ 2 + n ^ 2 ) ( m ^ 2 + 2 + n ^ 2 ) = 48$, find $m ^ 2 + n ^ 2$.", "answer": "6", "steps": "Let $m ^ { 2 } + n ^ { 2 } = y$, then the original equation can be transformed into $y ^ { 2 } + 2 y - 48 = 0$. Solving this equation yields $y _ { 1 } = - 8$ and $y _ { 2 } = 6$. Since $m ^ { 2 } + n ^ { 2 } \\ge 0$, we have $m ^ { 2 } + n ^ { 2 } = 6$.", "expr_cands": ["( m ^ { 2 } + n ^ { 2 } ) ( m ^ { 2 } + 2 + n ^ { 2 } ) = 48", "m", "n", "m ^ { 2 } + n ^ { 2 }", "m ^ { 2 } + n ^ { 2 } = y", "y", "y ^ { 2 } + 2 y - 48 = 0", "y = - 8", "y = 6", "y _ { 1 } = - 8", "y _ { 1 }", "y _ { 2 } = 6", "y _ { 2 }", "m ^ { 2 } + n ^ { 2 } \\ge 0", "m ^ { 2 } + n ^ { 2 } = 6", "6"], "exprs": ["m ^ { 2 } + n ^ { 2 } = y", "y ^ { 2 } + 2 y - 48 = 0", "6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "设 $m ^ { 2 } + n ^ { 2 } = y$"}, {"id": "m ^ { 2 } + n ^ { 2 } = y"}, {"id": "( m ^ { 2 } + n ^ { 2 } ) ( m ^ { 2 } + 2 + n ^ { 2 } ) = 48"}, {"id": "y ^ { 2 } + 2 y - 48 = 0"}, {"id": "6"}, {"id": "m ^ { 2 } + n ^ { 2 } \\ge 0"}, {"id": "多项式偶次方项恒大于等于0"}], "links": [{"rel": "假设描述", "source": "设 $m ^ { 2 } + n ^ { 2 } = y$", "target": "m ^ { 2 } + n ^ { 2 } = y"}, {"rel": "联立", "source": "m ^ { 2 } + n ^ { 2 } = y", "target": "y ^ { 2 } + 2 y - 48 = 0"}, {"rel": "被描述", "source": "m ^ { 2 } + n ^ { 2 } = y", "target": "6"}, {"rel": "联立", "source": "( m ^ { 2 } + n ^ { 2 } ) ( m ^ { 2 } + 2 + n ^ { 2 } ) = 48", "target": "y ^ { 2 } + 2 y - 48 = 0"}, {"rel": "被描述", "source": "y ^ { 2 } + 2 y - 48 = 0", "target": "6"}, {"rel": "被描述", "source": "m ^ { 2 } + n ^ { 2 } \\ge 0", "target": "6"}, {"rel": "属性描述", "source": "多项式偶次方项恒大于等于0", "target": "6"}]}}
{"content": "If the value of $\\frac { 2 ( 1 - x )} { 3 }$ is non-negative, then the range of possible values for $x$ is ____?", "answer": "x \\le 1", "steps": "According to the problem, we have $\\frac { 2 ( 1 - x )} { 3 } \\geq 0$. Thus, $2 - 2 x \\geq 0$, $- 2 x \\geq - 2$, and therefore $x \\leq 1$.", "expr_cands": ["\\frac { 2 ( 1 - x ) } { 3 }", "x", "\\frac { 2 ( 1 - x ) } { 3 } \\ge 0", "x \\le 1", "2 - 2 x \\ge 0", "- 2 x \\ge - 2"], "exprs": ["\\frac { 2 ( 1 - x ) } { 3 } \\ge 0", "x \\le 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 2 ( 1 - x ) } { 3 }"}, {"id": "\\frac { 2 ( 1 - x ) } { 3 } \\ge 0"}, {"id": "$\\frac { 2 ( 1 - x ) } { 3 }$ 的值是非负数"}, {"id": "x \\le 1"}], "links": [{"rel": "被描述", "source": "\\frac { 2 ( 1 - x ) } { 3 }", "target": "\\frac { 2 ( 1 - x ) } { 3 } \\ge 0"}, {"rel": "不等式方程求解", "source": "\\frac { 2 ( 1 - x ) } { 3 } \\ge 0", "target": "x \\le 1"}, {"rel": "限制性描述", "source": "$\\frac { 2 ( 1 - x ) } { 3 }$ 的值是非负数", "target": "\\frac { 2 ( 1 - x ) } { 3 } \\ge 0"}]}}
{"content": "Given that one root of the equation $x ^ 2 - 3 x + m = 0$ is $2$, the other root of this equation is ____?", "answer": "1", "steps": "Let the other root of the equation be $x _ 1$. According to the given condition, we have $2 + x _ 1 = 3$. Solving for $x _ 1$, we get $x _ 1 = 1$. Therefore, the other root of the equation is $1$.", "expr_cands": ["x", "x ^ { 2 } - 3 x + m = 0", "m", "2", "x _ { 1 }", "2 + x _ { 1 } = 3", "x_{1} = 1", "x _ { 1 } = 1", "1"], "exprs": ["x _ { 1 }", "2 + x _ { 1 } = 3", "x _ { 1 } = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "设方程另一个根为 $x _ { 1 }$"}, {"id": "x _ { 1 }"}, {"id": "2"}, {"id": "2 + x _ { 1 } = 3"}, {"id": "x ^ { 2 } - 3 x + m = 0"}, {"id": "关于 $x$ 的方程 $x ^ { 2 } - 3 x + m = 0$ 的一个根是 $2$"}, {"id": "一元二次方程根与系数关系，两根之和"}, {"id": "x _ { 1 } = 1"}], "links": [{"rel": "假设描述", "source": "设方程另一个根为 $x _ { 1 }$", "target": "x _ { 1 }"}, {"rel": "限制性描述", "source": "设方程另一个根为 $x _ { 1 }$", "target": "2 + x _ { 1 } = 3"}, {"rel": "被描述", "source": "x _ { 1 }", "target": "2 + x _ { 1 } = 3"}, {"rel": "被描述", "source": "2", "target": "2 + x _ { 1 } = 3"}, {"rel": "等式方程求解", "source": "2 + x _ { 1 } = 3", "target": "x _ { 1 } = 1"}, {"rel": "被描述", "source": "x ^ { 2 } - 3 x + m = 0", "target": "2 + x _ { 1 } = 3"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $x ^ { 2 } - 3 x + m = 0$ 的一个根是 $2$", "target": "2 + x _ { 1 } = 3"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "2 + x _ { 1 } = 3"}]}}
{"content": "Rewrite the quadratic function $y = x ^ { 2 } - 2 x - 3$ in the form of $y = ( x - h ) ^ { 2 } + k$. The value of $h + k$ is ____ ?", "answer": "- 3", "steps": "$y = x ^ { 2 } - 2 x - 3 = ( x ^ { 2 } - 2 x + 1 ) - 1 - 3 = ( x - 1 ) ^ { 2 } - 4$. Therefore, $h = 1$, $k = - 4$, and $h + k = - 3$.", "expr_cands": ["y = x ^ { 2 } - 2 x - 3", "y", "x", "y = ( x - h ) ^ { 2 } + k", "k", "h", "h + k", "y = ( x - 1 ) ^ { 2 } - 4", "h = 1", "k = - 4", "- 3"], "exprs": ["y = ( x - 1 ) ^ { 2 } - 4", "h = 1", "k = - 4", "- 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = x ^ { 2 } - 2 x - 3"}, {"id": "y = ( x - 1 ) ^ { 2 } - 4"}, {"id": "h = 1"}, {"id": "将二次函数 $y = x ^ { 2 } - 2 x - 3$ 化成 $y = ( x - h ) ^ { 2 } + k$ 形式"}, {"id": "k = - 4"}, {"id": "h + k"}, {"id": "- 3"}], "links": [{"rel": "提取因式", "source": "y = x ^ { 2 } - 2 x - 3", "target": "y = ( x - 1 ) ^ { 2 } - 4"}, {"rel": "被描述", "source": "y = ( x - 1 ) ^ { 2 } - 4", "target": "h = 1"}, {"rel": "被描述", "source": "y = ( x - 1 ) ^ { 2 } - 4", "target": "k = - 4"}, {"rel": "代入", "source": "h = 1", "target": "- 3"}, {"rel": "限制性描述", "source": "将二次函数 $y = x ^ { 2 } - 2 x - 3$ 化成 $y = ( x - h ) ^ { 2 } + k$ 形式", "target": "h = 1"}, {"rel": "限制性描述", "source": "将二次函数 $y = x ^ { 2 } - 2 x - 3$ 化成 $y = ( x - h ) ^ { 2 } + k$ 形式", "target": "k = - 4"}, {"rel": "代入", "source": "k = - 4", "target": "- 3"}, {"rel": "被代入", "source": "h + k", "target": "- 3"}]}}
{"content": "If $x - y = - 6$ and $xy = - 8$, then the value of the algebraic expression $( 4 x + 3 y - 2 xy ) - ( 2 x + 5 y + xy )$ is ____?", "answer": "12", "steps": "Original expression = $4 x + 3 y - 2 xy - 2 x - 5 y - xy = 2 x - 2 y - 3 xy = 2 ( x - y ) - 3 xy$ , when $x - y = - 6$ , $xy = - 8$ , the original expression = $- 12 + 24 = 12$.", "expr_cands": ["x - y = - 6", "y", "x", "xy = - 8", "( 4 x + 3 y - 2 xy ) - ( 2 x + 5 y + xy )", "4 x + 3 y - 2 xy - 2 x - 5 y - xy", "2 ( x - y ) - 3 xy", "- 12 + 24", "12"], "exprs": ["4 x + 3 y - 2 xy - 2 x - 5 y - xy", "2 ( x - y ) - 3 xy", "12"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( 4 x + 3 y - 2 xy ) - ( 2 x + 5 y + xy )"}, {"id": "4 x + 3 y - 2 xy - 2 x - 5 y - xy"}, {"id": "2 ( x - y ) - 3 xy"}, {"id": "x - y = - 6"}, {"id": "xy = - 8"}, {"id": "12"}], "links": [{"rel": "展开", "source": "( 4 x + 3 y - 2 xy ) - ( 2 x + 5 y + xy )", "target": "4 x + 3 y - 2 xy - 2 x - 5 y - xy"}, {"rel": "提取因式", "source": "4 x + 3 y - 2 xy - 2 x - 5 y - xy", "target": "2 ( x - y ) - 3 xy"}, {"rel": "被代入", "source": "2 ( x - y ) - 3 xy", "target": "12"}, {"rel": "提取因式参考", "source": "x - y = - 6", "target": "2 ( x - y ) - 3 xy"}, {"rel": "代入", "source": "x - y = - 6", "target": "12"}, {"rel": "提取因式参考", "source": "xy = - 8", "target": "2 ( x - y ) - 3 xy"}, {"rel": "代入", "source": "xy = - 8", "target": "12"}]}}
{"content": "Given that one root of the equation $x ^ 2 - 7 x + 15 = k$ is $2$, the value of $k$ is ____?", "answer": "5", "steps": "Substituting $x = 2$ into the equation $x ^ 2 - 7 x + 15 = k$, we get $4 - 14 + 15 = k$. Solving for $k$, we get $k = 5$.", "expr_cands": ["x", "x ^ { 2 } - 7 x + 15 = k", "k", "2", "x = 2", "5 = k", "4 - 14 + 15 = k", "k = 5"], "exprs": ["x = 2", "4 - 14 + 15 = k", "k = 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2"}, {"id": "x = 2"}, {"id": "x ^ { 2 } - 7 x + 15 = k"}, {"id": "x"}, {"id": "关于 $x$ 的方程 $x ^ { 2 } - 7 x + 15 = k$ 的一个根是 $2$"}, {"id": "4 - 14 + 15 = k"}, {"id": "k = 5"}], "links": [{"rel": "被描述", "source": "2", "target": "x = 2"}, {"rel": "代入", "source": "x = 2", "target": "4 - 14 + 15 = k"}, {"rel": "被描述", "source": "x ^ { 2 } - 7 x + 15 = k", "target": "x = 2"}, {"rel": "被代入", "source": "x ^ { 2 } - 7 x + 15 = k", "target": "4 - 14 + 15 = k"}, {"rel": "被描述", "source": "x", "target": "x = 2"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $x ^ { 2 } - 7 x + 15 = k$ 的一个根是 $2$", "target": "x = 2"}, {"rel": "等式方程求解", "source": "4 - 14 + 15 = k", "target": "k = 5"}]}}
{"content": "The proportional function $y = kx$ decreases by $2$ when the independent variable increases by $1$. The value of $k$ is ____?", "answer": "- 2", "steps": "According to the problem, we have $y - 2 = k ( x + 1 )$, which means $y - 2 = kx + k$. Since $y = kx$, we have $k = - 2$.", "expr_cands": ["y = kx", "k", "y", "x", "1", "2", "y - 2 = k ( x + 1 )", "y - 2", "kx + k", "k = - 2"], "exprs": ["y - 2 = k ( x + 1 )", "k = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = kx"}, {"id": "y - 2 = k ( x + 1 )"}, {"id": "正比例函数 $y = kx$ 的自变量取值增加 $1$"}, {"id": "函数值就相应减少 $2$"}, {"id": "k = - 2"}], "links": [{"rel": "被描述", "source": "y = kx", "target": "y - 2 = k ( x + 1 )"}, {"rel": "联立", "source": "y = kx", "target": "k = - 2"}, {"rel": "联立", "source": "y - 2 = k ( x + 1 )", "target": "k = - 2"}, {"rel": "限制性描述", "source": "正比例函数 $y = kx$ 的自变量取值增加 $1$", "target": "y - 2 = k ( x + 1 )"}, {"rel": "限制性描述", "source": "函数值就相应减少 $2$", "target": "y - 2 = k ( x + 1 )"}]}}
{"content": "If $| a - 1 | + | b + 2 | = 0$, then the value of $b - a$ is ____?", "answer": "- 3", "steps": "Since $| a - 1 | \\ge 0$ and $| b + 2 | \\ge 0$, and also $| a - 1 | + | b + 2 | = 0$, therefore $| a - 1 | = 0$ and $| b + 2 | = 0$. Hence, $a = 1$ and $b = - 2$. Therefore, $b - a = - 2 - 1 = - 3$.", "expr_cands": ["| a - 1 | + | b + 2 | = 0", "b", "a", "b - a", "| a - 1 | \\ge 0", "| b + 2 | \\ge 0", "| a - 1 | = 0", "a = 1", "| b + 2 | = 0", "b = - 2", "- 3"], "exprs": ["| a - 1 | = 0", "| b + 2 | = 0", "a = 1", "b = - 2", "- 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| a - 1 | + | b + 2 | = 0"}, {"id": "| a - 1 | = 0"}, {"id": "绝对值恒大于等于0"}, {"id": "| b + 2 | = 0"}, {"id": "a = 1"}, {"id": "b = - 2"}, {"id": "b - a"}, {"id": "- 3"}], "links": [{"rel": "被描述", "source": "| a - 1 | + | b + 2 | = 0", "target": "| a - 1 | = 0"}, {"rel": "被描述", "source": "| a - 1 | + | b + 2 | = 0", "target": "| b + 2 | = 0"}, {"rel": "等式方程求解", "source": "| a - 1 | = 0", "target": "a = 1"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "| a - 1 | = 0"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "| b + 2 | = 0"}, {"rel": "等式方程求解", "source": "| b + 2 | = 0", "target": "b = - 2"}, {"rel": "代入", "source": "a = 1", "target": "- 3"}, {"rel": "代入", "source": "b = - 2", "target": "- 3"}, {"rel": "被代入", "source": "b - a", "target": "- 3"}]}}
{"content": "Given $a$, $b$, and $c$ satisfy $a - b = 8$, $ab + c ^ 2 + 16 = 0$, then the value of $2 a + b + c$ is ____?", "answer": "4", "steps": "Since $a - b = 8$, therefore $a = b + 8$, therefore $ab + c ^ 2 + 16 = b ( b + 8 ) + c ^ 2 + 16 = ( b + 4 ) ^ 2 + c ^ 2 = 0$, therefore $b + 4 = 0$, $c = 0$, solving gives: $b = - 4$, therefore $a = 4$, therefore $2 a + b + c = 4$.", "expr_cands": ["a", "b", "c", "a - b = 8", "ab + c ^ { 2 } + 16 = 0", "2 a + b + c", "a = b + 8", "b + 4 = 0", "b = - 4", "c = 0", "a = 4", "4"], "exprs": ["a = b + 8", "b + 4 = 0", "c = 0", "b = - 4", "a = 4", "4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a - b = 8"}, {"id": "a = b + 8"}, {"id": "ab + c ^ { 2 } + 16 = 0"}, {"id": "b + 4 = 0"}, {"id": "多项式偶次方项恒大于等于0"}, {"id": "b = - 4"}, {"id": "c = 0"}, {"id": "a = 4"}, {"id": "2 a + b + c"}, {"id": "4"}], "links": [{"rel": "移项", "source": "a - b = 8", "target": "a = b + 8"}, {"rel": "被描述", "source": "a = b + 8", "target": "b + 4 = 0"}, {"rel": "被描述", "source": "a = b + 8", "target": "c = 0"}, {"rel": "被代入", "source": "a = b + 8", "target": "a = 4"}, {"rel": "被描述", "source": "ab + c ^ { 2 } + 16 = 0", "target": "b + 4 = 0"}, {"rel": "被描述", "source": "ab + c ^ { 2 } + 16 = 0", "target": "c = 0"}, {"rel": "等式方程求解", "source": "b + 4 = 0", "target": "b = - 4"}, {"rel": "限制性描述", "source": "多项式偶次方项恒大于等于0", "target": "b + 4 = 0"}, {"rel": "限制性描述", "source": "多项式偶次方项恒大于等于0", "target": "c = 0"}, {"rel": "代入", "source": "b = - 4", "target": "a = 4"}, {"rel": "代入", "source": "b = - 4", "target": "4"}, {"rel": "代入", "source": "c = 0", "target": "4"}, {"rel": "代入", "source": "a = 4", "target": "4"}, {"rel": "被代入", "source": "2 a + b + c", "target": "4"}]}}
{"content": "If the equation $2 x ^ { m - 1 } + y ^ { 2 n + m } = \\frac { 1 } { 2 }$ is a quadratic equation, then $mn$ = ____?", "answer": "- 1", "steps": "From the given information, we have $m - 1 = 1$ and $2 n + m = 1$. Solving for $m$ and $n$, we get $m = 2$ and $n = - \\frac { 1 } { 2 }$. Therefore, $mn = - 1$.", "expr_cands": ["2 x ^ { m - 1 } + y ^ { 2 n + m } = \\frac { 1 } { 2 }", "n", "y", "m", "x", "mn", "m - 1 = 1", "m = 2", "2 n + m = 1", "n = - \\frac { 1 } { 2 }", "- 1"], "exprs": ["m - 1 = 1", "2 n + m = 1", "m = 2", "n = - \\frac { 1 } { 2 }", "- 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x ^ { m - 1 } + y ^ { 2 n + m } = \\frac { 1 } { 2 }"}, {"id": "m - 1 = 1"}, {"id": "方程 $2 x ^ { m - 1 } + y ^ { 2 n + m } = \\frac { 1 } { 2 }$ 是二元一次方程"}, {"id": "2 n + m = 1"}, {"id": "m = 2"}, {"id": "n = - \\frac { 1 } { 2 }"}, {"id": "mn"}, {"id": "- 1"}], "links": [{"rel": "被描述", "source": "2 x ^ { m - 1 } + y ^ { 2 n + m } = \\frac { 1 } { 2 }", "target": "m - 1 = 1"}, {"rel": "被描述", "source": "2 x ^ { m - 1 } + y ^ { 2 n + m } = \\frac { 1 } { 2 }", "target": "2 n + m = 1"}, {"rel": "等式方程求解", "source": "m - 1 = 1", "target": "m = 2"}, {"rel": "限制性描述", "source": "方程 $2 x ^ { m - 1 } + y ^ { 2 n + m } = \\frac { 1 } { 2 }$ 是二元一次方程", "target": "m - 1 = 1"}, {"rel": "限制性描述", "source": "方程 $2 x ^ { m - 1 } + y ^ { 2 n + m } = \\frac { 1 } { 2 }$ 是二元一次方程", "target": "2 n + m = 1"}, {"rel": "联立", "source": "2 n + m = 1", "target": "n = - \\frac { 1 } { 2 }"}, {"rel": "联立", "source": "m = 2", "target": "n = - \\frac { 1 } { 2 }"}, {"rel": "代入", "source": "m = 2", "target": "- 1"}, {"rel": "代入", "source": "n = - \\frac { 1 } { 2 }", "target": "- 1"}, {"rel": "被代入", "source": "mn", "target": "- 1"}]}}
{"content": "If $m - 3 n - 5 = 0$, then the value of the algebraic expression $2018 - 3 m + 9 n$ is ____?", "answer": "2003", "steps": "Since $m - 3 n - 5 = 0$, it follows that $m - 3 n = 5$. Therefore, $2018 - 3 m + 9 n = 2018 - 3 ( m - 3 n ) = 2018 - 15 = 2003$.", "expr_cands": ["m - 3 n - 5 = 0", "m", "n", "2018 - 3 m + 9 n", "m - 3 n = 5", "2018 - 3 ( m - 3 n )", "2003"], "exprs": ["m - 3 n = 5", "2018 - 3 ( m - 3 n )", "2003"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "m - 3 n - 5 = 0"}, {"id": "m - 3 n = 5"}, {"id": "2018 - 3 m + 9 n"}, {"id": "2018 - 3 ( m - 3 n )"}, {"id": "2003"}], "links": [{"rel": "移项", "source": "m - 3 n - 5 = 0", "target": "m - 3 n = 5"}, {"rel": "提取因式参考", "source": "m - 3 n = 5", "target": "2018 - 3 ( m - 3 n )"}, {"rel": "代入", "source": "m - 3 n = 5", "target": "2003"}, {"rel": "提取因式", "source": "2018 - 3 m + 9 n", "target": "2018 - 3 ( m - 3 n )"}, {"rel": "被代入", "source": "2018 - 3 ( m - 3 n )", "target": "2003"}]}}
{"content": "If the equation $\\frac { x } { 3 - x } - x = \\frac { a } { x - 3 }$ has a positive root with respect to $x$, then $a$ = ____?", "answer": "- 3", "steps": "The original equation can be simplified by eliminating the denominator: $x - x ( 3 - x ) = - ax ^ 2 - 2 x = - a$. Since the equation has a fractional root at $x = 3$, we have $9 - 6 = - a$, which gives $a = - 3$.", "expr_cands": ["x", "\\frac { x } { 3 - x } - x = \\frac { a } { x - 3 }", "a", "x - x ( 3 - x ) = - a", "x = 3", "9 - 6 = - a", "a = - 3"], "exprs": ["x - x ( 3 - x ) = - a", "x = 3", "a = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x } { 3 - x } - x = \\frac { a } { x - 3 }"}, {"id": "x - x ( 3 - x ) = - a"}, {"id": "x = 3"}, {"id": "关于 $x$ 的方程 $\\frac { x } { 3 - x } - x = \\frac { a } { x - 3 }$ 有增根"}, {"id": "分式有增根，则分母为0"}, {"id": "a = - 3"}], "links": [{"rel": "同乘除", "source": "\\frac { x } { 3 - x } - x = \\frac { a } { x - 3 }", "target": "x - x ( 3 - x ) = - a"}, {"rel": "被描述", "source": "\\frac { x } { 3 - x } - x = \\frac { a } { x - 3 }", "target": "x = 3"}, {"rel": "被代入", "source": "x - x ( 3 - x ) = - a", "target": "a = - 3"}, {"rel": "代入", "source": "x = 3", "target": "a = - 3"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $\\frac { x } { 3 - x } - x = \\frac { a } { x - 3 }$ 有增根", "target": "x = 3"}, {"rel": "属性描述", "source": "分式有增根，则分母为0", "target": "x = 3"}]}}
{"content": "If the sum of two monomials $- 3 x ^ { m } y ^ { 2 }$ and $- \\frac { 1 } { 2 } xy ^ { n }$ is still a monomial, then the degree of this sum is ____?", "answer": "3", "steps": "Because the sum of the two monomials $- 3 x ^ { m } y ^ { 2 }$ and $- \\frac { 1 } { 2 } xy ^ { n }$ is still a monomial, we have $m = 1$ and $n = 2$. Therefore, the degree of this sum is $1 + 2 = 3$.", "expr_cands": ["- 3 x ^ { m } y ^ { 2 }", "y", "m", "x", "- \\frac { 1 } { 2 } xy ^ { n }", "n", "m = 1", "n = 2", "1 + 2", "3"], "exprs": ["m = 1", "n = 2", "3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 3 x ^ { m } y ^ { 2 }"}, {"id": "m = 1"}, {"id": "- \\frac { 1 } { 2 } xy ^ { n }"}, {"id": "两个单项式 $- 3 x ^ { m } y ^ { 2 }$ 与 $- \\frac { 1 } { 2 } xy ^ { n }$ 的和仍然是单项式"}, {"id": "n = 2"}, {"id": "3"}, {"id": "这个和的次数"}], "links": [{"rel": "被描述", "source": "- 3 x ^ { m } y ^ { 2 }", "target": "m = 1"}, {"rel": "被描述", "source": "- 3 x ^ { m } y ^ { 2 }", "target": "n = 2"}, {"rel": "被描述", "source": "- 3 x ^ { m } y ^ { 2 }", "target": "3"}, {"rel": "被描述", "source": "m = 1", "target": "3"}, {"rel": "被描述", "source": "- \\frac { 1 } { 2 } xy ^ { n }", "target": "m = 1"}, {"rel": "被描述", "source": "- \\frac { 1 } { 2 } xy ^ { n }", "target": "n = 2"}, {"rel": "被描述", "source": "- \\frac { 1 } { 2 } xy ^ { n }", "target": "3"}, {"rel": "限制性描述", "source": "两个单项式 $- 3 x ^ { m } y ^ { 2 }$ 与 $- \\frac { 1 } { 2 } xy ^ { n }$ 的和仍然是单项式", "target": "m = 1"}, {"rel": "限制性描述", "source": "两个单项式 $- 3 x ^ { m } y ^ { 2 }$ 与 $- \\frac { 1 } { 2 } xy ^ { n }$ 的和仍然是单项式", "target": "n = 2"}, {"rel": "限制性描述", "source": "两个单项式 $- 3 x ^ { m } y ^ { 2 }$ 与 $- \\frac { 1 } { 2 } xy ^ { n }$ 的和仍然是单项式", "target": "3"}, {"rel": "被描述", "source": "n = 2", "target": "3"}, {"rel": "限制性描述", "source": "这个和的次数", "target": "3"}]}}
{"content": "Given $\\frac { a } { 2 } = \\frac { b } { 3 } \\neq 0$, find the value of $\\frac { 5 a - 2 b } { a ^ 2 } - 4 b ^ 2 \\cdot ( a - 2 b )$.", "answer": "\\frac { 1 } { 2 }", "steps": "Let $\\frac { a } { 2 } = \\frac { b } { 3 } = k$, then $a = 2 k$, $b = 3 k$. The original expression is $\\frac { 5 a - 2 b } { a + 2 b } = \\frac { 5 \\cdot 2 k - 2 \\cdot 3 k } { 2 k + 2 \\cdot 3 k } = \\frac { 4 k } { 8 k } = \\frac { 1 } { 2 }$.", "expr_cands": ["\\frac { a } { 2 } = \\frac { b } { 3 } \\neq 0", "\\frac { 5 a - 2 b } { a } ^ { 2 } - 4 { b } ^ { 2 } \\cdot ( a - 2 b )", "a", "b", "\\frac { a } { 2 } = k", "k", "a = 2 k", "b = 3 k", "\\frac { 5 a - 2 b } { a + 2 b }", "\\frac { 1 } { 2 }"], "exprs": ["a = 2 k", "b = 3 k", "\\frac { 1 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "设 $\\frac { a } { 2 } = \\frac { b } { 3 } = k$"}, {"id": "a = 2 k"}, {"id": "b = 3 k"}, {"id": "\\frac { 5 a - 2 b } { a + 2 b }"}, {"id": "\\frac { 1 } { 2 }"}], "links": [{"rel": "假设描述", "source": "设 $\\frac { a } { 2 } = \\frac { b } { 3 } = k$", "target": "a = 2 k"}, {"rel": "假设描述", "source": "设 $\\frac { a } { 2 } = \\frac { b } { 3 } = k$", "target": "b = 3 k"}, {"rel": "代入", "source": "a = 2 k", "target": "\\frac { 1 } { 2 }"}, {"rel": "代入", "source": "b = 3 k", "target": "\\frac { 1 } { 2 }"}, {"rel": "被代入", "source": "\\frac { 5 a - 2 b } { a + 2 b }", "target": "\\frac { 1 } { 2 }"}]}}
{"content": "Given that $x = 4$ is a solution to the equation $mx - 8 = 20$, what is the value of $m$?", "answer": "7", "steps": "Substituting $x = 4$ into the equation $mx - 8 = 20$, we get $4 m - 8 = 20$. Solving for $m$, we get $m = 7$.", "expr_cands": ["x = 4", "x", "mx - 8 = 20", "m", "4 m - 8 = 20", "m = 7"], "exprs": ["4 m - 8 = 20", "m = 7"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "mx - 8 = 20"}, {"id": "4 m - 8 = 20"}, {"id": "x = 4"}, {"id": "m = 7"}], "links": [{"rel": "被代入", "source": "mx - 8 = 20", "target": "4 m - 8 = 20"}, {"rel": "等式方程求解", "source": "4 m - 8 = 20", "target": "m = 7"}, {"rel": "代入", "source": "x = 4", "target": "4 m - 8 = 20"}]}}
{"content": "If the expression $\\sqrt { a - 2 }$ is a quadratic radical, then the range of values for $a$ is ____?", "answer": "a \\ge 2", "steps": "According to the problem, we have $a - 2 \\ge 0$. Solving for $a$, we get $a \\ge 2$.", "expr_cands": ["\\sqrt { a - 2 }", "a", "a - 2 \\ge 0", "2 \\le a", "a \\ge 2"], "exprs": ["a - 2 \\ge 0", "a \\ge 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { a - 2 }"}, {"id": "a - 2 \\ge 0"}, {"id": "式子 $\\sqrt { a - 2 }$ 是二次根式"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "a \\ge 2"}], "links": [{"rel": "被描述", "source": "\\sqrt { a - 2 }", "target": "a - 2 \\ge 0"}, {"rel": "不等式方程求解", "source": "a - 2 \\ge 0", "target": "a \\ge 2"}, {"rel": "限制性描述", "source": "式子 $\\sqrt { a - 2 }$ 是二次根式", "target": "a - 2 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "a - 2 \\ge 0"}]}}
{"content": "If $\\frac { a } { b } = \\frac { 2 } { 3 }$, then the value of $\\frac { b } { a + b }$ is ____?", "answer": "\\frac { 3 } { 5 }", "steps": "Because $\\frac { a } { b } = \\frac { 2 } { 3 }$, therefore $a = \\frac { 2 } { 3 } b$, therefore $\\frac { b } { a + b } = \\frac { b } { \\frac { 2 } { 3 } b + b } = \\frac { 3 } { 5 }$.", "expr_cands": ["\\frac { a } { b } = \\frac { 2 } { 3 }", "b", "a", "\\frac { b } { a + b }", "a = \\frac { 2 } { 3 } b", "\\frac { 3 } { 5 }"], "exprs": ["a = \\frac { 2 } { 3 } b", "\\frac { 3 } { 5 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { a } { b } = \\frac { 2 } { 3 }"}, {"id": "a = \\frac { 2 } { 3 } b"}, {"id": "\\frac { b } { a + b }"}, {"id": "\\frac { 3 } { 5 }"}], "links": [{"rel": "等式方程部分求解", "source": "\\frac { a } { b } = \\frac { 2 } { 3 }", "target": "a = \\frac { 2 } { 3 } b"}, {"rel": "代入", "source": "a = \\frac { 2 } { 3 } b", "target": "\\frac { 3 } { 5 }"}, {"rel": "被代入", "source": "\\frac { b } { a + b }", "target": "\\frac { 3 } { 5 }"}]}}
{"content": "If $\\sqrt { x - 1 } = 2$, then the value of $x$ is ____?", "answer": "5", "steps": "From $\\sqrt { x - 1 } = 2$, we obtain $x - 1 = 4$, solving for $x$, we get: $x = 5$.", "expr_cands": ["\\sqrt { x - 1 } = 2", "x", "x = 5", "x - 1 = 4"], "exprs": ["x = 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x - 1 } = 2"}, {"id": "x = 5"}], "links": [{"rel": "等式方程求解", "source": "\\sqrt { x - 1 } = 2", "target": "x = 5"}]}}
{"content": "Given $a = - 2$, then $\\sqrt { a ^ 2 } + a$ = ____ ?", "answer": "0", "steps": "When $a = - 2$, the original expression becomes $| a | + a = - a + a = 0$.", "expr_cands": ["a = - 2", "a", "\\sqrt { a ^ { 2 } } + a", "| a | + a", "0"], "exprs": ["0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a = - 2"}, {"id": "0"}, {"id": "\\sqrt { a ^ { 2 } } + a"}], "links": [{"rel": "代入", "source": "a = - 2", "target": "0"}, {"rel": "被代入", "source": "\\sqrt { a ^ { 2 } } + a", "target": "0"}]}}
{"content": "When is the value of the algebraic expression $- 3 x + 4$ non-positive?", "answer": "x \\ge \\frac { 4 } { 3 }", "steps": "From the given condition, we have $- 3 x + 4 \\leq 0$, which implies $x \\geq \\frac { 4 } { 3 }$.", "expr_cands": ["x", "- 3 x + 4", "- 3 x + 4 \\le 0", "\\frac { 4 } { 3 } \\le x", "x \\ge \\frac { 4 } { 3 }"], "exprs": ["- 3 x + 4 \\le 0", "x \\ge \\frac { 4 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 3 x + 4"}, {"id": "- 3 x + 4 \\le 0"}, {"id": "代数式 $- 3 x + 4$ 的值是非正数"}, {"id": "x \\ge \\frac { 4 } { 3 }"}], "links": [{"rel": "被描述", "source": "- 3 x + 4", "target": "- 3 x + 4 \\le 0"}, {"rel": "不等式方程求解", "source": "- 3 x + 4 \\le 0", "target": "x \\ge \\frac { 4 } { 3 }"}, {"rel": "限制性描述", "source": "代数式 $- 3 x + 4$ 的值是非正数", "target": "- 3 x + 4 \\le 0"}]}}
{"content": "The square root of a number is $2 m + 1$ and $m - 4$. What is the number?", "answer": "9", "steps": "According to the problem, we have $2 m + 1 + m - 4 = 0$, which gives $m = 1$. Therefore, $2 m + 1 = 3$, and the number is $3 ^ 2 = 9$.", "expr_cands": ["2 m + 1", "m", "m - 4", "2 m + 1 + m - 4 = 0", "m = 1", "3", "3 ^ { 2 }", "9"], "exprs": ["2 m + 1 + m - 4 = 0", "m = 1", "3", "9"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 m + 1"}, {"id": "2 m + 1 + m - 4 = 0"}, {"id": "m - 4"}, {"id": "一个数的平方根是 $2 m + 1$ 和 $m - 4$"}, {"id": "平方根互为相反数"}, {"id": "m = 1"}, {"id": "3"}, {"id": "9"}, {"id": "平方"}], "links": [{"rel": "被描述", "source": "2 m + 1", "target": "2 m + 1 + m - 4 = 0"}, {"rel": "被代入", "source": "2 m + 1", "target": "3"}, {"rel": "等式方程求解", "source": "2 m + 1 + m - 4 = 0", "target": "m = 1"}, {"rel": "被描述", "source": "m - 4", "target": "2 m + 1 + m - 4 = 0"}, {"rel": "限制性描述", "source": "一个数的平方根是 $2 m + 1$ 和 $m - 4$", "target": "2 m + 1 + m - 4 = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "2 m + 1 + m - 4 = 0"}, {"rel": "代入", "source": "m = 1", "target": "3"}, {"rel": "被描述", "source": "3", "target": "9"}, {"rel": "限制性描述", "source": "平方", "target": "9"}]}}
{"content": "When $x = 3$, the value of the expression $2 x + 2$ is equal to the value of the expression $5 x + k$. What is the value of $k$?", "answer": "- 7", "steps": "According to the problem, we have $2 x + 2 = 5 x + k$. Substituting $x = 3$, we get $6 + 2 = 15 + k$. Solving for $k$, we get $k = - 7$.", "expr_cands": ["x = 3", "x", "2 x + 2", "5 x + k", "k", "2 x + 2 = 5 x + k", "6 + 2 = 15 + k", "k = - 7"], "exprs": ["2 x + 2 = 5 x + k", "k = - 7"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x + 2"}, {"id": "2 x + 2 = 5 x + k"}, {"id": "5 x + k"}, {"id": "式子 $2 x + 2$ 与 $5 x + k$ 的值相"}, {"id": "x = 3"}, {"id": "k = - 7"}], "links": [{"rel": "被描述", "source": "2 x + 2", "target": "2 x + 2 = 5 x + k"}, {"rel": "联立", "source": "2 x + 2 = 5 x + k", "target": "k = - 7"}, {"rel": "被描述", "source": "5 x + k", "target": "2 x + 2 = 5 x + k"}, {"rel": "限制性描述", "source": "式子 $2 x + 2$ 与 $5 x + k$ 的值相", "target": "2 x + 2 = 5 x + k"}, {"rel": "联立", "source": "x = 3", "target": "k = - 7"}]}}
{"content": "Given $y = x ^ { a - 1 }$ is a proportional function, what is the value of $a$?", "answer": "2", "steps": "$\\because y = x ^ { a - 1 }$ is a proportional function, $\\therefore a - 1 = 1$, solving for $a$ gives $a = 2$.", "expr_cands": ["y = x ^ { a - 1 }", "a", "x", "y", "a - 1 = 1", "a = 2"], "exprs": ["a - 1 = 1", "a = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = x ^ { a - 1 }"}, {"id": "a - 1 = 1"}, {"id": "$y = x ^ { a - 1 }$ 是正比例函数"}, {"id": "a = 2"}], "links": [{"rel": "被描述", "source": "y = x ^ { a - 1 }", "target": "a - 1 = 1"}, {"rel": "等式方程求解", "source": "a - 1 = 1", "target": "a = 2"}, {"rel": "限制性描述", "source": "$y = x ^ { a - 1 }$ 是正比例函数", "target": "a - 1 = 1"}]}}
{"content": "If the value of the algebraic expression $1 - \\frac { x - 2 } { 3 }$ is not greater than the value of $\\frac { 1 + 3 x } { 3 }$, then the range of possible values for $x$ is ____ ?", "answer": "x \\ge 1", "steps": "According to the problem, we have $1 - \\frac { x - 2 } { 3 } \\le \\frac { 1 + 3 x } { 3 }$. Simplifying by eliminating the denominators, we get $3 - ( x - 2 ) \\le 1 + 3 x$. Expanding the brackets, we get $3 - x + 2 \\le 1 + 3 x$. Rearranging, we get $x + 3 x \\ge 3 + 2 - 1$. Combining like terms, we get $4 x \\ge 4$. Dividing by 4, we get $x \\ge 1$.", "expr_cands": ["1 - \\frac { x - 2 } { 3 }", "x", "\\frac { 1 + 3 x } { 3 }", "1 - \\frac { x - 2 } { 3 } \\le \\frac { 1 + 3 x } { 3 }", "1 \\le x", "3 - ( x - 2 ) \\le 1 + 3 x", "3 - x + 2 \\le 1 + 3 x", "x + 3 x \\ge 3 + 2 - 1", "4 x \\ge 4", "1", "x \\ge 1"], "exprs": ["1 - \\frac { x - 2 } { 3 } \\le \\frac { 1 + 3 x } { 3 }", "x \\ge 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "1 - \\frac { x - 2 } { 3 }"}, {"id": "1 - \\frac { x - 2 } { 3 } \\le \\frac { 1 + 3 x } { 3 }"}, {"id": "\\frac { 1 + 3 x } { 3 }"}, {"id": "代数式 $1 - \\frac { x - 2 } { 3 }$ 的值不大于 $\\frac { 1 + 3 x } { 3 }$ 的值"}, {"id": "x \\ge 1"}], "links": [{"rel": "被描述", "source": "1 - \\frac { x - 2 } { 3 }", "target": "1 - \\frac { x - 2 } { 3 } \\le \\frac { 1 + 3 x } { 3 }"}, {"rel": "不等式方程求解", "source": "1 - \\frac { x - 2 } { 3 } \\le \\frac { 1 + 3 x } { 3 }", "target": "x \\ge 1"}, {"rel": "被描述", "source": "\\frac { 1 + 3 x } { 3 }", "target": "1 - \\frac { x - 2 } { 3 } \\le \\frac { 1 + 3 x } { 3 }"}, {"rel": "限制性描述", "source": "代数式 $1 - \\frac { x - 2 } { 3 }$ 的值不大于 $\\frac { 1 + 3 x } { 3 }$ 的值", "target": "1 - \\frac { x - 2 } { 3 } \\le \\frac { 1 + 3 x } { 3 }"}]}}
{"content": "If $y = \\frac { 1 } { x } ^ { 2 n - 5 }$ is an inverse proportion function, then $n$ = ____ ?", "answer": "3", "steps": "Since $y = \\frac { 1 } { x } ^ { 2 n - 5 }$ is an inverse proportion function, therefore $2 n - 5 = 1$, solving for $n$ gives $n = 3$.", "expr_cands": ["y = \\frac { 1 } { x } ^ { 2 n - 5 }", "y", "x", "n", "2 n - 5 = 1", "n = 3"], "exprs": ["2 n - 5 = 1", "n = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = \\frac { 1 } { x } ^ { 2 n - 5 }"}, {"id": "2 n - 5 = 1"}, {"id": "$y = \\frac { 1 } { x } ^ { 2 n - 5 }$ 是反比例函数"}, {"id": "n = 3"}], "links": [{"rel": "被描述", "source": "y = \\frac { 1 } { x } ^ { 2 n - 5 }", "target": "2 n - 5 = 1"}, {"rel": "等式方程求解", "source": "2 n - 5 = 1", "target": "n = 3"}, {"rel": "限制性描述", "source": "$y = \\frac { 1 } { x } ^ { 2 n - 5 }$ 是反比例函数", "target": "2 n - 5 = 1"}]}}
{"content": "If the solution of the fractional equation $\\frac { ax } { a - 1 } + \\frac { x } { 1 + x } = 2$ is the same as the solution of the fractional equation $\\frac { 7 } { 5 + x } = 1$, then the value of $a$ is ____?", "answer": "- 2", "steps": "Solve the equation $\\frac { 7 } { 5 + x } = 1$ to get $x = 2$. Substitute $x = 2$ into the fractional equation $\\frac { ax } { a - 1 } + \\frac { x } { 1 + x } = 2$ to get $\\frac { 2 a } { a - 1 } + \\frac { 2 } { 1 + 2 } = 2$. Solve for $a$ to get $a = - 2$.", "expr_cands": ["\\frac { ax } { a - 1 } + \\frac { x } { 1 + x } = 2", "x", "a", "\\frac { 7 } { 5 + x } = 1", "x = 2", "\\frac { 2 a } { a - 1 } + \\frac { 2 } { 3 } = 2", "\\frac { 2 a } { a - 1 } + \\frac { 2 } { 1 + 2 } = 2", "a = - 2"], "exprs": ["x = 2", "\\frac { 2 a } { a - 1 } + \\frac { 2 } { 1 + 2 } = 2", "a = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 7 } { 5 + x } = 1"}, {"id": "x = 2"}, {"id": "\\frac { ax } { a - 1 } + \\frac { x } { 1 + x } = 2"}, {"id": "\\frac { 2 a } { a - 1 } + \\frac { 2 } { 1 + 2 } = 2"}, {"id": "分式方程 $\\frac { ax } { a - 1 } + \\frac { x } { 1 + x } = 2$ 的解与分式方程 $\\frac { 7 } { 5 + x } = 1$ 的解相同"}, {"id": "a = - 2"}], "links": [{"rel": "等式方程求解", "source": "\\frac { 7 } { 5 + x } = 1", "target": "x = 2"}, {"rel": "被描述", "source": "x = 2", "target": "\\frac { 2 a } { a - 1 } + \\frac { 2 } { 1 + 2 } = 2"}, {"rel": "被描述", "source": "\\frac { ax } { a - 1 } + \\frac { x } { 1 + x } = 2", "target": "\\frac { 2 a } { a - 1 } + \\frac { 2 } { 1 + 2 } = 2"}, {"rel": "等式方程求解", "source": "\\frac { 2 a } { a - 1 } + \\frac { 2 } { 1 + 2 } = 2", "target": "a = - 2"}, {"rel": "限制性描述", "source": "分式方程 $\\frac { ax } { a - 1 } + \\frac { x } { 1 + x } = 2$ 的解与分式方程 $\\frac { 7 } { 5 + x } = 1$ 的解相同", "target": "\\frac { 2 a } { a - 1 } + \\frac { 2 } { 1 + 2 } = 2"}]}}
{"content": "Given that the equation $3 a + x = \\frac { ax } { 3 } + 3$ has a solution of $x = 6$, the value of $a ^ 2 - 2 a + 1$ is ____?", "answer": "16", "steps": "Substituting $x = 6$ into the equation $3 a + x = \\frac { ax } { 3 } + 3$, we get $3 a + 6 = 2 a + 3$. Solving for $a$, we get $a = - 3$. Therefore, $a ^ 2 - 2 a + 1 = ( - 3 ) ^ 2 - 2 * ( - 3 ) + 1 = 16$.", "expr_cands": ["x", "3 a + x = \\frac { { ax } } { 3 } + 3", "a", "x = 6", "a ^ { 2 } - 2 a + 1", "3 a + x = \\frac { ax } { 3 } + 3", "3 a + 6 = 2 a + 3", "a = - 3", "16"], "exprs": ["3 a + 6 = 2 a + 3", "a = - 3", "16"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 a + x = \\frac { ax } { 3 } + 3"}, {"id": "3 a + 6 = 2 a + 3"}, {"id": "x = 6"}, {"id": "a = - 3"}, {"id": "a ^ { 2 } - 2 a + 1"}, {"id": "16"}], "links": [{"rel": "被代入", "source": "3 a + x = \\frac { ax } { 3 } + 3", "target": "3 a + 6 = 2 a + 3"}, {"rel": "等式方程求解", "source": "3 a + 6 = 2 a + 3", "target": "a = - 3"}, {"rel": "代入", "source": "x = 6", "target": "3 a + 6 = 2 a + 3"}, {"rel": "代入", "source": "a = - 3", "target": "16"}, {"rel": "被代入", "source": "a ^ { 2 } - 2 a + 1", "target": "16"}]}}
{"content": "Four distinct integers $a$, $b$, $c$, $d$ have a product of $a * b * c * d = 169$. What is the value of $a + b + c + d$?", "answer": "0", "steps": "Since $a * b * c * d = 169 = 13 * 13$, it follows that $a = 1$, $b = - 1$, $c = 13$, and $d = - 13$. Therefore, $a + b + c + d = 0$.", "expr_cands": ["a", "b", "c", "d", "a * b * c * d = 169", "a + b + c + d", "a * b * c * d = 13 * 13", "a = 1", "b = - 1", "c = 13", "d = - 13", "0"], "exprs": ["a = 1", "b = - 1", "c = 13", "d = - 13", "0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a * b * c * d = 169"}, {"id": "a = 1"}, {"id": "四个不相等的整数 $a$ , $b$ , $c$ , $d$"}, {"id": "它们的积 $a * b * c * d = 169$"}, {"id": "b = - 1"}, {"id": "c = 13"}, {"id": "d = - 13"}, {"id": "a + b + c + d"}, {"id": "0"}], "links": [{"rel": "被描述", "source": "a * b * c * d = 169", "target": "a = 1"}, {"rel": "被描述", "source": "a * b * c * d = 169", "target": "b = - 1"}, {"rel": "被描述", "source": "a * b * c * d = 169", "target": "c = 13"}, {"rel": "被描述", "source": "a * b * c * d = 169", "target": "d = - 13"}, {"rel": "代入", "source": "a = 1", "target": "0"}, {"rel": "限制性描述", "source": "四个不相等的整数 $a$ , $b$ , $c$ , $d$", "target": "a = 1"}, {"rel": "限制性描述", "source": "四个不相等的整数 $a$ , $b$ , $c$ , $d$", "target": "b = - 1"}, {"rel": "限制性描述", "source": "四个不相等的整数 $a$ , $b$ , $c$ , $d$", "target": "c = 13"}, {"rel": "限制性描述", "source": "四个不相等的整数 $a$ , $b$ , $c$ , $d$", "target": "d = - 13"}, {"rel": "限制性描述", "source": "它们的积 $a * b * c * d = 169$", "target": "a = 1"}, {"rel": "限制性描述", "source": "它们的积 $a * b * c * d = 169$", "target": "b = - 1"}, {"rel": "限制性描述", "source": "它们的积 $a * b * c * d = 169$", "target": "c = 13"}, {"rel": "限制性描述", "source": "它们的积 $a * b * c * d = 169$", "target": "d = - 13"}, {"rel": "代入", "source": "b = - 1", "target": "0"}, {"rel": "代入", "source": "c = 13", "target": "0"}, {"rel": "代入", "source": "d = - 13", "target": "0"}, {"rel": "被代入", "source": "a + b + c + d", "target": "0"}]}}
{"content": "If $( x + 8 ) ( x - 1 ) = x ^ 2 + mx + n$ holds for any $x$, then $m + n$ = ____?", "answer": "- 1", "steps": "Because $( x + 8 ) ( x - 1 ) = x ^ 2 - x + 8 x - 8 = x ^ 2 + 7 x - 8 = x ^ 2 + mx + n$, therefore $m = 7$, $n = - 8$, and $m + n = 7 - 8 = - 1$.", "expr_cands": ["( x + 8 ) ( x - 1 ) = x ^ { 2 } + mx + n", "m", "n", "x", "m + n", "m = 7", "n = - 8", "- 1"], "exprs": ["m = 7", "n = - 8", "- 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( x + 8 ) ( x - 1 ) = x ^ { 2 } + mx + n"}, {"id": "m = 7"}, {"id": "n = - 8"}, {"id": "m + n"}, {"id": "- 1"}], "links": [{"rel": "移项", "source": "( x + 8 ) ( x - 1 ) = x ^ { 2 } + mx + n", "target": "m = 7"}, {"rel": "移项", "source": "( x + 8 ) ( x - 1 ) = x ^ { 2 } + mx + n", "target": "n = - 8"}, {"rel": "代入", "source": "m = 7", "target": "- 1"}, {"rel": "代入", "source": "n = - 8", "target": "- 1"}, {"rel": "被代入", "source": "m + n", "target": "- 1"}]}}
{"content": "$- 2 { a } ^ { 2 } { y } ^ { n - 1 }$ and $\\frac { 2 } { 3 } { a } ^ { 2 } { y } ^ { 3 }$ are like terms, then $n$ = ____ ?", "answer": "4", "steps": "According to the problem, we have $n - 1 = 3$, which implies $n = 4$.", "expr_cands": ["- 2 { a } ^ { 2 } { y } ^ { n - 1 }", "a", "n", "y", "\\frac { 2 } { 3 } { a } ^ { 2 } { y } ^ { 3 }", "n - 1 = 3", "n = 4"], "exprs": ["n - 1 = 3", "n = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 2 { a } ^ { 2 } { y } ^ { n - 1 }"}, {"id": "n - 1 = 3"}, {"id": "\\frac { 2 } { 3 } { a } ^ { 2 } { y } ^ { 3 }"}, {"id": "$- 2 { a } ^ { 2 } { y } ^ { n - 1 }$ 与 $\\frac { 2 } { 3 } { a } ^ { 2 } { y } ^ { 3 }$ 是同类项"}, {"id": "n = 4"}], "links": [{"rel": "被描述", "source": "- 2 { a } ^ { 2 } { y } ^ { n - 1 }", "target": "n - 1 = 3"}, {"rel": "等式方程求解", "source": "n - 1 = 3", "target": "n = 4"}, {"rel": "被描述", "source": "\\frac { 2 } { 3 } { a } ^ { 2 } { y } ^ { 3 }", "target": "n - 1 = 3"}, {"rel": "限制性描述", "source": "$- 2 { a } ^ { 2 } { y } ^ { n - 1 }$ 与 $\\frac { 2 } { 3 } { a } ^ { 2 } { y } ^ { 3 }$ 是同类项", "target": "n - 1 = 3"}]}}
{"content": "If the calculation result of $( x + m ) ( x - 1 )$ does not contain a linear term of $x$, then the value of $m$ is ____?", "answer": "1", "steps": "$( x + m ) ( x - 1 ) = x ^ { 2 } + ( m - 1 ) x - m$, from the absence of the linear term in x in the result, we get m - 1 = 0, which gives m = 1.", "expr_cands": ["( x + m ) ( x - 1 )", "x", "m", "x ^ { 2 } + ( m - 1 ) x - m", "m - 1 = 0", "m = 1"], "exprs": ["x ^ { 2 } + ( m - 1 ) x - m", "m - 1 = 0", "m = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( x + m ) ( x - 1 )"}, {"id": "x ^ { 2 } + ( m - 1 ) x - m"}, {"id": "x"}, {"id": "m - 1 = 0"}, {"id": "$( x + m ) ( x - 1 )$ 的计算结果中不含 $x$ 的一次项"}, {"id": "m = 1"}], "links": [{"rel": "提取因式", "source": "( x + m ) ( x - 1 )", "target": "x ^ { 2 } + ( m - 1 ) x - m"}, {"rel": "被描述", "source": "x ^ { 2 } + ( m - 1 ) x - m", "target": "m - 1 = 0"}, {"rel": "提取因式参考", "source": "x", "target": "x ^ { 2 } + ( m - 1 ) x - m"}, {"rel": "等式方程求解", "source": "m - 1 = 0", "target": "m = 1"}, {"rel": "限制性描述", "source": "$( x + m ) ( x - 1 )$ 的计算结果中不含 $x$ 的一次项", "target": "m - 1 = 0"}]}}
{"content": "Given the inequality $- \\frac { 1 } { 2 } x \\ge - 1$, what is the largest integer solution of $x$?", "answer": "2", "steps": "Since $- \\frac { 1 } { 2 } x \\ge - 1$, it follows that $x \\le 2$. The largest integer solution to this inequality is $2$.", "expr_cands": ["- \\frac { 1 } { 2 } x \\ge - 1", "x", "x \\le 2", "2"], "exprs": ["x \\le 2", "2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- \\frac { 1 } { 2 } x \\ge - 1"}, {"id": "x \\le 2"}, {"id": "2"}, {"id": "不等式 $- \\frac { 1 } { 2 } x \\ge - 1$"}, {"id": "它的最大整数解"}], "links": [{"rel": "不等式方程求解", "source": "- \\frac { 1 } { 2 } x \\ge - 1", "target": "x \\le 2"}, {"rel": "被描述", "source": "x \\le 2", "target": "2"}, {"rel": "限制性描述", "source": "不等式 $- \\frac { 1 } { 2 } x \\ge - 1$", "target": "2"}, {"rel": "限制性描述", "source": "它的最大整数解", "target": "2"}]}}
{"content": "Given $a$, $b$ are opposite numbers, $m$, $n$ are reciprocal numbers, and the absolute value of $x$ is equal to $1$, then the value of $( a + b ) \\times mn + x ^ { 2 }$ is ____?", "answer": "1", "steps": "$\\because$ $a$ and $b$ are opposite numbers, $m$ and $n$ are reciprocal, and the absolute value of $x$ is equal to $1$, $\\therefore$ $a + b = 0$, $mn = 1$, $x ^ 2 = 1$, $\\therefore$ the original expression $= 0 \\times 1 + 1 = 1$.", "expr_cands": ["a", "b", "m", "n", "x", "1", "( a + b ) \\times mn + x ^ { 2 }", "a + b = 0", "mn = 1", "x ^ { 2 } = 1", "x = - 1", "x = 1", "0 * 1 + 1"], "exprs": ["a + b = 0", "mn = 1", "x ^ { 2 } = 1", "0 * 1 + 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a + b = 0"}, {"id": "b"}, {"id": "$a$ , $b$ 互为相反数"}, {"id": "m"}, {"id": "mn = 1"}, {"id": "n"}, {"id": "$m$ , $n$ 互为倒数"}, {"id": "x"}, {"id": "x ^ { 2 } = 1"}, {"id": "1"}, {"id": "$x$ 的绝对值等于 $1$"}, {"id": "绝对值恒大于等于0"}, {"id": "0 * 1 + 1"}, {"id": "( a + b ) \\times mn + x ^ { 2 }"}], "links": [{"rel": "被描述", "source": "a", "target": "a + b = 0"}, {"rel": "代入", "source": "a + b = 0", "target": "0 * 1 + 1"}, {"rel": "被描述", "source": "b", "target": "a + b = 0"}, {"rel": "限制性描述", "source": "$a$ , $b$ 互为相反数", "target": "a + b = 0"}, {"rel": "被描述", "source": "m", "target": "mn = 1"}, {"rel": "代入", "source": "mn = 1", "target": "0 * 1 + 1"}, {"rel": "被描述", "source": "n", "target": "mn = 1"}, {"rel": "限制性描述", "source": "$m$ , $n$ 互为倒数", "target": "mn = 1"}, {"rel": "被描述", "source": "x", "target": "x ^ { 2 } = 1"}, {"rel": "代入", "source": "x ^ { 2 } = 1", "target": "0 * 1 + 1"}, {"rel": "被描述", "source": "1", "target": "x ^ { 2 } = 1"}, {"rel": "限制性描述", "source": "$x$ 的绝对值等于 $1$", "target": "x ^ { 2 } = 1"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "x ^ { 2 } = 1"}, {"rel": "被代入", "source": "( a + b ) \\times mn + x ^ { 2 }", "target": "0 * 1 + 1"}]}}
{"content": "Given that the value of $\\frac {( x - 2 ) ( x + 1 )} { \\sqrt { 1 - x }}$ is $0$, what is the value of $x$?", "answer": "- 1", "steps": "From the given information, we have $( x - 2 ) ( x + 1 ) = 0$ and $1 - x > 0$. Solving for $x$, we get $x = - 1$.", "expr_cands": ["\\frac { ( x - 2 ) ( x + 1 ) } { \\sqrt { 1 - x } }", "x", "0", "( x - 2 ) ( x + 1 ) = 0", "x = - 1", "x = 2", "1 - x > 0", "x < 1"], "exprs": ["( x - 2 ) ( x + 1 ) = 0", "1 - x > 0", "x = - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { ( x - 2 ) ( x + 1 ) } { \\sqrt { 1 - x } }"}, {"id": "( x - 2 ) ( x + 1 ) = 0"}, {"id": "0"}, {"id": "$\\frac { ( x - 2 ) ( x + 1 ) } { \\sqrt { 1 - x } }$ 的值为 $0$"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "1 - x > 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x = - 1"}], "links": [{"rel": "被描述", "source": "\\frac { ( x - 2 ) ( x + 1 ) } { \\sqrt { 1 - x } }", "target": "( x - 2 ) ( x + 1 ) = 0"}, {"rel": "被描述", "source": "\\frac { ( x - 2 ) ( x + 1 ) } { \\sqrt { 1 - x } }", "target": "1 - x > 0"}, {"rel": "联立", "source": "( x - 2 ) ( x + 1 ) = 0", "target": "x = - 1"}, {"rel": "被描述", "source": "0", "target": "( x - 2 ) ( x + 1 ) = 0"}, {"rel": "被描述", "source": "0", "target": "1 - x > 0"}, {"rel": "限制性描述", "source": "$\\frac { ( x - 2 ) ( x + 1 ) } { \\sqrt { 1 - x } }$ 的值为 $0$", "target": "( x - 2 ) ( x + 1 ) = 0"}, {"rel": "限制性描述", "source": "$\\frac { ( x - 2 ) ( x + 1 ) } { \\sqrt { 1 - x } }$ 的值为 $0$", "target": "1 - x > 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "( x - 2 ) ( x + 1 ) = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "1 - x > 0"}, {"rel": "联立", "source": "1 - x > 0", "target": "x = - 1"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "1 - x > 0"}]}}
{"content": "Given that the equation $x ^ 2 - 5 x + m = 0$ has two roots $\\alpha$ and $\\beta$, and $3 \\alpha + 4 \\beta = 10$, the value of $\\beta$ is ____?", "answer": "- 5", "steps": "According to the relationship between the roots and coefficients, we can obtain $\\alpha + \\beta = 5$. Since $3 \\alpha + 4 \\beta = 10$, we have $3 ( \\alpha + \\beta ) + \\beta = 10$, which means $3 * 5 + \\beta = 10$. Therefore, $\\beta = - 5$.", "expr_cands": ["{ x } ^ { 2 } - 5 x + m = 0", "m", "x", "3 \\alpha + 4 \\beta = 10", "\\alpha", "\\beta", "\\alpha + \\beta = 5", "3 ( \\alpha + \\beta ) + \\beta = 10", "\\beta + 15 = 10", "3 * 5 + \\beta = 10", "beta = - 5", "\\beta = - 5"], "exprs": ["\\alpha + \\beta = 5", "3 ( \\alpha + \\beta ) + \\beta = 10", "3 * 5 + \\beta = 10", "\\beta = - 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "{ x } ^ { 2 } - 5 x + m = 0"}, {"id": "\\alpha + \\beta = 5"}, {"id": "\\alpha"}, {"id": "\\beta"}, {"id": "方程 ${ x } ^ { 2 } - 5 x + m = 0$ 的两根为 \\alpha , \\beta"}, {"id": "一元二次方程根与系数关系，两根之和"}, {"id": "3 \\alpha + 4 \\beta = 10"}, {"id": "3 ( \\alpha + \\beta ) + \\beta = 10"}, {"id": "3 * 5 + \\beta = 10"}, {"id": "\\beta = - 5"}], "links": [{"rel": "被描述", "source": "{ x } ^ { 2 } - 5 x + m = 0", "target": "\\alpha + \\beta = 5"}, {"rel": "提取因式参考", "source": "\\alpha + \\beta = 5", "target": "3 ( \\alpha + \\beta ) + \\beta = 10"}, {"rel": "代入", "source": "\\alpha + \\beta = 5", "target": "3 * 5 + \\beta = 10"}, {"rel": "被描述", "source": "\\alpha", "target": "\\alpha + \\beta = 5"}, {"rel": "被描述", "source": "\\beta", "target": "\\alpha + \\beta = 5"}, {"rel": "限制性描述", "source": "方程 ${ x } ^ { 2 } - 5 x + m = 0$ 的两根为 \\alpha , \\beta", "target": "\\alpha + \\beta = 5"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "\\alpha + \\beta = 5"}, {"rel": "提取因式", "source": "3 \\alpha + 4 \\beta = 10", "target": "3 ( \\alpha + \\beta ) + \\beta = 10"}, {"rel": "被代入", "source": "3 ( \\alpha + \\beta ) + \\beta = 10", "target": "3 * 5 + \\beta = 10"}, {"rel": "等式方程求解", "source": "3 * 5 + \\beta = 10", "target": "\\beta = - 5"}]}}
{"content": "If $x = \\frac { 1 } { 2 }$ is a root of the quadratic equation $x ^ 2 + bx + 2 = 0$, then the value of $b$ is ____?", "answer": "- \\frac { 9 } { 2 }", "steps": "Substituting $x = \\frac { 1 } { 2 }$ into $x ^ 2 + bx + 2 = 0$, we get $\\frac { 1 } { 4 } + \\frac { 1 } { 2 } b + 2 = 0$. Solving for $b$, we get $b = - \\frac { 9 } { 2 }$.", "expr_cands": ["x = \\frac { 1 } { 2 }", "x", "x ^ { 2 } + bx + 2 = 0", "b", "\\frac { b } { 2 } + \\frac { 9 } { 4 } = 0", "\\frac { 1 } { 4 } + \\frac { 1 } { 2 } b + 2 = 0", "b = - \\frac { 9 } { 2 }"], "exprs": ["\\frac { 1 } { 4 } + \\frac { 1 } { 2 } b + 2 = 0", "b = - \\frac { 9 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } + bx + 2 = 0"}, {"id": "\\frac { 1 } { 4 } + \\frac { 1 } { 2 } b + 2 = 0"}, {"id": "x = \\frac { 1 } { 2 }"}, {"id": "b = - \\frac { 9 } { 2 }"}], "links": [{"rel": "被代入", "source": "x ^ { 2 } + bx + 2 = 0", "target": "\\frac { 1 } { 4 } + \\frac { 1 } { 2 } b + 2 = 0"}, {"rel": "等式方程求解", "source": "\\frac { 1 } { 4 } + \\frac { 1 } { 2 } b + 2 = 0", "target": "b = - \\frac { 9 } { 2 }"}, {"rel": "代入", "source": "x = \\frac { 1 } { 2 }", "target": "\\frac { 1 } { 4 } + \\frac { 1 } { 2 } b + 2 = 0"}]}}
{"content": "If $x = - 2$ is a root of the quadratic equation $x ^ 2 - 4 mx - 8 = 0$ in terms of $x$, then the other root is ____?", "answer": "4", "steps": "The quadratic equation $x ^ 2 - 4 mx - 8 = 0$ has another root $\\alpha$, then $- 2 \\alpha = - 8$, solving for $\\alpha$ gives $\\alpha = 4$.", "expr_cands": ["x = - 2", "x", "x ^ { 2 } - 4 mx - 8 = 0", "m", "- 2 \\alpha = - 8", "alpha = 4", "\\alpha", "\\alpha = 4"], "exprs": ["- 2 \\alpha = - 8", "\\alpha = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x = - 2"}, {"id": "- 2 \\alpha = - 8"}, {"id": "x ^ { 2 } - 4 mx - 8 = 0"}, {"id": "$x = - 2$ 是关 $x$ 的一元二次方程 $x ^ { 2 } - 4 mx - 8 = 0$ 的一个根"}, {"id": "另一个根"}, {"id": "设一元二次方程 $x ^ { 2 } - 4 mx - 8 = 0$ 的另一根为 \\alpha"}, {"id": "一元二次方程根与系数关系，两根之积"}, {"id": "\\alpha = 4"}], "links": [{"rel": "被描述", "source": "x = - 2", "target": "- 2 \\alpha = - 8"}, {"rel": "等式方程求解", "source": "- 2 \\alpha = - 8", "target": "\\alpha = 4"}, {"rel": "被描述", "source": "x ^ { 2 } - 4 mx - 8 = 0", "target": "- 2 \\alpha = - 8"}, {"rel": "限制性描述", "source": "$x = - 2$ 是关 $x$ 的一元二次方程 $x ^ { 2 } - 4 mx - 8 = 0$ 的一个根", "target": "- 2 \\alpha = - 8"}, {"rel": "限制性描述", "source": "另一个根", "target": "- 2 \\alpha = - 8"}, {"rel": "假设描述", "source": "设一元二次方程 $x ^ { 2 } - 4 mx - 8 = 0$ 的另一根为 \\alpha", "target": "- 2 \\alpha = - 8"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之积", "target": "- 2 \\alpha = - 8"}]}}
{"content": "The positive integer solution of the inequality $X + 3 < 5$ is ____ ?", "answer": "1", "steps": "$\\because$ The solution set of the inequality $x + 3 < 5$ is $x < 2$, $\\therefore$ the positive integer solution of the inequality is $1$.", "expr_cands": ["X + 3 < 5", "X", "x + 3 < 5", "x < 2", "x", "1"], "exprs": ["x < 2", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x + 3 < 5"}, {"id": "x < 2"}, {"id": "1"}, {"id": "不等式的正整数解是 $1$"}, {"id": "不等式 $x + 3 < 5$ 的解集是 $x < 2$"}], "links": [{"rel": "不等式方程求解", "source": "x + 3 < 5", "target": "x < 2"}, {"rel": "被描述", "source": "x < 2", "target": "1"}, {"rel": "限制性描述", "source": "不等式的正整数解是 $1$", "target": "1"}, {"rel": "限制性描述", "source": "不等式 $x + 3 < 5$ 的解集是 $x < 2$", "target": "1"}]}}
{"content": "If the cube root of $5 x - 2$ is $- 3$, then the value of $x$ is ____?", "answer": "- 5", "steps": "$\\because$ The cube root of $5 x - 2$ is $- 3$, $\\therefore$ $5 x - 2 = - 27$, solving for $x$ gives $x = - 5$.", "expr_cands": ["5 x - 2", "x", "- 3", "5 x - 2 = - 27", "x = - 5"], "exprs": ["5 x - 2 = - 27", "x = - 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5 x - 2"}, {"id": "5 x - 2 = - 27"}, {"id": "- 3"}, {"id": "$5 x - 2$ 的立方根是 $- 3$"}, {"id": "x = - 5"}], "links": [{"rel": "被描述", "source": "5 x - 2", "target": "5 x - 2 = - 27"}, {"rel": "等式方程求解", "source": "5 x - 2 = - 27", "target": "x = - 5"}, {"rel": "被描述", "source": "- 3", "target": "5 x - 2 = - 27"}, {"rel": "限制性描述", "source": "$5 x - 2$ 的立方根是 $- 3$", "target": "5 x - 2 = - 27"}]}}
{"content": "Given that $2 a + 3$ is the opposite of $5$, what is the value of $a$?", "answer": "- 4", "steps": "According to the problem, we have $2 a + 3 + 5 = 0$. By rearranging and combining terms, we get $2 a = - 8$. Solving for $a$, we get $a = - 4$.", "expr_cands": ["2 a + 3", "a", "5", "2 a + 3 + 5 = 0", "a = - 4", "2 a = - 8"], "exprs": ["2 a + 3 + 5 = 0", "a = - 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 a + 3"}, {"id": "2 a + 3 + 5 = 0"}, {"id": "5"}, {"id": "$2 a + 3$ 与 $5$ 互为相反数"}, {"id": "a = - 4"}], "links": [{"rel": "被描述", "source": "2 a + 3", "target": "2 a + 3 + 5 = 0"}, {"rel": "等式方程求解", "source": "2 a + 3 + 5 = 0", "target": "a = - 4"}, {"rel": "被描述", "source": "5", "target": "2 a + 3 + 5 = 0"}, {"rel": "限制性描述", "source": "$2 a + 3$ 与 $5$ 互为相反数", "target": "2 a + 3 + 5 = 0"}]}}
{"content": "The equation in terms of $x$, $ax - 2 = 4$, has a solution of $x = 2$. What is the value of $a$?", "answer": "3", "steps": "Substituting $x = 2$ into the equation, we get $2 a - 2 = 4$. Solving for $a$, we get $a = 3$.", "expr_cands": ["x", "ax - 2 = 4", "a", "x = 2", "2 a - 2 = 4", "a = 3"], "exprs": ["2 a - 2 = 4", "a = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "ax - 2 = 4"}, {"id": "2 a - 2 = 4"}, {"id": "x = 2"}, {"id": "a = 3"}], "links": [{"rel": "被代入", "source": "ax - 2 = 4", "target": "2 a - 2 = 4"}, {"rel": "等式方程求解", "source": "2 a - 2 = 4", "target": "a = 3"}, {"rel": "代入", "source": "x = 2", "target": "2 a - 2 = 4"}]}}
{"content": "If the polynomial $2 xy ^ 2 + 4 kxy - 6 x ^ 2 y + xy - 1$ does not contain the term $xy$, then $k$ = ____?", "answer": "- \\frac { 1 } { 4 }", "steps": "Original expression = $2 xy ^ { 2 } + 4 kxy - 6 x ^ { 2 } y + xy - 1$. Since there is no $xy$ term in the result, we get $4 k + 1 = 0$, so $k = - \\frac { 1 } { 4 }$.", "expr_cands": ["2 xy ^ { 2 } + 4 kxy - 6 x ^ { 2 } y + xy - 1", "x", "k", "y", "xy", "4 k + 1 = 0", "k = - \\frac { 1 } { 4 }"], "exprs": ["4 k + 1 = 0", "k = - \\frac { 1 } { 4 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 xy ^ { 2 } + 4 kxy - 6 x ^ { 2 } y + xy - 1"}, {"id": "4 k + 1 = 0"}, {"id": "多项式 $2 xy ^ { 2 } + 4 kxy - 6 x ^ { 2 } y + xy - 1$ 不含 $xy$ 项"}, {"id": "k = - \\frac { 1 } { 4 }"}], "links": [{"rel": "被描述", "source": "2 xy ^ { 2 } + 4 kxy - 6 x ^ { 2 } y + xy - 1", "target": "4 k + 1 = 0"}, {"rel": "等式方程求解", "source": "4 k + 1 = 0", "target": "k = - \\frac { 1 } { 4 }"}, {"rel": "限制性描述", "source": "多项式 $2 xy ^ { 2 } + 4 kxy - 6 x ^ { 2 } y + xy - 1$ 不含 $xy$ 项", "target": "4 k + 1 = 0"}]}}
{"content": "If $x : y = 3 : 2$ and $3 x + 2 y = 13$, then $x$ = ____ ?", "answer": "3", "steps": "Since $x : y = 3 : 2$, therefore $y = \\frac { 2 } { 3 } x$. Substituting into $3 x + 2 y = 13$, we get $3 x + \\frac { 4 } { 3 } x = 13$. Solving for $x$, we get $x = 3$.", "expr_cands": ["x : y = 3 : 2", "y", "x", "3 x + 2 y = 13", "y = \\frac { 2 } { 3 } x", "\\frac { 13 x } { 3 } = 13", "3 x + \\frac { 4 } { 3 } x = 13", "x = 3"], "exprs": ["y = \\frac { 2 } { 3 } x", "3 x + \\frac { 4 } { 3 } x = 13", "x = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x : y = 3 : 2"}, {"id": "y = \\frac { 2 } { 3 } x"}, {"id": "3 x + 2 y = 13"}, {"id": "3 x + \\frac { 4 } { 3 } x = 13"}, {"id": "x = 3"}], "links": [{"rel": "等式方程部分求解", "source": "x : y = 3 : 2", "target": "y = \\frac { 2 } { 3 } x"}, {"rel": "代入", "source": "y = \\frac { 2 } { 3 } x", "target": "3 x + \\frac { 4 } { 3 } x = 13"}, {"rel": "被代入", "source": "3 x + 2 y = 13", "target": "3 x + \\frac { 4 } { 3 } x = 13"}, {"rel": "等式方程求解", "source": "3 x + \\frac { 4 } { 3 } x = 13", "target": "x = 3"}]}}
{"content": "If the equation $2 mx ^ 2 + 4 mx + 3 m - 2 = 3 x ^ 2 + x$ is a quadratic equation in $x$, then the range of values for $m$ is _____.", "answer": "m \\neq \\frac { 3 } { 2 }", "steps": "Moving terms and combining like terms, we get: $( 2 m - 3 ) x ^ 2 + ( 4 m - 1 ) x + 3 m - 2 = 0$. According to the problem, we have $2 m - 3 \\neq 0$. Solving for $m$, we get $m \\neq \\frac { 3 } { 2 }$.", "expr_cands": ["x", "2 mx ^ { 2 } + 4 mx + 3 m - 2 = 3 x ^ { 2 } + x", "m", "( 2 m - 3 ) x ^ { 2 } + ( 4 m - 1 ) x + 3 m - 2 = 0", "2 m - 3 \\neq 0", "m \\neq \\frac { 3 } { 2 }"], "exprs": ["( 2 m - 3 ) x ^ { 2 } + ( 4 m - 1 ) x + 3 m - 2 = 0", "2 m - 3 \\neq 0", "m \\neq \\frac { 3 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 mx ^ { 2 } + 4 mx + 3 m - 2 = 3 x ^ { 2 } + x"}, {"id": "( 2 m - 3 ) x ^ { 2 } + ( 4 m - 1 ) x + 3 m - 2 = 0"}, {"id": "2 m - 3 \\neq 0"}, {"id": "关于 $x$ 的方程 $2 mx ^ { 2 } + 4 mx + 3 m - 2 = 3 x ^ { 2 } + x$ 是一元二次方程"}, {"id": "m \\neq \\frac { 3 } { 2 }"}], "links": [{"rel": "移项", "source": "2 mx ^ { 2 } + 4 mx + 3 m - 2 = 3 x ^ { 2 } + x", "target": "( 2 m - 3 ) x ^ { 2 } + ( 4 m - 1 ) x + 3 m - 2 = 0"}, {"rel": "被描述", "source": "( 2 m - 3 ) x ^ { 2 } + ( 4 m - 1 ) x + 3 m - 2 = 0", "target": "2 m - 3 \\neq 0"}, {"rel": "不等式方程求解", "source": "2 m - 3 \\neq 0", "target": "m \\neq \\frac { 3 } { 2 }"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $2 mx ^ { 2 } + 4 mx + 3 m - 2 = 3 x ^ { 2 } + x$ 是一元二次方程", "target": "2 m - 3 \\neq 0"}]}}
{"content": "The fraction $\\frac { 1 } { x - 1 }$ is undefined, then the possible values of $x$ are:", "answer": "x = 1", "steps": "Since x minus 1 equals 0, therefore x equals 1.", "expr_cands": ["\\frac { 1 } { x - { 1 } }", "x", "x - 1 = 0", "x = 1"], "exprs": ["x - 1 = 0", "x = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 1 } { x - { 1 } }"}, {"id": "x - 1 = 0"}, {"id": "分式 $\\frac { 1 } { x - { 1 } }$ 没有意义"}, {"id": "分式有意义，则分母不为0"}, {"id": "x = 1"}], "links": [{"rel": "被描述", "source": "\\frac { 1 } { x - { 1 } }", "target": "x - 1 = 0"}, {"rel": "等式方程求解", "source": "x - 1 = 0", "target": "x = 1"}, {"rel": "限制性描述", "source": "分式 $\\frac { 1 } { x - { 1 } }$ 没有意义", "target": "x - 1 = 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "x - 1 = 0"}]}}
{"content": "Given the line $y = ( 3 m + 2 ) x - m + 3$ passes through the origin, what is the value of $m$?", "answer": "3", "steps": "From the given information, we have $- m + 3 = 0$ and $3 m + 2 \\neq 0$. Solving for $m$, we get $m = 3$.", "expr_cands": ["y = ( 3 m + 2 ) x - m + 3", "y", "m", "x", "- m + 3 = 0", "m = 3", "3 m + 2 \\neq 0", "m \\neq - \\frac { 2 } { 3 }"], "exprs": ["- m + 3 = 0", "3 m + 2 \\neq 0", "m = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = ( 3 m + 2 ) x - m + 3"}, {"id": "- m + 3 = 0"}, {"id": "直线 $y = ( 3 m + 2 ) x - m + 3$ 经过原点"}, {"id": "3 m + 2 \\neq 0"}, {"id": "m = 3"}], "links": [{"rel": "被描述", "source": "y = ( 3 m + 2 ) x - m + 3", "target": "- m + 3 = 0"}, {"rel": "被描述", "source": "y = ( 3 m + 2 ) x - m + 3", "target": "3 m + 2 \\neq 0"}, {"rel": "联立", "source": "- m + 3 = 0", "target": "m = 3"}, {"rel": "限制性描述", "source": "直线 $y = ( 3 m + 2 ) x - m + 3$ 经过原点", "target": "- m + 3 = 0"}, {"rel": "限制性描述", "source": "直线 $y = ( 3 m + 2 ) x - m + 3$ 经过原点", "target": "3 m + 2 \\neq 0"}, {"rel": "联立", "source": "3 m + 2 \\neq 0", "target": "m = 3"}]}}
{"content": "In the direct proportion function $y = ( k - 2 ) x$, as $x$ increases, $y$ increases. The range of values for $k$ is ____?", "answer": "k > 2", "steps": "$\\because$ In the proportional function $y = ( k - 2 ) x$, $y$ increases as $x$ increases. $\\therefore$ $k - 2 > 0$. $\\therefore$ $k > 2$.", "expr_cands": ["y = ( k - 2 ) x", "k", "x", "y", "k - 2 > 0", "2 < k", "k > 2"], "exprs": ["k - 2 > 0", "k > 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = ( k - 2 ) x"}, {"id": "k - 2 > 0"}, {"id": "在正比例函数 $y = ( k - 2 ) x$ 中"}, {"id": "$y$ 随 $x$ 的增大而增大"}, {"id": "k > 2"}], "links": [{"rel": "被描述", "source": "y = ( k - 2 ) x", "target": "k - 2 > 0"}, {"rel": "不等式方程求解", "source": "k - 2 > 0", "target": "k > 2"}, {"rel": "限制性描述", "source": "在正比例函数 $y = ( k - 2 ) x$ 中", "target": "k - 2 > 0"}, {"rel": "限制性描述", "source": "$y$ 随 $x$ 的增大而增大", "target": "k - 2 > 0"}]}}
{"content": "Regarding the linear function of $y = 2 x + 3 m - 6$ with respect to $x$, in order to make it a proportional function, $m$ should be ____?", "answer": "2", "steps": "$\\because$ The linear function about $x$ is $y = 2 x + 3 m - 6$. To make it a proportional function, $\\therefore$ $3 m - 6 = 0$. Solving for $m$, we get $m = 2$.", "expr_cands": ["x", "y = 2 x + 3 m - 6", "m", "y", "3 m - 6 = 0", "m = 2"], "exprs": ["3 m - 6 = 0", "m = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = 2 x + 3 m - 6"}, {"id": "3 m - 6 = 0"}, {"id": "使其成为正比例函数"}, {"id": "关于 $x$ 的一次函数 $y = 2 x + 3 m - 6$"}, {"id": "m = 2"}], "links": [{"rel": "被描述", "source": "y = 2 x + 3 m - 6", "target": "3 m - 6 = 0"}, {"rel": "等式方程求解", "source": "3 m - 6 = 0", "target": "m = 2"}, {"rel": "限制性描述", "source": "使其成为正比例函数", "target": "3 m - 6 = 0"}, {"rel": "限制性描述", "source": "关于 $x$ 的一次函数 $y = 2 x + 3 m - 6$", "target": "3 m - 6 = 0"}]}}
{"content": "If the solution to the equation $2 x - m = x - 2$ with respect to $x$ is $3$, then the value of $m$ is ____?", "answer": "5", "steps": "$\\because$ The solution to the equation $2 x - m = x - 2$ with respect to $x$ is $3$, $\\therefore$ $6 - m = 3 - 2$, solving for $m$ yields $m = 5$.", "expr_cands": ["x", "2 x - m = x - 2", "m", "3", "6 - m = 3 - 2", "m = 5"], "exprs": ["6 - m = 3 - 2", "m = 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3"}, {"id": "6 - m = 3 - 2"}, {"id": "x"}, {"id": "2 x - m = x - 2"}, {"id": "关于 $x$ 的方程 $2 x - m = x - 2$ 的解是 $3$"}, {"id": "m = 5"}], "links": [{"rel": "被描述", "source": "3", "target": "6 - m = 3 - 2"}, {"rel": "等式方程求解", "source": "6 - m = 3 - 2", "target": "m = 5"}, {"rel": "被描述", "source": "x", "target": "6 - m = 3 - 2"}, {"rel": "被描述", "source": "2 x - m = x - 2", "target": "6 - m = 3 - 2"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $2 x - m = x - 2$ 的解是 $3$", "target": "6 - m = 3 - 2"}]}}
{"content": "If the line $y = k ^ { 2 } x - 2$ does not intersect with the line $y = 4 x + k$, then $k$ = ____ ?", "answer": "2", "steps": "From the given condition, we can obtain that $k ^ 2 = 4$ and $k \\neq - 2$. Solving this equation, we get $k = 2$.", "expr_cands": ["y = k ^ { 2 } x - 2", "k", "y", "x", "y = 4 x + k", "k ^ { 2 } = 4", "k = - 2", "k = 2", "k \\neq - 2"], "exprs": ["k ^ { 2 } = 4", "k \\neq - 2", "k = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = k ^ { 2 } x - 2"}, {"id": "k ^ { 2 } = 4"}, {"id": "y = 4 x + k"}, {"id": "直线 $y = k ^ { 2 } x - 2$ 与直线 $y = 4 x + k$ 没有交点"}, {"id": "k = 2"}, {"id": "k \\neq - 2"}], "links": [{"rel": "被描述", "source": "y = k ^ { 2 } x - 2", "target": "k ^ { 2 } = 4"}, {"rel": "被描述", "source": "y = k ^ { 2 } x - 2", "target": "k \\neq - 2"}, {"rel": "联立", "source": "k ^ { 2 } = 4", "target": "k = 2"}, {"rel": "被描述", "source": "y = 4 x + k", "target": "k ^ { 2 } = 4"}, {"rel": "限制性描述", "source": "直线 $y = k ^ { 2 } x - 2$ 与直线 $y = 4 x + k$ 没有交点", "target": "k ^ { 2 } = 4"}, {"rel": "限制性描述", "source": "直线 $y = k ^ { 2 } x - 2$ 与直线 $y = 4 x + k$ 没有交点", "target": "k \\neq - 2"}, {"rel": "联立", "source": "k \\neq - 2", "target": "k = 2"}]}}
{"content": "Given that the equation $2 x - 3 = 3 x - 2 + k$ has a solution of $x = 2$, the value of $k$ is ____?", "answer": "- 3", "steps": "Substituting $x = 2$ into the equation gives: $4 - 3 = 6 - 2 + k$, solving for $k$ gives: $k = - 3$.", "expr_cands": ["2 x - 3 = 3 x - 2 + k", "k", "x", "x = 2", "4 - 3 = 6 - 2 + k", "k = - 3"], "exprs": ["4 - 3 = 6 - 2 + k", "k = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x - 3 = 3 x - 2 + k"}, {"id": "4 - 3 = 6 - 2 + k"}, {"id": "x = 2"}, {"id": "k = - 3"}], "links": [{"rel": "被代入", "source": "2 x - 3 = 3 x - 2 + k", "target": "4 - 3 = 6 - 2 + k"}, {"rel": "等式方程求解", "source": "4 - 3 = 6 - 2 + k", "target": "k = - 3"}, {"rel": "代入", "source": "x = 2", "target": "4 - 3 = 6 - 2 + k"}]}}
{"content": "What is the result of moving the factor outside the square root of the quadratic radical $a \\sqrt { - \\frac { 1 } { a } }$ into the square root?", "answer": "- \\sqrt { - a }", "steps": "To make $\\sqrt { - \\frac { 1 } { a } }$ meaningful, we must have $- \\frac { 1 } { a } > 0$, which means $a < 0$. Therefore, $a \\sqrt { - \\frac { 1 } { a } } = - \\sqrt { ( - a ) ^ { 2 } ( - \\frac { 1 } { a } ) } = - \\sqrt { - a }$.", "expr_cands": ["a \\sqrt { - \\frac { 1 } { a } }", "a", "\\sqrt { - \\frac { 1 } { a } }", "- \\frac { 1 } { a } > 0", "a < 0", "a \\sqrt { - \\frac { 1 } { a } } = - \\sqrt { - a }", "- \\sqrt { - a }"], "exprs": ["- \\frac { 1 } { a } > 0", "a < 0", "- \\sqrt { - a }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a \\sqrt { - \\frac { 1 } { a } }"}, {"id": "- \\frac { 1 } { a } > 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "分式有意义，则分母不为0"}, {"id": "a < 0"}, {"id": "- \\sqrt { - a }"}, {"id": "把二次根式 $a \\sqrt { - \\frac { 1 } { a } }$ 根号外的因式移入根号内"}], "links": [{"rel": "被描述", "source": "a \\sqrt { - \\frac { 1 } { a } }", "target": "- \\frac { 1 } { a } > 0"}, {"rel": "被描述", "source": "a \\sqrt { - \\frac { 1 } { a } }", "target": "- \\sqrt { - a }"}, {"rel": "不等式方程求解", "source": "- \\frac { 1 } { a } > 0", "target": "a < 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "- \\frac { 1 } { a } > 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "- \\frac { 1 } { a } > 0"}, {"rel": "被描述", "source": "a < 0", "target": "- \\sqrt { - a }"}, {"rel": "限制性描述", "source": "把二次根式 $a \\sqrt { - \\frac { 1 } { a } }$ 根号外的因式移入根号内", "target": "- \\sqrt { - a }"}]}}
{"content": "If ${ a } ^ { 2 m - 1 } \\cdot { a } ^ { m + 2 } = { a } ^ { 7 }$, then the value of $m$ is ____?", "answer": "2", "steps": "According to the problem, we have $2 m - 1 + ( m + 2 ) = 7$, which gives us $m = 2$.", "expr_cands": ["{ a } ^ { 2 m - 1 } \\cdot { a } ^ { m + 2 } = { a } ^ { 7 }", "a", "m", "2 m - 1 + ( m + 2 ) = 7", "m = 2"], "exprs": ["2 m - 1 + ( m + 2 ) = 7", "m = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "{ a } ^ { 2 m - 1 } \\cdot { a } ^ { m + 2 } = { a } ^ { 7 }"}, {"id": "2 m - 1 + ( m + 2 ) = 7"}, {"id": "m = 2"}], "links": [{"rel": "同取对数", "source": "{ a } ^ { 2 m - 1 } \\cdot { a } ^ { m + 2 } = { a } ^ { 7 }", "target": "2 m - 1 + ( m + 2 ) = 7"}, {"rel": "等式方程求解", "source": "2 m - 1 + ( m + 2 ) = 7", "target": "m = 2"}]}}
{"content": "Given $( 3 x + 2 ) - 3 = 5$, then $6 x + 4$ = ____?", "answer": "16", "steps": "According to the problem, we have $3 x + 2 = 8$, so the original expression is $2 ( 3 x + 2 ) = 16$.", "expr_cands": ["( 3 x + 2 ) - 3 = 5", "x", "6 x + 4", "3 x + 2 = 8", "x = 2", "2 ( 3 x + 2 )", "16"], "exprs": ["x = 2", "16"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( 3 x + 2 ) - 3 = 5"}, {"id": "x = 2"}, {"id": "6 x + 4"}, {"id": "16"}], "links": [{"rel": "等式方程求解", "source": "( 3 x + 2 ) - 3 = 5", "target": "x = 2"}, {"rel": "代入", "source": "x = 2", "target": "16"}, {"rel": "被代入", "source": "6 x + 4", "target": "16"}]}}
{"content": "When $x < 0$, the values of the functions $y = ( k - 1 ) x$ and $y = \\frac { 2 - k } { 3 x }$ both increase as $x$ increases. What is the range of possible values for $k$?", "answer": "k > 2", "steps": "When $x < 0$, the values of the functions $y = ( k - 1 ) x$ and $y = \\frac { 2 - k } { 3 x }$ increase as $x$ increases. Therefore, $k - 1 > 0$ and $\\frac { 2 - k } { 3 } < 0$. Solving for $k$, we get $k > 2$.", "expr_cands": ["x < 0", "x", "y = ( k - 1 ) x", "y", "k", "y = \\frac { 2 - k } { 3 x }", "x ( k - 1 ) = \\frac { 2 - k } { 3 x }", "k - 1 > 0", "1 < k", "\\frac { 2 - k } { 3 } < 0", "2 < k", "k > 2"], "exprs": ["k - 1 > 0", "\\frac { 2 - k } { 3 } < 0", "k > 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = ( k - 1 ) x"}, {"id": "k - 1 > 0"}, {"id": "当 $x < 0$ 时"}, {"id": "函数 $y = ( k - 1 ) x$ 与 $y = \\frac { 2 - k } { 3 x }$ 的 $y$ 值都随 $x$ 的增大而增大"}, {"id": "y = \\frac { 2 - k } { 3 x }"}, {"id": "\\frac { 2 - k } { 3 } < 0"}, {"id": "k > 2"}], "links": [{"rel": "被描述", "source": "y = ( k - 1 ) x", "target": "k - 1 > 0"}, {"rel": "联立", "source": "k - 1 > 0", "target": "k > 2"}, {"rel": "限制性描述", "source": "当 $x < 0$ 时", "target": "k - 1 > 0"}, {"rel": "限制性描述", "source": "当 $x < 0$ 时", "target": "\\frac { 2 - k } { 3 } < 0"}, {"rel": "限制性描述", "source": "函数 $y = ( k - 1 ) x$ 与 $y = \\frac { 2 - k } { 3 x }$ 的 $y$ 值都随 $x$ 的增大而增大", "target": "k - 1 > 0"}, {"rel": "限制性描述", "source": "函数 $y = ( k - 1 ) x$ 与 $y = \\frac { 2 - k } { 3 x }$ 的 $y$ 值都随 $x$ 的增大而增大", "target": "\\frac { 2 - k } { 3 } < 0"}, {"rel": "被描述", "source": "y = \\frac { 2 - k } { 3 x }", "target": "\\frac { 2 - k } { 3 } < 0"}, {"rel": "联立", "source": "\\frac { 2 - k } { 3 } < 0", "target": "k > 2"}]}}
{"content": "Given $a$, $b$ are opposite numbers, and $m$, $n$ are reciprocal, what is the value of $5 a + 5 b - nm$?", "answer": "- 1", "steps": "Since $a$ and $b$ are opposite numbers and $m$ and $n$ are reciprocal, therefore $a + b = 0$, $mn = 1$. Thus, $5 a + 5 b - nm = 5 ( a + b ) - mn = 5 \\times 0 - 1 = - 1$.", "expr_cands": ["a", "b", "m", "n", "5 a + 5 b - nm", "a + b = 0", "mn = 1", "5 ( a + b ) - mn", "- 1"], "exprs": ["a + b = 0", "mn = 1", "5 ( a + b ) - mn", "- 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a + b = 0"}, {"id": "b"}, {"id": "$a$ , $b$ 互为相反数"}, {"id": "m"}, {"id": "mn = 1"}, {"id": "n"}, {"id": "$m$ , $n$ 互为倒数"}, {"id": "5 a + 5 b - nm"}, {"id": "5 ( a + b ) - mn"}, {"id": "- 1"}], "links": [{"rel": "被描述", "source": "a", "target": "a + b = 0"}, {"rel": "提取因式参考", "source": "a + b = 0", "target": "5 ( a + b ) - mn"}, {"rel": "代入", "source": "a + b = 0", "target": "- 1"}, {"rel": "被描述", "source": "b", "target": "a + b = 0"}, {"rel": "限制性描述", "source": "$a$ , $b$ 互为相反数", "target": "a + b = 0"}, {"rel": "被描述", "source": "m", "target": "mn = 1"}, {"rel": "提取因式参考", "source": "mn = 1", "target": "5 ( a + b ) - mn"}, {"rel": "代入", "source": "mn = 1", "target": "- 1"}, {"rel": "被描述", "source": "n", "target": "mn = 1"}, {"rel": "限制性描述", "source": "$m$ , $n$ 互为倒数", "target": "mn = 1"}, {"rel": "提取因式", "source": "5 a + 5 b - nm", "target": "5 ( a + b ) - mn"}, {"rel": "被代入", "source": "5 ( a + b ) - mn", "target": "- 1"}]}}
{"content": "If the irrational equation about $x$, $\\sqrt { x + 2 } - 1 + k = 0$, has no real roots, then the range of values for $k$ is ____?", "answer": "k > 1", "steps": "$\\because$ For the irrational equation about $x$, $\\sqrt { x + 2 } - 1 + k = 0$, $k$ has no real roots. $\\therefore$ $1 - k < 0$. Solving, we get $k > 1$.", "expr_cands": ["x", "\\sqrt { x + 2 } - 1 + k = 0", "k", "1 - k < 0", "1 < k", "k > 1"], "exprs": ["1 - k < 0", "k > 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x"}, {"id": "1 - k < 0"}, {"id": "\\sqrt { x + 2 } - 1 + k = 0"}, {"id": "关于 $x$ 的无理方程 $\\sqrt { x + 2 } - 1 + k = 0$ 没有实数根"}, {"id": "k > 1"}], "links": [{"rel": "被描述", "source": "x", "target": "1 - k < 0"}, {"rel": "不等式方程求解", "source": "1 - k < 0", "target": "k > 1"}, {"rel": "被描述", "source": "\\sqrt { x + 2 } - 1 + k = 0", "target": "1 - k < 0"}, {"rel": "限制性描述", "source": "关于 $x$ 的无理方程 $\\sqrt { x + 2 } - 1 + k = 0$ 没有实数根", "target": "1 - k < 0"}]}}
{"content": "If the line $l$ is symmetric about the $y$-axis with the line $y = 2 x + 3$, then the expression of the line $l$ is ____?", "answer": "y = - 2 x + 3", "steps": "The coordinates of the point symmetric to the line $y = 2 x + 3$ about the $y$-axis have opposite $x$-coordinates and the same $y$-coordinate. Therefore, $y = - 2 x + 3$ is the equation of the line $l$.", "expr_cands": ["l", "y = 2 x + 3", "x", "y", "y = 2 ( - x ) + 3", "2 x + 3 = 2 ( - x ) + 3", "- 2 x + 3"], "exprs": ["y = 2 ( - x ) + 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "l"}, {"id": "y = 2 ( - x ) + 3"}, {"id": "y = 2 x + 3"}, {"id": "y"}, {"id": "直线 $l$ 与直线 $y = 2 x + 3$ 关于 $y$ 轴对称"}], "links": [{"rel": "被描述", "source": "l", "target": "y = 2 ( - x ) + 3"}, {"rel": "被描述", "source": "y = 2 x + 3", "target": "y = 2 ( - x ) + 3"}, {"rel": "被描述", "source": "y", "target": "y = 2 ( - x ) + 3"}, {"rel": "限制性描述", "source": "直线 $l$ 与直线 $y = 2 x + 3$ 关于 $y$ 轴对称", "target": "y = 2 ( - x ) + 3"}]}}
{"content": "When doing his homework on solving equations, Xiaoming accidentally made one of the constants unclear. The equation is: $2 y + \\frac { 1 } { 2 } = - y - \\mdlgblksquare$. After looking through the book, Xiaoming found the answer that the solution to this equation is $y = - \\frac { 1 } { 2 }$. What is the constant? ", "answer": "1", "steps": "Let the missing part be $x$. Then $2 y + \\frac { 1 } { 2 } = - y - x$. Substituting $y = - \\frac { 1 } { 2 }$, we get $2 * ( - \\frac { 1 } { 2 }) + \\frac { 1 } { 2 } = -- \\frac { 1 } { 2 } - x$, which gives $x = 1$.", "expr_cands": ["2 y + \\frac { 1 } { 2 } = - y - \\mdlgblksquare", "y", "mdlgblksquare", "y = - \\frac { 1 } { 2 }", "x", "2 y + \\frac { 1 } { 2 } = - y - x", "2 * ( - \\frac { 1 } { 2 } ) + \\frac { 1 } { 2 } = - ( - \\frac { 1 } { 2 } ) - x", "x = 1"], "exprs": ["2 y + \\frac { 1 } { 2 } = - y - x", "2 * ( - \\frac { 1 } { 2 } ) + \\frac { 1 } { 2 } = - ( - \\frac { 1 } { 2 } ) - x", "x = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "设所缺的部分为 $x$"}, {"id": "2 y + \\frac { 1 } { 2 } = - y - x"}, {"id": "y = - \\frac { 1 } { 2 }"}, {"id": "2 * ( - \\frac { 1 } { 2 } ) + \\frac { 1 } { 2 } = - ( - \\frac { 1 } { 2 } ) - x"}, {"id": "x = 1"}], "links": [{"rel": "假设描述", "source": "设所缺的部分为 $x$", "target": "2 y + \\frac { 1 } { 2 } = - y - x"}, {"rel": "被代入", "source": "2 y + \\frac { 1 } { 2 } = - y - x", "target": "2 * ( - \\frac { 1 } { 2 } ) + \\frac { 1 } { 2 } = - ( - \\frac { 1 } { 2 } ) - x"}, {"rel": "代入", "source": "y = - \\frac { 1 } { 2 }", "target": "2 * ( - \\frac { 1 } { 2 } ) + \\frac { 1 } { 2 } = - ( - \\frac { 1 } { 2 } ) - x"}, {"rel": "等式方程求解", "source": "2 * ( - \\frac { 1 } { 2 } ) + \\frac { 1 } { 2 } = - ( - \\frac { 1 } { 2 } ) - x", "target": "x = 1"}]}}
{"content": "Given that $x = m$ is a solution to the equation $2 x + m = 6$, the value of $m$ is ____?", "answer": "2", "steps": "$\\because x = m$ is a solution to the equation $2 x + m = 6$ in terms of $x$, $\\therefore 2 m + m = 6$, $\\therefore m = 2$.", "expr_cands": ["x = m", "m", "x", "2 x + m = 6", "3 m = 6", "2 m + m = 6", "m = 2"], "exprs": ["2 m + m = 6", "m = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x + m = 6"}, {"id": "2 m + m = 6"}, {"id": "x = m"}, {"id": "m = 2"}], "links": [{"rel": "被代入", "source": "2 x + m = 6", "target": "2 m + m = 6"}, {"rel": "等式方程求解", "source": "2 m + m = 6", "target": "m = 2"}, {"rel": "代入", "source": "x = m", "target": "2 m + m = 6"}]}}
{"content": "If there is a root of the quadratic equation $kx ^ 2 + 1 = x - x ^ 2$ with respect to $x$, which is $2$, then the value of $k$ is ____?", "answer": "- \\frac { 3 } { 4 }", "steps": "Substituting $x = 2$ into the equation, we get $4 k + 1 = 2 - 4$, which yields $k = - \\frac { 3 } { 4 }$.", "expr_cands": ["x", "kx ^ { 2 } + 1 = x - x ^ { 2 }", "k", "2", "x = 2", "4 k + 1 = 2 - 4", "k = - \\frac { 3 } { 4 }"], "exprs": ["x = 2", "4 k + 1 = 2 - 4", "k = - \\frac { 3 } { 4 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2"}, {"id": "x = 2"}, {"id": "x"}, {"id": "kx ^ { 2 } + 1 = x - x ^ { 2 }"}, {"id": "关于 $x$ 的一元二次方程 $kx ^ { 2 } + 1 = x - x ^ { 2 }$ 有一个根为 $2$"}, {"id": "4 k + 1 = 2 - 4"}, {"id": "k = - \\frac { 3 } { 4 }"}], "links": [{"rel": "被描述", "source": "2", "target": "x = 2"}, {"rel": "代入", "source": "x = 2", "target": "4 k + 1 = 2 - 4"}, {"rel": "被描述", "source": "x", "target": "x = 2"}, {"rel": "被描述", "source": "kx ^ { 2 } + 1 = x - x ^ { 2 }", "target": "x = 2"}, {"rel": "被代入", "source": "kx ^ { 2 } + 1 = x - x ^ { 2 }", "target": "4 k + 1 = 2 - 4"}, {"rel": "限制性描述", "source": "关于 $x$ 的一元二次方程 $kx ^ { 2 } + 1 = x - x ^ { 2 }$ 有一个根为 $2$", "target": "x = 2"}, {"rel": "等式方程求解", "source": "4 k + 1 = 2 - 4", "target": "k = - \\frac { 3 } { 4 }"}]}}
{"content": "When $a = 3$, $\\sqrt { 2 a - 2 } - a$ = ____?", "answer": "- 1", "steps": "When $a = 3$, the original expression equals $2 - 3 = - 1$.", "expr_cands": ["a = 3", "a", "\\sqrt { 2 a - 2 } - a", "2 - 3", "- 1"], "exprs": ["- 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a = 3"}, {"id": "- 1"}, {"id": "\\sqrt { 2 a - 2 } - a"}], "links": [{"rel": "代入", "source": "a = 3", "target": "- 1"}, {"rel": "被代入", "source": "\\sqrt { 2 a - 2 } - a", "target": "- 1"}]}}
{"content": "If the fraction $\\frac { 2 } { x - 3 }$ is meaningful, then the range of values for $x$ is ____?", "answer": "x \\neq 3", "steps": "From the given condition, we have $x - 3 \\neq 0$, which implies that $x \\neq 3$.", "expr_cands": ["\\frac { 2 } { x - 3 }", "x", "x - 3 \\neq 0", "x \\neq 3"], "exprs": ["x - 3 \\neq 0", "x \\neq 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 2 } { x - 3 }"}, {"id": "x - 3 \\neq 0"}, {"id": "分式 $\\frac { 2 } { x - 3 }$ 有意义"}, {"id": "分式有意义，则分母不为0"}, {"id": "x \\neq 3"}], "links": [{"rel": "被描述", "source": "\\frac { 2 } { x - 3 }", "target": "x - 3 \\neq 0"}, {"rel": "不等式方程求解", "source": "x - 3 \\neq 0", "target": "x \\neq 3"}, {"rel": "限制性描述", "source": "分式 $\\frac { 2 } { x - 3 }$ 有意义", "target": "x - 3 \\neq 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "x - 3 \\neq 0"}]}}
{"content": "The range of values of $x$ that make $\\sqrt { 4 x - 1 }$ meaningful is ____?", "answer": "x \\ge \\frac { 1 } { 4 }", "steps": "According to the problem, we have $4 x - 1 \\ge 0$, which implies $x \\ge \\frac { 1 } { 4 }$.", "expr_cands": ["\\sqrt { 4 x - 1 }", "x", "4 x - 1 \\ge 0", "\\frac { 1 } { 4 } \\le x", "x \\ge \\frac { 1 } { 4 }"], "exprs": ["4 x - 1 \\ge 0", "x \\ge \\frac { 1 } { 4 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { 4 x - 1 }"}, {"id": "4 x - 1 \\ge 0"}, {"id": "使 $\\sqrt { 4 x - 1 }$ 有意义的 $x$ 的取值范围"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x \\ge \\frac { 1 } { 4 }"}], "links": [{"rel": "被描述", "source": "\\sqrt { 4 x - 1 }", "target": "4 x - 1 \\ge 0"}, {"rel": "不等式方程求解", "source": "4 x - 1 \\ge 0", "target": "x \\ge \\frac { 1 } { 4 }"}, {"rel": "限制性描述", "source": "使 $\\sqrt { 4 x - 1 }$ 有意义的 $x$ 的取值范围", "target": "4 x - 1 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "4 x - 1 \\ge 0"}]}}
{"content": "If $a$, $b$ are reciprocals, $c$, $d$ are opposite numbers, and $m = - 3$, then the algebraic expression $2 abm - ( c + d ) - m$ = ____?", "answer": "- 3", "steps": "From the given information, we know that $ab = 1$ and $c + d = 0$. Therefore, the original expression is equal to $2 \\times 1 \\times ( - 3 ) - 0 - ( - 3 ) = - 6 + 3 = - 3$.", "expr_cands": ["a", "b", "c", "d", "m = - 3", "m", "2 abm - ( c + d ) - m", "ab = 1", "c + d = 0", "2 * 1 * ( - 3 ) - 0 - ( - 3 )", "- 3"], "exprs": ["ab = 1", "c + d = 0", "- 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "ab = 1"}, {"id": "b"}, {"id": "$a$ , $b$ 互为倒数"}, {"id": "c"}, {"id": "c + d = 0"}, {"id": "d"}, {"id": "$c$ , $d$ 互为相反数"}, {"id": "m = - 3"}, {"id": "- 3"}, {"id": "2 abm - ( c + d ) - m"}], "links": [{"rel": "被描述", "source": "a", "target": "ab = 1"}, {"rel": "代入", "source": "ab = 1", "target": "- 3"}, {"rel": "被描述", "source": "b", "target": "ab = 1"}, {"rel": "属性描述", "source": "$a$ , $b$ 互为倒数", "target": "ab = 1"}, {"rel": "被描述", "source": "c", "target": "c + d = 0"}, {"rel": "代入", "source": "c + d = 0", "target": "- 3"}, {"rel": "被描述", "source": "d", "target": "c + d = 0"}, {"rel": "属性描述", "source": "$c$ , $d$ 互为相反数", "target": "c + d = 0"}, {"rel": "代入", "source": "m = - 3", "target": "- 3"}, {"rel": "被代入", "source": "2 abm - ( c + d ) - m", "target": "- 3"}]}}
{"content": "When $x$ = ____ ?, the value of the quadratic radical $\\sqrt { x + 3 }$ is 0.", "answer": "- 3", "steps": "According to the problem, we have $x + 3 = 0$, which gives us the solution $x = - 3$. ", "expr_cands": ["x", "\\sqrt { x + 3 }", "0", "x + 3 = 0", "x = - 3"], "exprs": ["x + 3 = 0", "x = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x + 3 }"}, {"id": "x + 3 = 0"}, {"id": "二次根式 $\\sqrt { x + 3 }$ 的值为 $0$"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x = - 3"}], "links": [{"rel": "被描述", "source": "\\sqrt { x + 3 }", "target": "x + 3 = 0"}, {"rel": "等式方程求解", "source": "x + 3 = 0", "target": "x = - 3"}, {"rel": "限制性描述", "source": "二次根式 $\\sqrt { x + 3 }$ 的值为 $0$", "target": "x + 3 = 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x + 3 = 0"}]}}
{"content": "The rational number $a$ moves $4$ units to the left to become the opposite of $a$. What is the value of $a$?", "answer": "2", "steps": "From the given information, we have $a - 4 = - a$. Solving for $a$, we get $a = 2$.", "expr_cands": ["a", "4", "a - 4 = - a", "a = 2"], "exprs": ["a - 4 = - a", "a = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4"}, {"id": "a - 4 = - a"}, {"id": "a"}, {"id": "有理数 $a$ 向左移动 $4$ 个单位得到 $a$ 的相反数"}, {"id": "a = 2"}], "links": [{"rel": "被描述", "source": "4", "target": "a - 4 = - a"}, {"rel": "等式方程求解", "source": "a - 4 = - a", "target": "a = 2"}, {"rel": "被描述", "source": "a", "target": "a - 4 = - a"}, {"rel": "限制性描述", "source": "有理数 $a$ 向左移动 $4$ 个单位得到 $a$ 的相反数", "target": "a - 4 = - a"}]}}
{"content": "The equation of the line obtained by translating the line $y = 3 x + 1$ $2$ units to the left is _____.", "answer": "y = 3 x + 7", "steps": "According to the principle of adding to the left and subtracting from the right, we know that the equation of the line obtained by shifting the line $y = 3 x + 1$ 2 units to the left is $y = 3 ( x + 2 ) + 1 = 3 x + 7$. That is, $y = 3 x + 7$.", "expr_cands": ["y = 3 x + 1", "y", "x", "2", "y = 3 x + 7", "3 x + 7"], "exprs": ["y = 3 x + 7"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = 3 x + 1"}, {"id": "y = 3 x + 7"}, {"id": "2"}, {"id": "直线 $y = 3 x + 1$ 向左平移 $2$ 个单位长度后所得到的直线的解析式"}, {"id": "\" 左加右减 \" 的原可"}], "links": [{"rel": "被描述", "source": "y = 3 x + 1", "target": "y = 3 x + 7"}, {"rel": "被描述", "source": "2", "target": "y = 3 x + 7"}, {"rel": "限制性描述", "source": "直线 $y = 3 x + 1$ 向左平移 $2$ 个单位长度后所得到的直线的解析式", "target": "y = 3 x + 7"}, {"rel": "限制性描述", "source": "\" 左加右减 \" 的原可", "target": "y = 3 x + 7"}]}}
{"content": "If the algebraic expression $\\frac { 1 } { \\sqrt { x - 8 }}$ is meaningful, then the condition that $x$ should satisfy is ____?", "answer": "x > 8", "steps": "When the algebraic expression $\\frac { 1 } { \\sqrt { x - 8 }}$ is meaningful, $x - 8 > 0$, which can be solved as $x > 8$.", "expr_cands": ["\\frac { 1 } { \\sqrt { x - 8 } }", "x", "x - 8 > 0", "8 < x", "x > 8"], "exprs": ["x - 8 > 0", "x > 8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 1 } { \\sqrt { x - 8 } }"}, {"id": "x - 8 > 0"}, {"id": "分式有意义，则分母不为0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "代数式 $\\frac { 1 } { \\sqrt { x - 8 } }$ 有意义"}, {"id": "x > 8"}], "links": [{"rel": "被描述", "source": "\\frac { 1 } { \\sqrt { x - 8 } }", "target": "x - 8 > 0"}, {"rel": "不等式方程求解", "source": "x - 8 > 0", "target": "x > 8"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "x - 8 > 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 8 > 0"}, {"rel": "限制性描述", "source": "代数式 $\\frac { 1 } { \\sqrt { x - 8 } }$ 有意义", "target": "x - 8 > 0"}]}}
{"content": "If $a - 3 b = 4$, then the value of $1 - 2 a + 6 b$ is ____?", "answer": "- 7", "steps": "Since $a - 3 b = 4$, it follows that $1 - 2 a + 6 b = 1 - 2 ( a - 3 b ) = 1 - 2 * 4 = 1 - 8 = - 7$.", "expr_cands": ["a - 3 b = 4", "b", "a", "1 - 2 a + 6 b", "1 - 2 ( a - 3 b )", "- 7"], "exprs": ["1 - 2 ( a - 3 b )", "- 7"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "1 - 2 a + 6 b"}, {"id": "1 - 2 ( a - 3 b )"}, {"id": "a - 3 b = 4"}, {"id": "- 7"}], "links": [{"rel": "提取因式", "source": "1 - 2 a + 6 b", "target": "1 - 2 ( a - 3 b )"}, {"rel": "被代入", "source": "1 - 2 ( a - 3 b )", "target": "- 7"}, {"rel": "提取因式参考", "source": "a - 3 b = 4", "target": "1 - 2 ( a - 3 b )"}, {"rel": "代入", "source": "a - 3 b = 4", "target": "- 7"}]}}
{"content": "If the equation $\\frac { x } { x - 2 } = \\frac { a } { 2 - x } - 1$ has no solution for $x$, then $a$ = ____?", "answer": "- 2", "steps": "$\\frac { x } { x - 2 } = \\frac { a } { 2 - x } - 1$ is an equation. To eliminate the denominators and obtain a polynomial equation, we multiply both sides by $( x - 2 ) ( 2 - x )$, which gives $2 x + a - 2 = 0$. Therefore, $a = 2 - 2 x$. Since the equation $\\frac { x } { x - 2 } = \\frac { a } { 2 - x } - 1$ has no solution, we conclude that $x = 2$. Hence, $a = 2 - 2 * 2 = - 2$.", "expr_cands": ["x", "\\frac { x } { x - 2 } = \\frac { a } { 2 - x } - 1", "a", "2 x + a - 2 = 0", "a = 2 - 2 x", "\\frac { x } { x - 2 } = \\frac { 2 - 2 x } { 2 - x } - 1", "x = 2", "a = - 2"], "exprs": ["2 x + a - 2 = 0", "x = 2", "a = 2 - 2 x", "a = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x } { x - 2 } = \\frac { a } { 2 - x } - 1"}, {"id": "2 x + a - 2 = 0"}, {"id": "a = 2 - 2 x"}, {"id": "x = 2"}, {"id": "关于 $x$ 的方程 $\\frac { x } { x - 2 } = \\frac { a } { 2 - x } - 1$ 无解"}, {"id": "分式方程无解，则分母为0"}, {"id": "a = - 2"}], "links": [{"rel": "移项", "source": "\\frac { x } { x - 2 } = \\frac { a } { 2 - x } - 1", "target": "2 x + a - 2 = 0"}, {"rel": "被描述", "source": "\\frac { x } { x - 2 } = \\frac { a } { 2 - x } - 1", "target": "x = 2"}, {"rel": "等式方程部分求解", "source": "2 x + a - 2 = 0", "target": "a = 2 - 2 x"}, {"rel": "被代入", "source": "a = 2 - 2 x", "target": "a = - 2"}, {"rel": "代入", "source": "x = 2", "target": "a = - 2"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $\\frac { x } { x - 2 } = \\frac { a } { 2 - x } - 1$ 无解", "target": "x = 2"}, {"rel": "属性描述", "source": "分式方程无解，则分母为0", "target": "x = 2"}]}}
{"content": "If $2 * 2 ^ { x } = 8$, what is the value of $x$?", "answer": "2", "steps": "$\\because$ $2 * 2 ^ { x } = 8$ , $\\therefore$ $2 ^ { x + 1 } = 2 ^ { 3 }$ , $\\therefore$ $x + 1 = 3$ , which gives $x = 2$.", "expr_cands": ["2 * 2 ^ { x } = 8", "x", "x = 2", "2 ^ { x + 1 } = 2 ^ { 3 }", "x + 1 = 3"], "exprs": ["x = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 * 2 ^ { x } = 8"}, {"id": "x = 2"}], "links": [{"rel": "等式方程求解", "source": "2 * 2 ^ { x } = 8", "target": "x = 2"}]}}
{"content": "When $m \\neq 0$, if $m ^ { 0 } * m ^ { - 5 } m ^ { n } = 1$, then $n$ = ____ ?", "answer": "5", "steps": "When $m \\neq 0$, from $m ^ { 0 } * m ^ { - 5 } m ^ { n } = 1$, we can get $m ^ { 0 - 5 + n } = m ^ { 0 } = 1$. Therefore, $0 - 5 + n = 0$, and we can solve for $n$ to get $n = 5$.", "expr_cands": ["m \\neq 0", "m", "m ^ { 0 } * m ^ { - 5 } m ^ { n } = 1", "n", "m ^ { 0 }", "1", "0 - 5 + n = 0", "n = 5"], "exprs": ["0 - 5 + n = 0", "n = 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "m ^ { 0 } * m ^ { - 5 } m ^ { n } = 1"}, {"id": "0 - 5 + n = 0"}, {"id": "n = 5"}], "links": [{"rel": "同取对数", "source": "m ^ { 0 } * m ^ { - 5 } m ^ { n } = 1", "target": "0 - 5 + n = 0"}, {"rel": "等式方程求解", "source": "0 - 5 + n = 0", "target": "n = 5"}]}}
{"content": "The equation $\\frac { x } { x - 3 } = \\frac { 2 } { 3 - x }$ has roots of _____.", "answer": "x = - 2", "steps": "The original equation can be rearranged as: $\\frac { x } { x - 3 } = \\frac { - 2 } { x - 3 }$. Simplifying by removing the denominators, we get: $x = - 2$. After checking, we find that $x = - 2$ is a solution to the fractional equation.", "expr_cands": ["\\frac { x } { x - 3 } = \\frac { 2 } { 3 - x }", "x", "\\frac { x } { x - 3 } = \\frac { - 2 } { x - 3 }", "x = - 2"], "exprs": ["x = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x } { x - 3 } = \\frac { 2 } { 3 - x }"}, {"id": "x = - 2"}], "links": [{"rel": "等式方程求解", "source": "\\frac { x } { x - 3 } = \\frac { 2 } { 3 - x }", "target": "x = - 2"}]}}
{"content": "If the two roots of the quadratic equation $( m ^ 2 - 2 ) x ^ 2 - ( m - 2 ) x - 1 = 0$ with respect to $x$ are opposite, then $m$ = ____?", "answer": "2", "steps": "$\\because$ The two roots of the quadratic equation in $x$, $( m ^ 2 - 2 ) x ^ 2 - ( m - 2 ) x - 1 = 0$, are opposite to each other. $\\therefore$ $\\frac { m - 2 } { m ^ 2 - 2 } = 0$. $\\therefore$ $m - 2 = 0$ and $m ^ 2 - 2 \\neq 0$. $\\therefore$ $m = 2$.", "expr_cands": ["x", "( m ^ { 2 } - 2 ) x ^ { 2 } - ( m - 2 ) x - 1 = 0", "m", "\\frac { m - 2 } { { m } ^ { 2 } - 2 } = 0", "m = 2", "m - 2 = 0", "m ^ { 2 } - 2 \\neq 0", "( - \\sqrt { 2 } < m \\wedge m < \\sqrt { 2 })", "\\sqrt { 2 } < m", "m < - \\sqrt { 2 }"], "exprs": ["\\frac { m - 2 } { { m } ^ { 2 } - 2 } = 0", "m - 2 = 0", "m ^ { 2 } - 2 \\neq 0", "m = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( m ^ { 2 } - 2 ) x ^ { 2 } - ( m - 2 ) x - 1 = 0"}, {"id": "\\frac { m - 2 } { { m } ^ { 2 } - 2 } = 0"}, {"id": "关于 $x$ 的一元二次方程 $( m ^ { 2 } - 2 ) x ^ { 2 } - ( m - 2 ) x - 1 = 0$ 的两个根互为相反数"}, {"id": "一元二次方程根与系数关系，两根之和"}, {"id": "m - 2 = 0"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "m ^ { 2 } - 2 \\neq 0"}, {"id": "m = 2"}], "links": [{"rel": "被描述", "source": "( m ^ { 2 } - 2 ) x ^ { 2 } - ( m - 2 ) x - 1 = 0", "target": "\\frac { m - 2 } { { m } ^ { 2 } - 2 } = 0"}, {"rel": "被描述", "source": "\\frac { m - 2 } { { m } ^ { 2 } - 2 } = 0", "target": "m - 2 = 0"}, {"rel": "被描述", "source": "\\frac { m - 2 } { { m } ^ { 2 } - 2 } = 0", "target": "m ^ { 2 } - 2 \\neq 0"}, {"rel": "限制性描述", "source": "关于 $x$ 的一元二次方程 $( m ^ { 2 } - 2 ) x ^ { 2 } - ( m - 2 ) x - 1 = 0$ 的两个根互为相反数", "target": "\\frac { m - 2 } { { m } ^ { 2 } - 2 } = 0"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "\\frac { m - 2 } { { m } ^ { 2 } - 2 } = 0"}, {"rel": "联立", "source": "m - 2 = 0", "target": "m = 2"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "m - 2 = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "m ^ { 2 } - 2 \\neq 0"}, {"rel": "联立", "source": "m ^ { 2 } - 2 \\neq 0", "target": "m = 2"}]}}
{"content": "If $a + 19 = b + 9 = c + 8$, then $( a - b ) ^ 2 + ( b - c ) ^ 2 + ( c - a ) ^ 2$ = ____ ?", "answer": "222", "steps": "$\\because a + 19 = b + 9 = c + 8$ , $\\therefore a - b = - 10$ , $b - c = - 1$ , $c - a = 11$ . $\\therefore$ The original expression $= ( - 10 ) ^ 2 + ( - 1 ) ^ 2 + 11 ^ 2 = 222$.", "expr_cands": ["a + 19 = b + 9 = c + 8", "( a - b ) ^ { 2 } + ( b - c ) ^ { 2 } + ( c - a ) ^ { 2 }", "c", "b", "a", "a + 19 = c + 8", "a - b = - 10", "b - c = - 1", "c - a = 11", "( - 10 ) ^ { 2 } + ( - 1 ) ^ { 2 } + 11 ^ { 2 }", "222"], "exprs": ["222"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a + 19 = b + 9 = c + 8"}, {"id": "222"}, {"id": "( a - b ) ^ { 2 } + ( b - c ) ^ { 2 } + ( c - a ) ^ { 2 }"}, {"id": "原式 = $( - 10 ) ^ { 2 } + ( - 1 ) ^ { 2 } + 11 ^ { 2 } = 222$"}], "links": [{"rel": "被描述", "source": "a + 19 = b + 9 = c + 8", "target": "222"}, {"rel": "被描述", "source": "( a - b ) ^ { 2 } + ( b - c ) ^ { 2 } + ( c - a ) ^ { 2 }", "target": "222"}, {"rel": "限制性描述", "source": "原式 = $( - 10 ) ^ { 2 } + ( - 1 ) ^ { 2 } + 11 ^ { 2 } = 222$", "target": "222"}]}}
{"content": "If the value of $m - 3 n + 1$ is $5$, then the value of the algebraic expression $5 - m + 3 n$ is ____?", "answer": "1", "steps": "According to the problem, we have $m - 3 n + 1 = 5$, which means $m - 3 n = 4$. Therefore, the original expression is equal to $5 - ( m - 3 n ) = 5 - 4 = 1$.", "expr_cands": ["m - 3 n + 1", "n", "m", "5", "5 - m + 3 n", "m - 3 n + 1 = 5", "m - 3 n = 4", "5 - ( m - 3 n )", "1"], "exprs": ["m - 3 n + 1 = 5", "m - 3 n = 4", "5 - ( m - 3 n )", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "m - 3 n + 1"}, {"id": "m - 3 n + 1 = 5"}, {"id": "5"}, {"id": "$m - 3 n + 1$ 的值为 $5$"}, {"id": "m - 3 n = 4"}, {"id": "5 - m + 3 n"}, {"id": "5 - ( m - 3 n )"}, {"id": "1"}], "links": [{"rel": "被描述", "source": "m - 3 n + 1", "target": "m - 3 n + 1 = 5"}, {"rel": "移项", "source": "m - 3 n + 1 = 5", "target": "m - 3 n = 4"}, {"rel": "被描述", "source": "5", "target": "m - 3 n + 1 = 5"}, {"rel": "限制性描述", "source": "$m - 3 n + 1$ 的值为 $5$", "target": "m - 3 n + 1 = 5"}, {"rel": "提取因式参考", "source": "m - 3 n = 4", "target": "5 - ( m - 3 n )"}, {"rel": "代入", "source": "m - 3 n = 4", "target": "1"}, {"rel": "提取因式", "source": "5 - m + 3 n", "target": "5 - ( m - 3 n )"}, {"rel": "被代入", "source": "5 - ( m - 3 n )", "target": "1"}]}}
{"content": "If $5 x ^ { 2 } y ^ { a + 1 }$ and $- 9 x ^ { b - 3 } y ^ { 4 }$ are like terms, then the value of $a - b$ is ____?", "answer": "- 2", "steps": "Because $5 x ^ { 2 } y ^ { a + 1 }$ and $- 9 x ^ { b - 3 } y ^ { 4 }$ are like terms, therefore $a + 1 = 4$, $b - 3 = 2$, solving for $a = 3$, $b = 5$, therefore $a - b = 3 - 5 = - 2$.", "expr_cands": ["5 x ^ { 2 } y ^ { a + 1 }", "a", "y", "x", "- 9 x ^ { b - 3 } y ^ { 4 }", "b", "a - b", "a + 1 = 4", "a = 3", "b - 3 = 2", "b = 5", "- 2"], "exprs": ["a + 1 = 4", "b - 3 = 2", "a = 3", "b = 5", "- 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5 x ^ { 2 } y ^ { a + 1 }"}, {"id": "a + 1 = 4"}, {"id": "- 9 x ^ { b - 3 } y ^ { 4 }"}, {"id": "$5 x ^ { 2 } y ^ { a + 1 }$ 和 $- 9 x ^ { b - 3 } y ^ { 4 }$ 是同类项"}, {"id": "b - 3 = 2"}, {"id": "a = 3"}, {"id": "b = 5"}, {"id": "a - b"}, {"id": "- 2"}], "links": [{"rel": "被描述", "source": "5 x ^ { 2 } y ^ { a + 1 }", "target": "a + 1 = 4"}, {"rel": "被描述", "source": "5 x ^ { 2 } y ^ { a + 1 }", "target": "b - 3 = 2"}, {"rel": "等式方程求解", "source": "a + 1 = 4", "target": "a = 3"}, {"rel": "被描述", "source": "- 9 x ^ { b - 3 } y ^ { 4 }", "target": "a + 1 = 4"}, {"rel": "被描述", "source": "- 9 x ^ { b - 3 } y ^ { 4 }", "target": "b - 3 = 2"}, {"rel": "限制性描述", "source": "$5 x ^ { 2 } y ^ { a + 1 }$ 和 $- 9 x ^ { b - 3 } y ^ { 4 }$ 是同类项", "target": "a + 1 = 4"}, {"rel": "限制性描述", "source": "$5 x ^ { 2 } y ^ { a + 1 }$ 和 $- 9 x ^ { b - 3 } y ^ { 4 }$ 是同类项", "target": "b - 3 = 2"}, {"rel": "等式方程求解", "source": "b - 3 = 2", "target": "b = 5"}, {"rel": "代入", "source": "a = 3", "target": "- 2"}, {"rel": "代入", "source": "b = 5", "target": "- 2"}, {"rel": "被代入", "source": "a - b", "target": "- 2"}]}}
{"content": "When $a = 2 + \\sqrt { 3 }$, what is the value of the fraction $\\frac { 9 } { a ^ { 2 } - 4 a + 4 }$?", "answer": "3", "steps": "Because $a = 2 + \\sqrt { 3 }$, therefore $\\frac { 9 } { a ^ 2 - 4 a + 4 } = \\frac { 9 } {( a - 2 ) ^ 2 } = \\frac { 9 } {( 2 + \\sqrt { 3 } - 2 ) ^ 2 } = 3$.", "expr_cands": ["a = 2 + \\sqrt { 3 }", "a", "\\frac { 9 } { a ^ { 2 } - 4 a + 4 }", "3"], "exprs": ["3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 9 } { a ^ { 2 } - 4 a + 4 }"}, {"id": "3"}, {"id": "a = 2 + \\sqrt { 3 }"}], "links": [{"rel": "被代入", "source": "\\frac { 9 } { a ^ { 2 } - 4 a + 4 }", "target": "3"}, {"rel": "代入", "source": "a = 2 + \\sqrt { 3 }", "target": "3"}]}}
{"content": "The value of the fraction $\\frac { 2 } { x - 3 }$ is positive, then the condition that $x$ should satisfy is ____?", "answer": "x > 3", "steps": "Because the value of the fraction $\\frac { 2 } { x - 3 }$ is positive, we know that $x - 3 > 0$, which means $x > 3$.", "expr_cands": ["\\frac { 2 } { x - 3 }", "x", "x - 3 > 0", "3 < x", "x > 3"], "exprs": ["x - 3 > 0", "x > 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 2 } { x - 3 }"}, {"id": "x - 3 > 0"}, {"id": "分式为正数，则分子分母同号"}, {"id": "x > 3"}], "links": [{"rel": "被描述", "source": "\\frac { 2 } { x - 3 }", "target": "x - 3 > 0"}, {"rel": "不等式方程求解", "source": "x - 3 > 0", "target": "x > 3"}, {"rel": "属性描述", "source": "分式为正数，则分子分母同号", "target": "x - 3 > 0"}]}}
{"content": "$8$. Given that the square root of a positive number is $3 x + 2$ and $4 x - 9$, find the number.", "answer": "25", "steps": "$\\because$ A positive number has two square roots, which are $3 x + 2$ and $4 x - 9$. $\\therefore$ $3 x + 2 + 4 x - 9 = 0$, solving for $x$ gives $x = 1$, $3 x + 2 = 5$. This positive number is $5 ^ 2 = 25$.", "expr_cands": ["8", "3 x + 2", "x", "4 x - 9", "4 X - 9", "X", "3 x + 2 + 4 x - 9 = 0", "x = 1", "5", "5 ^ { 2 }", "25"], "exprs": ["3 x + 2 + 4 x - 9 = 0", "x = 1", "5", "25"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 x + 2"}, {"id": "3 x + 2 + 4 x - 9 = 0"}, {"id": "4 x - 9"}, {"id": "一个正数的平方根分别是 $3 x + 2$ 和 $4 x - 9$"}, {"id": "平方根互为相反数"}, {"id": "x = 1"}, {"id": "5"}, {"id": "25"}, {"id": "平方"}], "links": [{"rel": "被描述", "source": "3 x + 2", "target": "3 x + 2 + 4 x - 9 = 0"}, {"rel": "被代入", "source": "3 x + 2", "target": "5"}, {"rel": "等式方程求解", "source": "3 x + 2 + 4 x - 9 = 0", "target": "x = 1"}, {"rel": "被描述", "source": "4 x - 9", "target": "3 x + 2 + 4 x - 9 = 0"}, {"rel": "限制性描述", "source": "一个正数的平方根分别是 $3 x + 2$ 和 $4 x - 9$", "target": "3 x + 2 + 4 x - 9 = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "3 x + 2 + 4 x - 9 = 0"}, {"rel": "代入", "source": "x = 1", "target": "5"}, {"rel": "被描述", "source": "5", "target": "25"}, {"rel": "限制性描述", "source": "平方", "target": "25"}]}}
{"content": "The square root of a positive number is $x - 5$ and $x + 1$, what is the value of $x$?", "answer": "2", "steps": "From the given information, we have $x - 5 + x + 1 = 0$, which yields $x = 2$ as the solution.", "expr_cands": ["x - 5", "x", "x + 1", "x - 5 + x + 1 = 0", "x = 2"], "exprs": ["x - 5 + x + 1 = 0", "x = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x - 5"}, {"id": "x - 5 + x + 1 = 0"}, {"id": "x + 1"}, {"id": "一个正数的平方根是 $x - 5$ 和 $x + 1$"}, {"id": "平方根互为相反数"}, {"id": "x = 2"}], "links": [{"rel": "被描述", "source": "x - 5", "target": "x - 5 + x + 1 = 0"}, {"rel": "等式方程求解", "source": "x - 5 + x + 1 = 0", "target": "x = 2"}, {"rel": "被描述", "source": "x + 1", "target": "x - 5 + x + 1 = 0"}, {"rel": "限制性描述", "source": "一个正数的平方根是 $x - 5$ 和 $x + 1$", "target": "x - 5 + x + 1 = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "x - 5 + x + 1 = 0"}]}}
{"content": "Given that the equation $2 x = 8$ has the same solution as $x + 2 = - k$, the value of the algebraic expression $\\frac { 2 - 3 | k | } { k ^ 2 }$ is ____?", "answer": "- \\frac { 4 } { 9 }", "steps": "Solve the equation $2 x = 8$ to get $x = 4$. Substitute $x = 4$ into $x + 2 = - k$ to get $4 + 2 = - k$. Solve for $k$ to get $k = - 6$. Substitute $k = - 6$ into $\\frac { 2 - 3 | k | } { k ^ 2 }$ to get the original expression $= \\frac { 2 - 3 | - 6 | } {( - 6 ) ^ 2 } = - \\frac { 4 } { 9 }$.", "expr_cands": ["x", "2 x = 8", "x + 2 = - k", "k", "\\frac { 2 - 3 | k | } { k ^ { 2 } }", "x = 4", "6 = - k", "4 + 2 = - k", "k = - 6", "- \\frac { 4 } { 9 }"], "exprs": ["x = 4", "4 + 2 = - k", "k = - 6", "- \\frac { 4 } { 9 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x = 8"}, {"id": "x = 4"}, {"id": "x + 2 = - k"}, {"id": "4 + 2 = - k"}, {"id": "k = - 6"}, {"id": "\\frac { 2 - 3 | k | } { k ^ { 2 } }"}, {"id": "- \\frac { 4 } { 9 }"}], "links": [{"rel": "等式方程求解", "source": "2 x = 8", "target": "x = 4"}, {"rel": "代入", "source": "x = 4", "target": "4 + 2 = - k"}, {"rel": "被代入", "source": "x + 2 = - k", "target": "4 + 2 = - k"}, {"rel": "等式方程求解", "source": "4 + 2 = - k", "target": "k = - 6"}, {"rel": "代入", "source": "k = - 6", "target": "- \\frac { 4 } { 9 }"}, {"rel": "被代入", "source": "\\frac { 2 - 3 | k | } { k ^ { 2 } }", "target": "- \\frac { 4 } { 9 }"}]}}
{"content": "If the result of calculating $( x ^ 2 + ax + 5 ) \\cdot ( - 2 x ) - 6 x ^ 2$ does not contain the term $x ^ 2$, then the value of $a$ is ____?", "answer": "- 3", "steps": "$( x ^ { 2 } + ax + 5 ) \\cdot ( - 2 x ) - 6 x ^ { 2 } = - 2 { x } ^ { 3 } - 2 a { x } ^ { 2 } - 10 x - 6 { x } ^ { 2 } = - 2 { x } ^ { 3 } - ( 2 a + 6 ) { x } ^ { 2 } - 10 x$ , from the term without $x ^ { 2 }$ in the result, we get $2 a + 6 = 0$, and solve for $a = - 3$.", "expr_cands": ["( x ^ { 2 } + ax + 5 ) \\cdot ( - 2 x ) - 6 x ^ { 2 }", "a", "x", "x ^ { 2 }", "- 2 { x } ^ { 3 } - ( 2 a + 6 ) { x } ^ { 2 } - 10 x", "2 a + 6 = 0", "a = - 3"], "exprs": ["- 2 { x } ^ { 3 } - ( 2 a + 6 ) { x } ^ { 2 } - 10 x", "2 a + 6 = 0", "a = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( x ^ { 2 } + ax + 5 ) \\cdot ( - 2 x ) - 6 x ^ { 2 }"}, {"id": "- 2 { x } ^ { 3 } - ( 2 a + 6 ) { x } ^ { 2 } - 10 x"}, {"id": "2 a + 6 = 0"}, {"id": "计算 $( x ^ { 2 } + ax + 5 ) \\cdot ( - 2 x ) - 6 x ^ { 2 }$ 的结果中不含有 $x ^ { 2 }$ 项"}, {"id": "a = - 3"}], "links": [{"rel": "提取因式", "source": "( x ^ { 2 } + ax + 5 ) \\cdot ( - 2 x ) - 6 x ^ { 2 }", "target": "- 2 { x } ^ { 3 } - ( 2 a + 6 ) { x } ^ { 2 } - 10 x"}, {"rel": "被描述", "source": "- 2 { x } ^ { 3 } - ( 2 a + 6 ) { x } ^ { 2 } - 10 x", "target": "2 a + 6 = 0"}, {"rel": "等式方程求解", "source": "2 a + 6 = 0", "target": "a = - 3"}, {"rel": "限制性描述", "source": "计算 $( x ^ { 2 } + ax + 5 ) \\cdot ( - 2 x ) - 6 x ^ { 2 }$ 的结果中不含有 $x ^ { 2 }$ 项", "target": "2 a + 6 = 0"}]}}
{"content": "The equation of the parabola $y = - 3 x ^ 2 - 5 x + 4$ is symmetric about the $x$-axis. The equation of a parabola with the form $y = ax ^ 2 + bx + c$ is given by $y = a ( x - h ) ^ 2 + k$, where $( h , k )$ is the vertex of the parabola. Since the given parabola is symmetric about the $x$-axis, the vertex is at $( 0 , 4 / 3 )$. Thus, we have $k = 4 / 3$.Substituting $x = 0$ and $x = 1$ into the equation of the given parabola, we get the system of equations:$$\\begin{cases} y = 4 \\\\ y = -3x^2 - 5x + 4 \\end{cases}$$Solving for $a$ and $b$, we get $a = - 3$ and $b = - 5$. Therefore, $c = 4 / 3$ and $a + b + c = - 3 - 5 + 4 / 3 = \\boxed { - 14 / 3 }$.", "answer": "4", "steps": "When $x = 1$, $y = - 3 x ^ 2 - 5 x + 4 = - 3 - 5 + 4 = - 4$. The equation of the parabola that is symmetric about the $x$-axis with the formula $y = ax ^ 2 + bx + c$. Therefore, when $x = 1$, $y = ax ^ 2 + bx + c = 4$, which means $a + b + c = 4$.", "expr_cands": ["y = - 3 x ^ { 2 } - 5 x + 4", "y", "x", "y = ax ^ { 2 } + bx + c", "c", "b", "a", "a + b + c", "x = 1", "y = - 4", "- 3 x ^ { 2 } - 5 x + 4 = a + b + c", "y = 4", "a + b + c = 4", "4"], "exprs": ["x = 1", "y = - 4", "y = 4", "4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "当 $x = 1$ 时"}, {"id": "x = 1"}, {"id": "y = - 3 x ^ { 2 } - 5 x + 4"}, {"id": "y = - 4"}, {"id": "x"}, {"id": "y = 4"}, {"id": "y = ax ^ { 2 } + bx + c"}, {"id": "抛物线 $y = - 3 x ^ { 2 } - 5 x + 4$ 关于 $x$ 轴对称的抛物线解析式是 $y = ax ^ { 2 } + bx + c$"}, {"id": "4"}, {"id": "即 $a + b + c = 4$"}], "links": [{"rel": "假设描述", "source": "当 $x = 1$ 时", "target": "x = 1"}, {"rel": "代入", "source": "x = 1", "target": "y = - 4"}, {"rel": "被描述", "source": "x = 1", "target": "4"}, {"rel": "被代入", "source": "y = - 3 x ^ { 2 } - 5 x + 4", "target": "y = - 4"}, {"rel": "被描述", "source": "y = - 4", "target": "y = 4"}, {"rel": "被描述", "source": "x", "target": "y = 4"}, {"rel": "被描述", "source": "y = 4", "target": "4"}, {"rel": "被描述", "source": "y = ax ^ { 2 } + bx + c", "target": "y = 4"}, {"rel": "被描述", "source": "y = ax ^ { 2 } + bx + c", "target": "4"}, {"rel": "限制性描述", "source": "抛物线 $y = - 3 x ^ { 2 } - 5 x + 4$ 关于 $x$ 轴对称的抛物线解析式是 $y = ax ^ { 2 } + bx + c$", "target": "y = 4"}, {"rel": "限制性描述", "source": "即 $a + b + c = 4$", "target": "4"}]}}
{"content": "If the solution set of the inequality $2 m + x > 5$ with respect to $x$ is $x > 1$, then the value of $m$ is ____?", "answer": "2", "steps": "Since $2 m + x > 5$, we have $x > 5 - 2 m$. Also, since $x > 1$, we have $5 - 2 m = 1$, which implies $m = 2$.", "expr_cands": ["x", "2 m + x > 5", "m", "x > 1", "x > 5 - 2 m", "5 - 2 m = 1", "m = 2"], "exprs": ["x > 5 - 2 m", "5 - 2 m = 1", "m = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 m + x > 5"}, {"id": "x > 5 - 2 m"}, {"id": "x > 1"}, {"id": "5 - 2 m = 1"}, {"id": "关于 $x$ 的不等式 $2 m + x > 5$ 的解集是 $x > 1$"}, {"id": "m = 2"}], "links": [{"rel": "不等式方程部分求解", "source": "2 m + x > 5", "target": "x > 5 - 2 m"}, {"rel": "被描述", "source": "x > 5 - 2 m", "target": "5 - 2 m = 1"}, {"rel": "被描述", "source": "x > 1", "target": "5 - 2 m = 1"}, {"rel": "等式方程求解", "source": "5 - 2 m = 1", "target": "m = 2"}, {"rel": "限制性描述", "source": "关于 $x$ 的不等式 $2 m + x > 5$ 的解集是 $x > 1$", "target": "5 - 2 m = 1"}]}}
{"content": "When $k$ = ____ ?, the polynomial $4 x ^ { 2 } - 3 xy + y ^ { 2 } + kx ^ { 2 }$ does not contain the term $x ^ { 2 }$.", "answer": "- 4", "steps": "From the given information, we have $4 + k = 0$, which can be solved to obtain $k = - 4$.", "expr_cands": ["k", "4 x ^ { 2 } - 3 xy + y ^ { 2 } + kx ^ { 2 }", "x", "y", "x ^ { 2 }", "4 + k = 0", "k = - 4"], "exprs": ["4 + k = 0", "k = - 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 x ^ { 2 } - 3 xy + y ^ { 2 } + kx ^ { 2 }"}, {"id": "4 + k = 0"}, {"id": "多项式 $4 x ^ { 2 } - 3 xy + y ^ { 2 } + kx ^ { 2 }$ 中不含 $x ^ { 2 }$ 项"}, {"id": "k = - 4"}], "links": [{"rel": "被描述", "source": "4 x ^ { 2 } - 3 xy + y ^ { 2 } + kx ^ { 2 }", "target": "4 + k = 0"}, {"rel": "等式方程求解", "source": "4 + k = 0", "target": "k = - 4"}, {"rel": "限制性描述", "source": "多项式 $4 x ^ { 2 } - 3 xy + y ^ { 2 } + kx ^ { 2 }$ 中不含 $x ^ { 2 }$ 项", "target": "4 + k = 0"}]}}
{"content": "The increasing root of the fractional equation $\\frac { x } { x - 1 } - 2 = \\frac { k } { 1 - x }$ with respect to $x$ is ____ ?", "answer": "x = 1", "steps": "Regarding the fractional equation in $x$, $\\frac { x } { x - 1 } - 2 = \\frac { k } { 1 - x }$, the increasing root is $x = 1$ since $x - 1 = 0$.", "expr_cands": ["x", "\\frac { x } { x - 1 } - 2 = \\frac { k } { 1 - x }", "k", "x - 1 = 0", "x = 1"], "exprs": ["x - 1 = 0", "x = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x } { x - 1 } - 2 = \\frac { k } { 1 - x }"}, {"id": "x - 1 = 0"}, {"id": "关于 $x$ 的分式方程 $\\frac { x } { x - 1 } - 2 = \\frac { k } { 1 - x }$ 的增根"}, {"id": "分式有增根，则分母为0"}, {"id": "x = 1"}], "links": [{"rel": "被描述", "source": "\\frac { x } { x - 1 } - 2 = \\frac { k } { 1 - x }", "target": "x - 1 = 0"}, {"rel": "等式方程求解", "source": "x - 1 = 0", "target": "x = 1"}, {"rel": "限制性描述", "source": "关于 $x$ 的分式方程 $\\frac { x } { x - 1 } - 2 = \\frac { k } { 1 - x }$ 的增根", "target": "x - 1 = 0"}, {"rel": "属性描述", "source": "分式有增根，则分母为0", "target": "x - 1 = 0"}]}}
{"content": "$11$, What is the largest integer solution of the inequality $5 x - 3 < 3 x + 5$?", "answer": "3", "steps": "$5 x - 3 < 3 x + 5$, therefore $5 x - 3 x < 5 + 3$, therefore $2 x < 8$, therefore $x < 4$, therefore the largest integer solution to the inequality is $3$.", "expr_cands": ["11", "5 x - 3 < 3 x + 5", "x", "x < 4", "5 x - 3 x < 5 + 3", "2 x < 8", "3"], "exprs": ["x < 4", "3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5 x - 3 < 3 x + 5"}, {"id": "x < 4"}, {"id": "3"}, {"id": "不等式 $5 x - 3 < 3 x + 5$ 的最大整数解"}], "links": [{"rel": "不等式方程求解", "source": "5 x - 3 < 3 x + 5", "target": "x < 4"}, {"rel": "被描述", "source": "x < 4", "target": "3"}, {"rel": "限制性描述", "source": "不等式 $5 x - 3 < 3 x + 5$ 的最大整数解", "target": "3"}]}}
{"content": "The square root of the product of the difference between 2 and x and the difference between x and 2 is equal to the product of the square root of the difference between 2 and x and the square root of the difference between x and 2. What is the value of x + 3?", "answer": "5", "steps": "From the given conditions, we know that $2 - x \\geq 0$ and $x - 2 \\leq 0$. Therefore, we can conclude that $x = 2$. Thus, we have $x + 3 = 5$.", "expr_cands": ["\\sqrt { ( 2 - x ) ( x - 2 ) } = \\sqrt { 2 - x } \\times \\sqrt { x - 2 }", "x", "x + 3", "2 - x \\ge 0", "x \\le 2", "x - 2 \\le 0", "x = 2", "5"], "exprs": ["2 - x \\ge 0", "x - 2 \\le 0", "x = 2", "5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { ( 2 - x ) ( x - 2 ) } = \\sqrt { 2 - x } \\times \\sqrt { x - 2 }"}, {"id": "2 - x \\ge 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x - 2 \\le 0"}, {"id": "x = 2"}, {"id": "x + 3"}, {"id": "5"}], "links": [{"rel": "被描述", "source": "\\sqrt { ( 2 - x ) ( x - 2 ) } = \\sqrt { 2 - x } \\times \\sqrt { x - 2 }", "target": "2 - x \\ge 0"}, {"rel": "被描述", "source": "\\sqrt { ( 2 - x ) ( x - 2 ) } = \\sqrt { 2 - x } \\times \\sqrt { x - 2 }", "target": "x - 2 \\le 0"}, {"rel": "联立", "source": "2 - x \\ge 0", "target": "x = 2"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "2 - x \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 2 \\le 0"}, {"rel": "联立", "source": "x - 2 \\le 0", "target": "x = 2"}, {"rel": "代入", "source": "x = 2", "target": "5"}, {"rel": "被代入", "source": "x + 3", "target": "5"}]}}
{"content": "If $a$ is the largest negative integer, $b$ is the rational number with the smallest absolute value, and $c$ is the natural number that is equal to its reciprocal, then the value of ${ a } ^ { 2018 } + 2019 b + { c } ^ { 2018 }$ is ____?", "answer": "2", "steps": "According to the problem, we have $a = - 1$, $b = 0$, and $c = 1$. Therefore, the original expression is equal to $1 + 0 + 1 = 2$.", "expr_cands": ["a", "b", "c", "{ a } ^ { 2018 } + 2019 b + { c } ^ { 2018 }", "a = - 1", "b = 0", "c = 1", "1 + 0 + 1", "2"], "exprs": ["a = - 1", "b = 0", "c = 1", "2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a = - 1"}, {"id": "$a$ 是最大的负整数"}, {"id": "b"}, {"id": "b = 0"}, {"id": "$b$ 是绝对值最小的有理数"}, {"id": "c"}, {"id": "c = 1"}, {"id": "$c$ 是倒数等于它本身的自然数"}, {"id": "{ a } ^ { 2018 } + 2019 b + { c } ^ { 2018 }"}, {"id": "2"}], "links": [{"rel": "被描述", "source": "a", "target": "a = - 1"}, {"rel": "代入", "source": "a = - 1", "target": "2"}, {"rel": "限制性描述", "source": "$a$ 是最大的负整数", "target": "a = - 1"}, {"rel": "被描述", "source": "b", "target": "b = 0"}, {"rel": "代入", "source": "b = 0", "target": "2"}, {"rel": "限制性描述", "source": "$b$ 是绝对值最小的有理数", "target": "b = 0"}, {"rel": "被描述", "source": "c", "target": "c = 1"}, {"rel": "代入", "source": "c = 1", "target": "2"}, {"rel": "限制性描述", "source": "$c$ 是倒数等于它本身的自然数", "target": "c = 1"}, {"rel": "被代入", "source": "{ a } ^ { 2018 } + 2019 b + { c } ^ { 2018 }", "target": "2"}]}}
{"content": "The solution set of the inequality $( 3.14 - \\pi ) x < \\pi - 3.14$ is ____ ?", "answer": "x > - 1", "steps": "$( 3.14 - \\pi ) x < \\pi - 3.14$ because $3.14 - \\pi < 0$, therefore $x > \\frac { \\pi - 3.14 } { 3.14 - \\pi }$, therefore $x > - 1$.", "expr_cands": ["( 3.14 - \\pi ) x < \\pi - 3.14", "x", "- 1 < x", "3.14 - \\pi < 0", "x > \\frac { \\pi - 3.14 } { 3.14 - \\pi }", "x > - 1"], "exprs": ["x > - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( 3.14 - \\pi ) x < \\pi - 3.14"}, {"id": "x > - 1"}], "links": [{"rel": "不等式方程求解", "source": "( 3.14 - \\pi ) x < \\pi - 3.14", "target": "x > - 1"}]}}
{"content": "If the two square roots of a positive number are $2 a - 3$ and $5 - a$, then the positive number is ____?", "answer": "49", "steps": "According to the problem, we have $2 a - 3 + 5 - a = 0$, which gives us $a = - 2$. Therefore, $2 a - 3 = - 7$. The positive integer we are looking for is $( - 7 ) ^ { 2 } = 49$.", "expr_cands": ["2 a - 3", "a", "5 - a", "2 a - 3 + 5 - a = 0", "a = - 2", "- 7", "( - 7 ) ^ { 2 }", "49"], "exprs": ["2 a - 3 + 5 - a = 0", "a = - 2", "- 7", "49"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 a - 3"}, {"id": "2 a - 3 + 5 - a = 0"}, {"id": "5 - a"}, {"id": "一个正数的两个平方根是 $2 a - 3$ 和 $5 - a$"}, {"id": "平方根互为相反数"}, {"id": "a = - 2"}, {"id": "- 7"}, {"id": "49"}, {"id": "平方"}], "links": [{"rel": "被描述", "source": "2 a - 3", "target": "2 a - 3 + 5 - a = 0"}, {"rel": "被代入", "source": "2 a - 3", "target": "- 7"}, {"rel": "等式方程求解", "source": "2 a - 3 + 5 - a = 0", "target": "a = - 2"}, {"rel": "被描述", "source": "5 - a", "target": "2 a - 3 + 5 - a = 0"}, {"rel": "限制性描述", "source": "一个正数的两个平方根是 $2 a - 3$ 和 $5 - a$", "target": "2 a - 3 + 5 - a = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "2 a - 3 + 5 - a = 0"}, {"rel": "代入", "source": "a = - 2", "target": "- 7"}, {"rel": "被描述", "source": "- 7", "target": "49"}, {"rel": "属性描述", "source": "平方", "target": "49"}]}}
{"content": "To make the function $y = \\sqrt { 2 x + 4 }$ meaningful, what is the range of values for $x$?", "answer": "x \\ge - 2", "steps": "From the given condition, we have $2 x + 4 \\geq 0$, which implies $x \\geq - 2$.", "expr_cands": ["y = \\sqrt { 2 x + 4 }", "x", "y", "2 x + 4 \\ge 0", "- 2 \\le x", "x \\ge - 2"], "exprs": ["2 x + 4 \\ge 0", "x \\ge - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = \\sqrt { 2 x + 4 }"}, {"id": "2 x + 4 \\ge 0"}, {"id": "要使函数 $y = \\sqrt { 2 x + 4 }$ 有意义"}, {"id": "$x$ 的取值范围"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x \\ge - 2"}], "links": [{"rel": "被描述", "source": "y = \\sqrt { 2 x + 4 }", "target": "2 x + 4 \\ge 0"}, {"rel": "不等式方程求解", "source": "2 x + 4 \\ge 0", "target": "x \\ge - 2"}, {"rel": "限制性描述", "source": "要使函数 $y = \\sqrt { 2 x + 4 }$ 有意义", "target": "2 x + 4 \\ge 0"}, {"rel": "限制性描述", "source": "$x$ 的取值范围", "target": "2 x + 4 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "2 x + 4 \\ge 0"}]}}
{"content": "Given the parabola $y = \\frac { 1 } { 2 } x ^ 2$, if we keep the parabola unchanged and shift the coordinate system to the left by $3$ units, what is the equation of the resulting parabola?", "answer": "y = \\frac { 1 } { 2 } ( x - 3 ) ^ { 2 }", "steps": "The equation of the parabola $y = \\frac { 1 } { 2 } x ^ { 2 }$ shifted $3$ units to the left is given by the equation: $y = \\frac { 1 } { 2 } ( x - 3 ) ^ { 2 }$.", "expr_cands": ["y = \\frac { 1 } { 2 } x ^ { 2 }", "x", "y", "3", "y = \\frac { 1 } { 2 } ( x - 3 ) ^ { 2 }", "\\frac { x ^ { 2 }} { 2 } = \\frac { 1 } { 2 } ( x - 3 ) ^ { 2 }", "\\frac { x ^ { 2 }} { 2 }"], "exprs": ["y = \\frac { 1 } { 2 } ( x - 3 ) ^ { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3"}, {"id": "y = \\frac { 1 } { 2 } ( x - 3 ) ^ { 2 }"}, {"id": "y = \\frac { 1 } { 2 } x ^ { 2 }"}, {"id": "保持抛物线不动"}, {"id": "将坐标系向左平移 $3$ 个单位"}], "links": [{"rel": "被描述", "source": "3", "target": "y = \\frac { 1 } { 2 } ( x - 3 ) ^ { 2 }"}, {"rel": "被描述", "source": "y = \\frac { 1 } { 2 } x ^ { 2 }", "target": "y = \\frac { 1 } { 2 } ( x - 3 ) ^ { 2 }"}, {"rel": "限制性描述", "source": "保持抛物线不动", "target": "y = \\frac { 1 } { 2 } ( x - 3 ) ^ { 2 }"}, {"rel": "限制性描述", "source": "将坐标系向左平移 $3$ 个单位", "target": "y = \\frac { 1 } { 2 } ( x - 3 ) ^ { 2 }"}]}}
{"content": "To make the value of the fraction $\\frac { 3 x - 6 } { x + 1 }$ equal to zero, what is the value of $x$?", "answer": "x = 2", "steps": "From the given information, we have $3 x - 6 = 0$ and $x + 1 \\neq 0$. Solving for $x$, we get $x = 2$.", "expr_cands": ["\\frac { 3 x - 6 } { x + 1 }", "x", "3 x - 6 = 0", "x = 2", "x + 1 \\neq 0", "x \\neq - 1"], "exprs": ["3 x - 6 = 0", "x + 1 \\neq 0", "x = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 3 x - 6 } { x + 1 }"}, {"id": "3 x - 6 = 0"}, {"id": "要使分式 $\\frac { 3 x - 6 } { x + 1 }$ 的值等于零"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "x + 1 \\neq 0"}, {"id": "x = 2"}], "links": [{"rel": "被描述", "source": "\\frac { 3 x - 6 } { x + 1 }", "target": "3 x - 6 = 0"}, {"rel": "被描述", "source": "\\frac { 3 x - 6 } { x + 1 }", "target": "x + 1 \\neq 0"}, {"rel": "联立", "source": "3 x - 6 = 0", "target": "x = 2"}, {"rel": "限制性描述", "source": "要使分式 $\\frac { 3 x - 6 } { x + 1 }$ 的值等于零", "target": "3 x - 6 = 0"}, {"rel": "限制性描述", "source": "要使分式 $\\frac { 3 x - 6 } { x + 1 }$ 的值等于零", "target": "x + 1 \\neq 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "3 x - 6 = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x + 1 \\neq 0"}, {"rel": "联立", "source": "x + 1 \\neq 0", "target": "x = 2"}]}}
{"content": "If $\\frac { b } { a - b } = \\frac { 1 } { 4 }$, then the value of $\\frac { a } { b }$ is ____?", "answer": "5", "steps": "Since $\\frac { b } { a - b } = \\frac { 1 } { 4 }$, it follows that $a - b = 4 b$. Therefore, $a = 5 b$, and $\\frac { a } { b } = \\frac { 5 b } { b } = 5$.", "expr_cands": ["\\frac { b } { a - b } = \\frac { 1 } { 4 }", "a", "b", "\\frac { a } { b }", "a - b = 4 b", "a = 5 b", "5"], "exprs": ["a - b = 4 b", "a = 5 b", "5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { b } { a - b } = \\frac { 1 } { 4 }"}, {"id": "a - b = 4 b"}, {"id": "a = 5 b"}, {"id": "\\frac { a } { b }"}, {"id": "5"}], "links": [{"rel": "同乘除", "source": "\\frac { b } { a - b } = \\frac { 1 } { 4 }", "target": "a - b = 4 b"}, {"rel": "移项", "source": "a - b = 4 b", "target": "a = 5 b"}, {"rel": "代入", "source": "a = 5 b", "target": "5"}, {"rel": "被代入", "source": "\\frac { a } { b }", "target": "5"}]}}
{"content": "If the algebraic expressions $- a ^ { m } b ^ { 4 }$ and $3 ab ^ { n }$ are like terms, then $m + n$ = ____ ?", "answer": "5", "steps": "$\\because$ The algebraic expressions $- a ^ m b ^ 4$ and $3 ab ^ n$ are like terms, $\\therefore$ $m = 1$, $n = 4$, $\\therefore$ $m + n = 1 + 4 = 5$.", "expr_cands": ["- a ^ { m } b ^ { 4 }", "b", "m", "a", "3 ab ^ { n }", "n", "m + n", "m = 1", "n = 4", "5"], "exprs": ["m = 1", "n = 4", "5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- a ^ { m } b ^ { 4 }"}, {"id": "m = 1"}, {"id": "3 ab ^ { n }"}, {"id": "代数式 $- a ^ { m } b ^ { 4 }$ 和 $3 ab ^ { n }$ 是同类项"}, {"id": "n = 4"}, {"id": "m + n"}, {"id": "5"}], "links": [{"rel": "被描述", "source": "- a ^ { m } b ^ { 4 }", "target": "m = 1"}, {"rel": "被描述", "source": "- a ^ { m } b ^ { 4 }", "target": "n = 4"}, {"rel": "代入", "source": "m = 1", "target": "5"}, {"rel": "被描述", "source": "3 ab ^ { n }", "target": "m = 1"}, {"rel": "被描述", "source": "3 ab ^ { n }", "target": "n = 4"}, {"rel": "限制性描述", "source": "代数式 $- a ^ { m } b ^ { 4 }$ 和 $3 ab ^ { n }$ 是同类项", "target": "m = 1"}, {"rel": "限制性描述", "source": "代数式 $- a ^ { m } b ^ { 4 }$ 和 $3 ab ^ { n }$ 是同类项", "target": "n = 4"}, {"rel": "代入", "source": "n = 4", "target": "5"}, {"rel": "被代入", "source": "m + n", "target": "5"}]}}
{"content": "If $m + n = 7$, $2 n - p = 4$, then $m + 3 n - p$ = ____ ?", "answer": "11", "steps": "Since $m + n = 7$ and $2 n - p = 4$, we can conclude that $m + 3 n - p = ( m + n ) + ( 2 n - p ) = 7 + 4 = 11$.", "expr_cands": ["m + n = 7", "m", "n", "2 n - p = 4", "p", "m + 3 n - p", "( m + n ) + ( 2 n - p )", "11"], "exprs": ["( m + n ) + ( 2 n - p )", "11"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "m + 3 n - p"}, {"id": "( m + n ) + ( 2 n - p )"}, {"id": "11"}, {"id": "m + n = 7"}, {"id": "2 n - p = 4"}], "links": [{"rel": "展开", "source": "m + 3 n - p", "target": "( m + n ) + ( 2 n - p )"}, {"rel": "被代入", "source": "( m + n ) + ( 2 n - p )", "target": "11"}, {"rel": "代入", "source": "m + n = 7", "target": "11"}, {"rel": "代入", "source": "2 n - p = 4", "target": "11"}]}}
{"content": "The solution set of the inequality $1 - 2 x < 5$ for negative integers is ____ ?", "answer": "- 1", "steps": "By moving terms in the original inequality, we get $- 2 x < 4$. Dividing both sides of the inequality by $- 2$, we get $x > - 2$. Therefore, the set of negative integer solutions to the original inequality is ${ - 1 }$.", "expr_cands": ["1 - 2 x < 5", "x", "- 2 x < 4", "- 2 < x", "- 2", "x > - 2", "{ - 1 }"], "exprs": ["x > - 2", "{ - 1 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "1 - 2 x < 5"}, {"id": "x > - 2"}, {"id": "{ - 1 }"}, {"id": "不等式 $1 - 2 x < 5$ 的负整数解集"}], "links": [{"rel": "不等式方程求解", "source": "1 - 2 x < 5", "target": "x > - 2"}, {"rel": "被描述", "source": "x > - 2", "target": "{ - 1 }"}, {"rel": "限制性描述", "source": "不等式 $1 - 2 x < 5$ 的负整数解集", "target": "{ - 1 }"}]}}
{"content": "Given the equation $\\sqrt { 2 - x } + \\sqrt { x - 2 } - y + 3 = 0$ holds true in the real number range, the value of $x ^ y$ is ____?", "answer": "8", "steps": "According to the problem, we have $2 - x \\geq 0$ and $x - 2 \\geq 0$, which implies $x \\leq 2$ and $x \\geq 2$. Therefore, $x = 2$ and $y = 3$, so $x ^ y = 2 ^ 3 = 8$.", "expr_cands": ["\\sqrt { 2 - x } + \\sqrt { x - 2 } - y + 3 = 0", "y", "x", "x ^ { y }", "2 - x \\ge 0", "x \\le 2", "x - 2 \\ge 0", "2 \\le x", "x \\ge 2", "x = 2", "y = 3", "8"], "exprs": ["2 - x \\ge 0", "x - 2 \\ge 0", "x = 2", "y = 3", "8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { 2 - x } + \\sqrt { x - 2 } - y + 3 = 0"}, {"id": "2 - x \\ge 0"}, {"id": "在实数范围内等式 $\\sqrt { 2 - x } + \\sqrt { x - 2 } - y + 3 = 0$ 成立"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x - 2 \\ge 0"}, {"id": "x = 2"}, {"id": "y = 3"}, {"id": "x ^ { y }"}, {"id": "8"}], "links": [{"rel": "被描述", "source": "\\sqrt { 2 - x } + \\sqrt { x - 2 } - y + 3 = 0", "target": "2 - x \\ge 0"}, {"rel": "被描述", "source": "\\sqrt { 2 - x } + \\sqrt { x - 2 } - y + 3 = 0", "target": "x - 2 \\ge 0"}, {"rel": "联立", "source": "\\sqrt { 2 - x } + \\sqrt { x - 2 } - y + 3 = 0", "target": "y = 3"}, {"rel": "联立", "source": "2 - x \\ge 0", "target": "x = 2"}, {"rel": "限制性描述", "source": "在实数范围内等式 $\\sqrt { 2 - x } + \\sqrt { x - 2 } - y + 3 = 0$ 成立", "target": "2 - x \\ge 0"}, {"rel": "限制性描述", "source": "在实数范围内等式 $\\sqrt { 2 - x } + \\sqrt { x - 2 } - y + 3 = 0$ 成立", "target": "x - 2 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "2 - x \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 2 \\ge 0"}, {"rel": "联立", "source": "x - 2 \\ge 0", "target": "x = 2"}, {"rel": "联立", "source": "x = 2", "target": "y = 3"}, {"rel": "代入", "source": "x = 2", "target": "8"}, {"rel": "代入", "source": "y = 3", "target": "8"}, {"rel": "被代入", "source": "x ^ { y }", "target": "8"}]}}
{"content": "Given that the equation $2 x + a = x - 1$ has a solution of $- 4$, what is the value of $a$?", "answer": "3", "steps": "Substituting $x = - 4$ into the equation, we get $- 8 + a = - 4 - 1$, which yields $a = 3$.", "expr_cands": ["x", "2 x + a = x - 1", "a", "- 4", "x = - 4", "- 8 + a = - 4 - 1", "a = 3"], "exprs": ["x = - 4", "- 8 + a = - 4 - 1", "a = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 4"}, {"id": "x = - 4"}, {"id": "x"}, {"id": "2 x + a = x - 1"}, {"id": "关于 $x$ 的方程 $2 x + a = x - 1$ 的解为 $- 4$"}, {"id": "- 8 + a = - 4 - 1"}, {"id": "a = 3"}], "links": [{"rel": "被描述", "source": "- 4", "target": "x = - 4"}, {"rel": "代入", "source": "x = - 4", "target": "- 8 + a = - 4 - 1"}, {"rel": "被描述", "source": "x", "target": "x = - 4"}, {"rel": "被描述", "source": "2 x + a = x - 1", "target": "x = - 4"}, {"rel": "被代入", "source": "2 x + a = x - 1", "target": "- 8 + a = - 4 - 1"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $2 x + a = x - 1$ 的解为 $- 4$", "target": "x = - 4"}, {"rel": "等式方程求解", "source": "- 8 + a = - 4 - 1", "target": "a = 3"}]}}
{"content": "The new function obtained by translating the line $y = 2 x - 1$ one unit to the left is ____?", "answer": "y = 2 x + 1", "steps": "According to the principle of adding to the left and subtracting from the right, we know that the equation of the line obtained by shifting the line $y = 2 x - 1$ one unit to the left is $y = 2 ( x + 1 ) - 1$, which is $y = 2 x + 1$.", "expr_cands": ["y = 2 x - 1", "y", "x", "1", "y = 2 ( x + 1 ) - 1", "2 x - 1 = 2 ( x + 1 ) - 1", "2 x + 1"], "exprs": ["y = 2 ( x + 1 ) - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = 2 x - 1"}, {"id": "y = 2 ( x + 1 ) - 1"}, {"id": "将直线 $y = 2 x - 1$ 向左平移 $1$ 个单位后得到的新函数"}], "links": [{"rel": "被描述", "source": "y = 2 x - 1", "target": "y = 2 ( x + 1 ) - 1"}, {"rel": "限制性描述", "source": "将直线 $y = 2 x - 1$ 向左平移 $1$ 个单位后得到的新函数", "target": "y = 2 ( x + 1 ) - 1"}]}}
{"content": "When $x = 5$, the value of the polynomial $ax ^ 3 + x - 2$ is $10$. What is the value of the polynomial $ax ^ 3 + x - 2$ when $x = - 5$?", "answer": "- 14", "steps": "Substituting $x = 5$ into $a { x } ^ 3 + x - 2 = 10$ yields: $125 a + 5 - 2 = 10$, $125 a + 5 = 12$, $\\therefore - 125 a - 5 = - 12$. Substituting $x = - 5$ into $a { x } ^ 3 + x - 2$ yields: $- 125 a - 5 - 2 = - 12 - 2 = - 14$.", "expr_cands": ["x = 5", "x", "ax ^ { 3 } + x - 2", "a", "10", "x = - 5", "a { x } ^ { 3 } + x - 2 = 10", "125 a - 2 + 5 = 10", "125 a + 5 - 2 = 10", "a = \\frac { 7 } { 125 }", "125 a + 5 = 12", "- 125 a - 5 = - 12", "a { x } ^ { 3 } + x - 2", "- 125 a - 5 - 2", "- 14"], "exprs": ["125 a + 5 - 2 = 10", "a = \\frac { 7 } { 125 }", "- 14"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x = 5"}, {"id": "125 a + 5 - 2 = 10"}, {"id": "ax ^ { 3 } + x - 2"}, {"id": "整式 $ax ^ { 3 } + x - 2$ 的值是 $10$"}, {"id": "当 $x = 5$ 时"}, {"id": "a = \\frac { 7 } { 125 }"}, {"id": "x = - 5"}, {"id": "- 14"}, {"id": "a { x } ^ { 3 } + x - 2"}], "links": [{"rel": "被描述", "source": "x = 5", "target": "125 a + 5 - 2 = 10"}, {"rel": "等式方程求解", "source": "125 a + 5 - 2 = 10", "target": "a = \\frac { 7 } { 125 }"}, {"rel": "被描述", "source": "ax ^ { 3 } + x - 2", "target": "125 a + 5 - 2 = 10"}, {"rel": "限制性描述", "source": "整式 $ax ^ { 3 } + x - 2$ 的值是 $10$", "target": "125 a + 5 - 2 = 10"}, {"rel": "限制性描述", "source": "当 $x = 5$ 时", "target": "125 a + 5 - 2 = 10"}, {"rel": "代入", "source": "a = \\frac { 7 } { 125 }", "target": "- 14"}, {"rel": "代入", "source": "x = - 5", "target": "- 14"}, {"rel": "被代入", "source": "a { x } ^ { 3 } + x - 2", "target": "- 14"}]}}
{"content": "If $- 5 x ^ { a } yz ^ { b }$ and $2 x ^ { 3 } y ^ { c } z ^ { 2 }$ are like terms, then the value of $abc$ is ____?", "answer": "6", "steps": "Because $- 5 x ^ { a } yz ^ { b }$ and $2 x ^ { 3 } y ^ { c } z ^ { 2 }$ are like terms, therefore $a = 3$, $b = 2$, $c = 1$. Therefore, $abc = 3 * 2 * 1 = 6$.", "expr_cands": ["- 5 x ^ { a } yz ^ { b }", "b", "y", "x", "a", "z", "2 x ^ { 3 } y ^ { c } z ^ { 2 }", "c", "abc", "a = 3", "b = 2", "c = 1", "6"], "exprs": ["a = 3", "b = 2", "c = 1", "6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 5 x ^ { a } yz ^ { b }"}, {"id": "a = 3"}, {"id": "2 x ^ { 3 } y ^ { c } z ^ { 2 }"}, {"id": "$- 5 x ^ { a } yz ^ { b }$ 与 $2 x ^ { 3 } y ^ { c } z ^ { 2 }$ 是同类项"}, {"id": "b = 2"}, {"id": "c = 1"}, {"id": "abc"}, {"id": "6"}], "links": [{"rel": "被描述", "source": "- 5 x ^ { a } yz ^ { b }", "target": "a = 3"}, {"rel": "被描述", "source": "- 5 x ^ { a } yz ^ { b }", "target": "b = 2"}, {"rel": "被描述", "source": "- 5 x ^ { a } yz ^ { b }", "target": "c = 1"}, {"rel": "代入", "source": "a = 3", "target": "6"}, {"rel": "被描述", "source": "2 x ^ { 3 } y ^ { c } z ^ { 2 }", "target": "a = 3"}, {"rel": "被描述", "source": "2 x ^ { 3 } y ^ { c } z ^ { 2 }", "target": "b = 2"}, {"rel": "被描述", "source": "2 x ^ { 3 } y ^ { c } z ^ { 2 }", "target": "c = 1"}, {"rel": "限制性描述", "source": "$- 5 x ^ { a } yz ^ { b }$ 与 $2 x ^ { 3 } y ^ { c } z ^ { 2 }$ 是同类项", "target": "a = 3"}, {"rel": "限制性描述", "source": "$- 5 x ^ { a } yz ^ { b }$ 与 $2 x ^ { 3 } y ^ { c } z ^ { 2 }$ 是同类项", "target": "b = 2"}, {"rel": "限制性描述", "source": "$- 5 x ^ { a } yz ^ { b }$ 与 $2 x ^ { 3 } y ^ { c } z ^ { 2 }$ 是同类项", "target": "c = 1"}, {"rel": "代入", "source": "b = 2", "target": "6"}, {"rel": "代入", "source": "c = 1", "target": "6"}, {"rel": "被代入", "source": "abc", "target": "6"}]}}
{"content": "If $16 x ^ { m } y ^ { 5 }$ and $x ^ { 2 } y ^ { n + 1 }$ are similar terms, then the value of $2 m + n$ is ____?", "answer": "8", "steps": "Since the monomials $16 x ^ m y ^ 5$ and $x ^ 2 y ^ { n + 1 }$ are like terms, it follows that $m = 2$ and $n + 1 = 5$, so $n = 4$. Therefore, $2 m + n = 4 + 4 = 8$.", "expr_cands": ["16 x ^ { m } y ^ { 5 }", "y", "m", "x", "x ^ { 2 } y ^ { n + 1 }", "n", "2 m + n", "m = 2", "n + 1 = 5", "n = 4", "8"], "exprs": ["m = 2", "n + 1 = 5", "n = 4", "8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "16 x ^ { m } y ^ { 5 }"}, {"id": "m = 2"}, {"id": "x ^ { 2 } y ^ { n + 1 }"}, {"id": "$16 x ^ { m } y ^ { 5 }$ 和 $x ^ { 2 } y ^ { n + 1 }$ 是同类项"}, {"id": "n + 1 = 5"}, {"id": "n = 4"}, {"id": "2 m + n"}, {"id": "8"}], "links": [{"rel": "被描述", "source": "16 x ^ { m } y ^ { 5 }", "target": "m = 2"}, {"rel": "被描述", "source": "16 x ^ { m } y ^ { 5 }", "target": "n + 1 = 5"}, {"rel": "代入", "source": "m = 2", "target": "8"}, {"rel": "被描述", "source": "x ^ { 2 } y ^ { n + 1 }", "target": "m = 2"}, {"rel": "被描述", "source": "x ^ { 2 } y ^ { n + 1 }", "target": "n + 1 = 5"}, {"rel": "限制性描述", "source": "$16 x ^ { m } y ^ { 5 }$ 和 $x ^ { 2 } y ^ { n + 1 }$ 是同类项", "target": "m = 2"}, {"rel": "限制性描述", "source": "$16 x ^ { m } y ^ { 5 }$ 和 $x ^ { 2 } y ^ { n + 1 }$ 是同类项", "target": "n + 1 = 5"}, {"rel": "等式方程求解", "source": "n + 1 = 5", "target": "n = 4"}, {"rel": "代入", "source": "n = 4", "target": "8"}, {"rel": "被代入", "source": "2 m + n", "target": "8"}]}}
{"content": "Let $a$ be the integer part of $2 + \\sqrt { 6 }$, and let $b$ be the decimal part of $2 - \\sqrt { 6 }$. Find $a + b$.", "answer": "6 - \\sqrt { 6 }", "steps": "Let $a$ be the integer part of $2 + \\sqrt { 6 }$, and let $b$ be the decimal part of $2 - \\sqrt { 6 }$. Given that $a = 4$ and $b = 2 - \\sqrt { 6 }$, we have $a + b = 4 + 2 - \\sqrt { 6 } = 6 - \\sqrt { 6 }$.", "expr_cands": ["a", "2 + \\sqrt { 6 }", "b", "2 - \\sqrt { 6 }", "a + b", "a = 4", "b = 2 - \\sqrt { 6 }", "6 - \\sqrt { 6 }"], "exprs": ["a = 4", "b = 2 - \\sqrt { 6 }", "6 - \\sqrt { 6 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a = 4"}, {"id": "设 $a$ 是 $2 + \\sqrt { 6 }$ 的整数部分"}, {"id": "b"}, {"id": "b = 2 - \\sqrt { 6 }"}, {"id": "$b$ 是 $2 - \\sqrt { 6 }$ 的小数部分"}, {"id": "a + b"}, {"id": "6 - \\sqrt { 6 }"}], "links": [{"rel": "被描述", "source": "a", "target": "a = 4"}, {"rel": "代入", "source": "a = 4", "target": "6 - \\sqrt { 6 }"}, {"rel": "限制性描述", "source": "设 $a$ 是 $2 + \\sqrt { 6 }$ 的整数部分", "target": "a = 4"}, {"rel": "被描述", "source": "b", "target": "b = 2 - \\sqrt { 6 }"}, {"rel": "代入", "source": "b = 2 - \\sqrt { 6 }", "target": "6 - \\sqrt { 6 }"}, {"rel": "限制性描述", "source": "$b$ 是 $2 - \\sqrt { 6 }$ 的小数部分", "target": "b = 2 - \\sqrt { 6 }"}, {"rel": "被代入", "source": "a + b", "target": "6 - \\sqrt { 6 }"}]}}
{"content": "The coefficient of $x ^ 2$ in the result of $( 1 + x ) ( 2 x ^ 2 + ax + 1 )$ is $- 2$. Find the value of $a$.", "answer": "- 4", "steps": "$( 1 + x ) ( 2 x ^ 2 + ax + 1 ) = 2 x ^ 2 + ax + 1 + 2 x ^ 3 + ax ^ 2 + x = 2 x ^ 3 + ( 2 + a ) x ^ 2 + ( a + 1 ) x + 1$. Because the coefficient of $x ^ 2$ term is $- 2$, therefore $2 + a = - 2$, therefore $a = - 4$.", "expr_cands": ["( 1 + x ) ( 2 { x } ^ { 2 } + ax + 1 )", "a", "x", "x ^ { 2 }", "- 2", "( 1 + x ) ( 2 x ^ { 2 } + ax + 1 )", "2 x ^ { 3 } + ( 2 + a ) x ^ { 2 } + ( a + 1 ) x + 1", "2 + a = - 2", "a = - 4"], "exprs": ["2 x ^ { 3 } + ( 2 + a ) x ^ { 2 } + ( a + 1 ) x + 1", "2 + a = - 2", "a = - 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( 1 + x ) ( 2 x ^ { 2 } + ax + 1 )"}, {"id": "2 x ^ { 3 } + ( 2 + a ) x ^ { 2 } + ( a + 1 ) x + 1"}, {"id": "x ^ { 2 }"}, {"id": "2 + a = - 2"}, {"id": "- 2"}, {"id": "$( 1 + x ) ( 2 { x } ^ { 2 } + ax + 1 )$ 的结果中 $x ^ { 2 }$ 的项的系数为 $- 2$"}, {"id": "a = - 4"}], "links": [{"rel": "提取因式", "source": "( 1 + x ) ( 2 x ^ { 2 } + ax + 1 )", "target": "2 x ^ { 3 } + ( 2 + a ) x ^ { 2 } + ( a + 1 ) x + 1"}, {"rel": "被描述", "source": "2 x ^ { 3 } + ( 2 + a ) x ^ { 2 } + ( a + 1 ) x + 1", "target": "2 + a = - 2"}, {"rel": "提取因式参考", "source": "x ^ { 2 }", "target": "2 x ^ { 3 } + ( 2 + a ) x ^ { 2 } + ( a + 1 ) x + 1"}, {"rel": "等式方程求解", "source": "2 + a = - 2", "target": "a = - 4"}, {"rel": "被描述", "source": "- 2", "target": "2 + a = - 2"}, {"rel": "限制性描述", "source": "$( 1 + x ) ( 2 { x } ^ { 2 } + ax + 1 )$ 的结果中 $x ^ { 2 }$ 的项的系数为 $- 2$", "target": "2 + a = - 2"}]}}
{"content": "The number of negative integer solutions of the inequality $5 x + 16 > 0$ is _____.", "answer": "3", "steps": "The solution set of the inequality is $x > - ( 3 * 5 + 1 / 5 )$, so the negative integer solutions of the inequality $5 x + 16 > 0$ are $- 3$, $- 2$, $- 1$, totaling $3$.", "expr_cands": ["5 x + 16 > 0", "x", "x > - ( 3 * 5 + 1 / 5 )", "- \\frac { 16 } { 5 } < x", "- 3", "- 2", "- 1", "3"], "exprs": ["- \\frac { 16 } { 5 } < x", "- 3", "- 2", "- 1", "3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5 x + 16 > 0"}, {"id": "- \\frac { 16 } { 5 } < x"}, {"id": "- 3"}, {"id": "不等式 $5 x + 16 > 0$ 的负整数解为 : $- 3$ , $- 2$ , $- 1$"}, {"id": "- 2"}, {"id": "- 1"}, {"id": "3"}, {"id": "不等式 $5 x + 16 > 0$ 的负整数解有个"}], "links": [{"rel": "计算", "source": "5 x + 16 > 0", "target": "- \\frac { 16 } { 5 } < x"}, {"rel": "被描述", "source": "- \\frac { 16 } { 5 } < x", "target": "- 3"}, {"rel": "被描述", "source": "- \\frac { 16 } { 5 } < x", "target": "- 2"}, {"rel": "被描述", "source": "- \\frac { 16 } { 5 } < x", "target": "- 1"}, {"rel": "被描述", "source": "- 3", "target": "3"}, {"rel": "限制性描述", "source": "不等式 $5 x + 16 > 0$ 的负整数解为 : $- 3$ , $- 2$ , $- 1$", "target": "- 3"}, {"rel": "限制性描述", "source": "不等式 $5 x + 16 > 0$ 的负整数解为 : $- 3$ , $- 2$ , $- 1$", "target": "- 2"}, {"rel": "限制性描述", "source": "不等式 $5 x + 16 > 0$ 的负整数解为 : $- 3$ , $- 2$ , $- 1$", "target": "- 1"}, {"rel": "被描述", "source": "- 2", "target": "3"}, {"rel": "被描述", "source": "- 1", "target": "3"}, {"rel": "限制性描述", "source": "不等式 $5 x + 16 > 0$ 的负整数解有个", "target": "3"}]}}
{"content": "When $x$ = ____ ?, the polynomial $3 - \\frac { 11 } { 2 } x$ is equal to the polynomial $\\frac { 4 } { 3 } - 8 x$.", "answer": "- \\frac { 2 } { 3 }", "steps": "According to the problem, we have $3 - \\frac { 11 } { 2 } x = \\frac { 4 } { 3 } - 8 x$. Multiplying both sides by the least common multiple of the denominators, we get $18 - 33 x = 8 - 48 x$. Rearranging and combining like terms, we get $15 x = - 10$, so $x = - \\frac { 2 } { 3 }$.", "expr_cands": ["x", "3 - \\frac { 11 } { 2 } x", "\\frac { 4 } { 3 } - 8 x", "3 - \\frac { 11 } { 2 } x = \\frac { 4 } { 3 } - 8 x", "x = - \\frac { 2 } { 3 }", "18 - 33 x = 8 - 48 x", "15 x = - 10"], "exprs": ["3 - \\frac { 11 } { 2 } x = \\frac { 4 } { 3 } - 8 x", "x = - \\frac { 2 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 - \\frac { 11 } { 2 } x"}, {"id": "3 - \\frac { 11 } { 2 } x = \\frac { 4 } { 3 } - 8 x"}, {"id": "\\frac { 4 } { 3 } - 8 x"}, {"id": "多项式 $3 - \\frac { 11 } { 2 } x$ 与多项式 $\\frac { 4 } { 3 } - 8 x$ 的值相"}, {"id": "x = - \\frac { 2 } { 3 }"}], "links": [{"rel": "被描述", "source": "3 - \\frac { 11 } { 2 } x", "target": "3 - \\frac { 11 } { 2 } x = \\frac { 4 } { 3 } - 8 x"}, {"rel": "等式方程求解", "source": "3 - \\frac { 11 } { 2 } x = \\frac { 4 } { 3 } - 8 x", "target": "x = - \\frac { 2 } { 3 }"}, {"rel": "被描述", "source": "\\frac { 4 } { 3 } - 8 x", "target": "3 - \\frac { 11 } { 2 } x = \\frac { 4 } { 3 } - 8 x"}, {"rel": "限制性描述", "source": "多项式 $3 - \\frac { 11 } { 2 } x$ 与多项式 $\\frac { 4 } { 3 } - 8 x$ 的值相", "target": "3 - \\frac { 11 } { 2 } x = \\frac { 4 } { 3 } - 8 x"}]}}
{"content": "The monomial that is the product of $- 3 x ^ 2 y$ and equals $9 x ^ 6 y ^ 3$ is ____?", "answer": "- 3 x ^ { 4 } y ^ { 2 }", "steps": "From the given problem, we have $9 x ^ { 6 } y ^ { 3 } \\div ( - 3 x ^ { 2 } y ) = - 3 x ^ { 4 } y ^ { 2 }$. Therefore, the monomial that is the product of $- 3 x ^ { 2 } y$ and equals $9 x ^ { 6 } y ^ { 3 }$ is $- 3 x ^ { 4 } y ^ { 2 }$.", "expr_cands": ["- 3 x ^ { 2 } y", "y", "x", "9 x ^ { 6 } y ^ { 3 }", "9 x ^ { 6 } y ^ { 3 } \\div ( - 3 x ^ { 2 } y )", "- 3 x ^ { 4 } y ^ { 2 }"], "exprs": ["9 x ^ { 6 } y ^ { 3 } \\div ( - 3 x ^ { 2 } y )", "- 3 x ^ { 4 } y ^ { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 3 x ^ { 2 } y"}, {"id": "9 x ^ { 6 } y ^ { 3 } \\div ( - 3 x ^ { 2 } y )"}, {"id": "9 x ^ { 6 } y ^ { 3 }"}, {"id": "$- 3 x ^ { 2 } y$ 的乘积是 $9 x ^ { 6 } y ^ { 3 }$ 的单项式是 $- 3 x ^ { 4 } y ^ { 2 }$"}, {"id": "- 3 x ^ { 4 } y ^ { 2 }"}], "links": [{"rel": "被描述", "source": "- 3 x ^ { 2 } y", "target": "9 x ^ { 6 } y ^ { 3 } \\div ( - 3 x ^ { 2 } y )"}, {"rel": "计算", "source": "9 x ^ { 6 } y ^ { 3 } \\div ( - 3 x ^ { 2 } y )", "target": "- 3 x ^ { 4 } y ^ { 2 }"}, {"rel": "被描述", "source": "9 x ^ { 6 } y ^ { 3 }", "target": "9 x ^ { 6 } y ^ { 3 } \\div ( - 3 x ^ { 2 } y )"}, {"rel": "限制性描述", "source": "$- 3 x ^ { 2 } y$ 的乘积是 $9 x ^ { 6 } y ^ { 3 }$ 的单项式是 $- 3 x ^ { 4 } y ^ { 2 }$", "target": "9 x ^ { 6 } y ^ { 3 } \\div ( - 3 x ^ { 2 } y )"}]}}
{"content": "If the operation result of $( x ^ { 3 } + ax ^ { 2 } - x ) \\cdot ( - 8 x ^ { 4 } )$ does not contain the term $x ^ { 6 }$, then the value of $a$ should be ____?", "answer": "0", "steps": "$( x ^ 3 + ax ^ 2 - x ) \\times ( - 8 x ^ 4 ) = - 8 x ^ 7 - 8 ax ^ 6 + 8 x ^ 5$, because there is no term containing $x ^ 6$ in the result of the operation, therefore $- 8 a = 0$, which gives us $a = 0$.", "expr_cands": ["( x ^ { 3 } + ax ^ { 2 } - x ) \\cdot ( - 8 x ^ { 4 } )", "a", "x", "x ^ { 6 }", "( x ^ { 3 } + ax ^ { 2 } - x ) \\times ( - 8 x ^ { 4 } )", "- 8 x ^ { 7 } - 8 ax ^ { 6 } + 8 x ^ { 5 }", "- 8 a = 0", "a = 0"], "exprs": ["- 8 x ^ { 7 } - 8 ax ^ { 6 } + 8 x ^ { 5 }", "- 8 a = 0", "a = 0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( x ^ { 3 } + ax ^ { 2 } - x ) \\cdot ( - 8 x ^ { 4 } )"}, {"id": "- 8 x ^ { 7 } - 8 ax ^ { 6 } + 8 x ^ { 5 }"}, {"id": "- 8 a = 0"}, {"id": "要使 $( x ^ { 3 } + ax ^ { 2 } - x ) \\cdot ( - 8 x ^ { 4 } )$ 的运算结果中不含 $x ^ { 6 }$ 的项"}, {"id": "a = 0"}], "links": [{"rel": "展开", "source": "( x ^ { 3 } + ax ^ { 2 } - x ) \\cdot ( - 8 x ^ { 4 } )", "target": "- 8 x ^ { 7 } - 8 ax ^ { 6 } + 8 x ^ { 5 }"}, {"rel": "被描述", "source": "- 8 x ^ { 7 } - 8 ax ^ { 6 } + 8 x ^ { 5 }", "target": "- 8 a = 0"}, {"rel": "等式方程求解", "source": "- 8 a = 0", "target": "a = 0"}, {"rel": "限制性描述", "source": "要使 $( x ^ { 3 } + ax ^ { 2 } - x ) \\cdot ( - 8 x ^ { 4 } )$ 的运算结果中不含 $x ^ { 6 }$ 的项", "target": "- 8 a = 0"}]}}
{"content": "When $x = - 2$, the value of the algebraic expression $kx + 5$ is $- 1$. What is the value of $k$?", "answer": "3", "steps": "When $x = - 2$, therefore $- 2 k + 5 = - 1$, therefore $k = 3$.", "expr_cands": ["x = - 2", "x", "kx + 5", "k", "- 1", "- 2 k + 5 = - 1", "k = 3"], "exprs": ["- 2 k + 5 = - 1", "k = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "kx + 5"}, {"id": "- 2 k + 5 = - 1"}, {"id": "当 $x = - 2$ 时"}, {"id": "代数式 $kx + 5$ 的值为 $- 1$"}, {"id": "k = 3"}], "links": [{"rel": "被描述", "source": "kx + 5", "target": "- 2 k + 5 = - 1"}, {"rel": "等式方程求解", "source": "- 2 k + 5 = - 1", "target": "k = 3"}, {"rel": "限制性描述", "source": "当 $x = - 2$ 时", "target": "- 2 k + 5 = - 1"}, {"rel": "限制性描述", "source": "代数式 $kx + 5$ 的值为 $- 1$", "target": "- 2 k + 5 = - 1"}]}}
{"content": "Regarding the polynomial in $x$, $5 x ^ 3 - kx ^ 2 - 1 + 2 x ^ 2 - x$, if it does not contain the term $x ^ 2$, then the value of $k$ is ____?", "answer": "2", "steps": "From the given information, we have $- k + 2 = 0$, which implies $k = 2$.", "expr_cands": ["x", "5 x ^ { 3 } - kx ^ { 2 } - 1 + 2 x ^ { 2 } - x", "k", "x ^ { 2 }", "- k + 2 = 0", "k = 2"], "exprs": ["- k + 2 = 0", "k = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5 x ^ { 3 } - kx ^ { 2 } - 1 + 2 x ^ { 2 } - x"}, {"id": "- k + 2 = 0"}, {"id": "关于 $x$ 的多项式 $5 x ^ { 3 } - kx ^ { 2 } - 1 + 2 x ^ { 2 } - x$ 中"}, {"id": "不含 $x ^ { 2 }$ 项"}, {"id": "k = 2"}], "links": [{"rel": "被描述", "source": "5 x ^ { 3 } - kx ^ { 2 } - 1 + 2 x ^ { 2 } - x", "target": "- k + 2 = 0"}, {"rel": "等式方程求解", "source": "- k + 2 = 0", "target": "k = 2"}, {"rel": "限制性描述", "source": "关于 $x$ 的多项式 $5 x ^ { 3 } - kx ^ { 2 } - 1 + 2 x ^ { 2 } - x$ 中", "target": "- k + 2 = 0"}, {"rel": "限制性描述", "source": "不含 $x ^ { 2 }$ 项", "target": "- k + 2 = 0"}]}}
{"content": "If the polynomial $4 xy ^ 3 - 2 nx ^ 2 - 3 xy + 6 x ^ 2$ in terms of $x$ and $y$ does not contain the term $x ^ 2$, then $n$ = ____?", "answer": "3", "steps": "$4 xy ^ { 3 } - 2 nx ^ { 2 } - 3 xy + 6 x ^ { 2 } = 4 xy ^ { 3 } + ( 6 - 2 n ) x ^ { 2 } - 3 xy$ . Since the polynomial $4 xy ^ { 3 } - 2 nx ^ { 2 } - 3 xy + 6 x ^ { 2 }$ does not contain a term with $x ^ { 2 }$, it follows that $6 - 2 n = 0$, and thus $n = 3$.", "expr_cands": ["x", "y", "4 xy ^ { 3 } - 2 nx ^ { 2 } - 3 xy + 6 x ^ { 2 }", "n", "x ^ { 2 }", "4 xy ^ { 3 } + ( 6 - 2 n ) x ^ { 2 } - 3 xy", "6 - 2 n = 0", "n = 3"], "exprs": ["4 xy ^ { 3 } + ( 6 - 2 n ) x ^ { 2 } - 3 xy", "6 - 2 n = 0", "n = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 xy ^ { 3 } - 2 nx ^ { 2 } - 3 xy + 6 x ^ { 2 }"}, {"id": "4 xy ^ { 3 } + ( 6 - 2 n ) x ^ { 2 } - 3 xy"}, {"id": "x ^ { 2 }"}, {"id": "6 - 2 n = 0"}, {"id": "关于 $x$ , $y$ 的多项式 $4 xy ^ { 3 } - 2 nx ^ { 2 } - 3 xy + 6 x ^ { 2 }$ 不含 $x ^ { 2 }$ 项"}, {"id": "n = 3"}], "links": [{"rel": "提取因式", "source": "4 xy ^ { 3 } - 2 nx ^ { 2 } - 3 xy + 6 x ^ { 2 }", "target": "4 xy ^ { 3 } + ( 6 - 2 n ) x ^ { 2 } - 3 xy"}, {"rel": "被描述", "source": "4 xy ^ { 3 } + ( 6 - 2 n ) x ^ { 2 } - 3 xy", "target": "6 - 2 n = 0"}, {"rel": "提取因式参考", "source": "x ^ { 2 }", "target": "4 xy ^ { 3 } + ( 6 - 2 n ) x ^ { 2 } - 3 xy"}, {"rel": "等式方程求解", "source": "6 - 2 n = 0", "target": "n = 3"}, {"rel": "限制性描述", "source": "关于 $x$ , $y$ 的多项式 $4 xy ^ { 3 } - 2 nx ^ { 2 } - 3 xy + 6 x ^ { 2 }$ 不含 $x ^ { 2 }$ 项", "target": "6 - 2 n = 0"}]}}
{"content": "If $3 m + 5$ and $m - 2$ are opposite in sign, then the reciprocal of $m$ is ____?", "answer": "- \\frac { 4 } { 3 }", "steps": "According to the problem, we have $3 m + 5 + m - 2 = 0$, $4 m = - 3$, and $m = - \\frac { 3 } { 4 }$. Therefore, the reciprocal of $m$ is $- \\frac { 4 } { 3 }$.", "expr_cands": ["3 m + 5", "m", "m - 2", "3 m + 5 + m - 2 = 0", "m = - \\frac { 3 } { 4 }", "4 m = - 3", "- \\frac { 4 } { 3 }"], "exprs": ["3 m + 5 + m - 2 = 0", "m = - \\frac { 3 } { 4 }", "- \\frac { 4 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 m + 5"}, {"id": "3 m + 5 + m - 2 = 0"}, {"id": "m - 2"}, {"id": "$3 m + 5$ 与 $m - 2$ 互为相反数"}, {"id": "m = - \\frac { 3 } { 4 }"}, {"id": "- \\frac { 4 } { 3 }"}, {"id": "$m$ 的倒数是 $- \\frac { 4 } { 3 }$"}], "links": [{"rel": "被描述", "source": "3 m + 5", "target": "3 m + 5 + m - 2 = 0"}, {"rel": "等式方程求解", "source": "3 m + 5 + m - 2 = 0", "target": "m = - \\frac { 3 } { 4 }"}, {"rel": "被描述", "source": "m - 2", "target": "3 m + 5 + m - 2 = 0"}, {"rel": "限制性描述", "source": "$3 m + 5$ 与 $m - 2$ 互为相反数", "target": "3 m + 5 + m - 2 = 0"}, {"rel": "被描述", "source": "m = - \\frac { 3 } { 4 }", "target": "- \\frac { 4 } { 3 }"}, {"rel": "限制性描述", "source": "$m$ 的倒数是 $- \\frac { 4 } { 3 }$", "target": "- \\frac { 4 } { 3 }"}]}}
{"content": "Given $| a + 1 | + | b + 3 | = 0$, what is the value of $ab$?", "answer": "3", "steps": "Because $| a + 1 | + | b + 3 | = 0$, therefore $a + 1 = 0$, $b + 3 = 0$, which leads to $a = - 1$, $b = - 3$. Thus, $ab = - 1 * ( - 3 ) = 3$.", "expr_cands": ["| a + 1 | + | b + 3 | = 0", "a", "b", "ab", "a + 1 = 0", "a = - 1", "b + 3 = 0", "b = - 3", "3"], "exprs": ["a + 1 = 0", "b + 3 = 0", "a = - 1", "b = - 3", "3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| a + 1 | + | b + 3 | = 0"}, {"id": "a + 1 = 0"}, {"id": "绝对值恒大于等于0"}, {"id": "b + 3 = 0"}, {"id": "a = - 1"}, {"id": "b = - 3"}, {"id": "ab"}, {"id": "3"}], "links": [{"rel": "被描述", "source": "| a + 1 | + | b + 3 | = 0", "target": "a + 1 = 0"}, {"rel": "被描述", "source": "| a + 1 | + | b + 3 | = 0", "target": "b + 3 = 0"}, {"rel": "等式方程求解", "source": "a + 1 = 0", "target": "a = - 1"}, {"rel": "限制性描述", "source": "绝对值恒大于等于0", "target": "a + 1 = 0"}, {"rel": "限制性描述", "source": "绝对值恒大于等于0", "target": "b + 3 = 0"}, {"rel": "等式方程求解", "source": "b + 3 = 0", "target": "b = - 3"}, {"rel": "代入", "source": "a = - 1", "target": "3"}, {"rel": "代入", "source": "b = - 3", "target": "3"}, {"rel": "被代入", "source": "ab", "target": "3"}]}}
{"content": "If $( x - 3 ) ^ { 2 } + \\sqrt { y + 2 } + | 3 - z | = 0$, then $2 x - y - z$ = ____ ?", "answer": "5", "steps": "According to the given information, we have $x - 3 = 0$, $y + 2 = 0$, $3 - z = 0$, $x = 3$, $y = - 2$, and $z = 3$. Therefore, $2 x - y - z = 5$.", "expr_cands": ["( x - 3 ) ^ { 2 } + \\sqrt { y + 2 } + | 3 - z | = 0", "x", "z", "y", "2 x - y - z", "x - 3 = 0", "x = 3", "y + 2 = 0", "y = - 2", "3 - z = 3", "z = 0", "z = 3", "2 x - y - z = 5", "2 x + 2 = 5", "2 x + 2", "5"], "exprs": ["x - 3 = 0", "y + 2 = 0", "z = 3", "x = 3", "y = - 2", "5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( x - 3 ) ^ { 2 } + \\sqrt { y + 2 } + | 3 - z | = 0"}, {"id": "x - 3 = 0"}, {"id": "多项式偶次方项恒大于等于0"}, {"id": "x = 3"}, {"id": "y + 2 = 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "y = - 2"}, {"id": "z = 3"}, {"id": "绝对值恒大于等于0"}, {"id": "2 x - y - z"}, {"id": "5"}], "links": [{"rel": "被描述", "source": "( x - 3 ) ^ { 2 } + \\sqrt { y + 2 } + | 3 - z | = 0", "target": "x - 3 = 0"}, {"rel": "被描述", "source": "( x - 3 ) ^ { 2 } + \\sqrt { y + 2 } + | 3 - z | = 0", "target": "y + 2 = 0"}, {"rel": "被描述", "source": "( x - 3 ) ^ { 2 } + \\sqrt { y + 2 } + | 3 - z | = 0", "target": "z = 3"}, {"rel": "等式方程求解", "source": "x - 3 = 0", "target": "x = 3"}, {"rel": "属性描述", "source": "多项式偶次方项恒大于等于0", "target": "x - 3 = 0"}, {"rel": "代入", "source": "x = 3", "target": "5"}, {"rel": "等式方程求解", "source": "y + 2 = 0", "target": "y = - 2"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "y + 2 = 0"}, {"rel": "代入", "source": "y = - 2", "target": "5"}, {"rel": "代入", "source": "z = 3", "target": "5"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "z = 3"}, {"rel": "被代入", "source": "2 x - y - z", "target": "5"}]}}
{"content": "The solution to the equation $\\frac { x } { x + 2 } + \\frac { 3 } { x + 2 } = 0$ is ____ ?", "answer": "x = - 3", "steps": "Going to the denominator, we get $x ^ 2 + 2 x + 3 x + 6 = 0$. Solving this, we get $x = - 2$ or $x = - 3$. After checking, we find that $x = - 2$ is an extraneous root, so the solution to the fractional equation is $x = - 3$.", "expr_cands": ["\\frac { x } { x + 2 } + \\frac { 3 } { x + 2 } = 0", "x", "x ^ { 2 } + 2 x + 3 x + 6 = 0", "x = - 3", "x = - 2"], "exprs": ["x = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x } { x + 2 } + \\frac { 3 } { x + 2 } = 0"}, {"id": "x = - 3"}], "links": [{"rel": "等式方程求解", "source": "\\frac { x } { x + 2 } + \\frac { 3 } { x + 2 } = 0", "target": "x = - 3"}]}}
{"content": "Given the line $y = ( k - 2 ) x + 3$ is parallel to the line $y = 3 x - 2$, what is the value of $k$?", "answer": "5", "steps": "$\\because$ The line $y = ( k - 2 ) x + 3$ is parallel to the line $y = 3 x - 2$, $\\therefore$ $k - 2 = 3$, $\\therefore$ $k = 5$.", "expr_cands": ["y = ( k - 2 ) x + 3", "k", "x", "y", "y = 3 x - 2", "x ( k - 2 ) + 3 = 3 x - 2", "k - 2 = 3", "k = 5"], "exprs": ["k - 2 = 3", "k = 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = ( k - 2 ) x + 3"}, {"id": "k - 2 = 3"}, {"id": "y = 3 x - 2"}, {"id": "直线 $y = ( k - 2 ) x + 3$ 与直线 $y = 3 x - 2$ 平行"}, {"id": "k = 5"}], "links": [{"rel": "被描述", "source": "y = ( k - 2 ) x + 3", "target": "k - 2 = 3"}, {"rel": "等式方程求解", "source": "k - 2 = 3", "target": "k = 5"}, {"rel": "被描述", "source": "y = 3 x - 2", "target": "k - 2 = 3"}, {"rel": "限制性描述", "source": "直线 $y = ( k - 2 ) x + 3$ 与直线 $y = 3 x - 2$ 平行", "target": "k - 2 = 3"}]}}
{"content": "If the opposite of $a + 1$ is $- 5$, then $a$ = ____ ?", "answer": "4", "steps": "$\\because$ The opposite of $a + 1$ is $- 5$, $\\therefore$ $a + 1 = 5$, $\\therefore$ $a = 4$.", "expr_cands": ["a + 1", "a", "- 5", "a + 1 = 5", "a = 4"], "exprs": ["a + 1 = 5", "a = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a + 1"}, {"id": "a + 1 = 5"}, {"id": "- 5"}, {"id": "$a + 1$ 的相反数是 $- 5$"}, {"id": "a = 4"}], "links": [{"rel": "被描述", "source": "a + 1", "target": "a + 1 = 5"}, {"rel": "等式方程求解", "source": "a + 1 = 5", "target": "a = 4"}, {"rel": "被描述", "source": "- 5", "target": "a + 1 = 5"}, {"rel": "限制性描述", "source": "$a + 1$ 的相反数是 $- 5$", "target": "a + 1 = 5"}]}}
{"content": "Given that the two square roots of a positive number are $a + 1$ and $a - 2$, what is the number?", "answer": "\\frac { 9 } { 4 }", "steps": "$\\because$ The square root of a positive number is $a + 1$ and $a - 2$, $\\therefore$ $a + 1 + a - 2 = 0$. Solving for $a$, we get $a = \\frac { 1 } { 2 }$. $\\therefore$ $a + 1 = \\frac { 3 } { 2 }$. $\\because$ $( \\frac { 3 } { 2 }) ^ 2 = \\frac { 9 } { 4 }$, $\\therefore$ the positive number is $\\frac { 9 } { 4 }$.", "expr_cands": ["a + 1", "a", "a - 2", "a + 1 + a - 2 = 0", "a = \\frac { 1 } { 2 }", "\\frac { 3 } { 2 }", "( \\frac { 3 } { 2 } ) ^ { 2 }", "\\frac { 9 } { 4 }"], "exprs": ["a + 1 + a - 2 = 0", "a = \\frac { 1 } { 2 }", "\\frac { 3 } { 2 }", "\\frac { 9 } { 4 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a + 1"}, {"id": "a + 1 + a - 2 = 0"}, {"id": "a - 2"}, {"id": "一个正数的两个平方根分别为 $a + 1$ , $a - 2$"}, {"id": "平方根互为相反数"}, {"id": "a = \\frac { 1 } { 2 }"}, {"id": "\\frac { 3 } { 2 }"}, {"id": "\\frac { 9 } { 4 }"}, {"id": "这个正数"}], "links": [{"rel": "被描述", "source": "a + 1", "target": "a + 1 + a - 2 = 0"}, {"rel": "被代入", "source": "a + 1", "target": "\\frac { 3 } { 2 }"}, {"rel": "被描述", "source": "a + 1", "target": "\\frac { 9 } { 4 }"}, {"rel": "等式方程求解", "source": "a + 1 + a - 2 = 0", "target": "a = \\frac { 1 } { 2 }"}, {"rel": "被描述", "source": "a - 2", "target": "a + 1 + a - 2 = 0"}, {"rel": "被描述", "source": "a - 2", "target": "\\frac { 9 } { 4 }"}, {"rel": "限制性描述", "source": "一个正数的两个平方根分别为 $a + 1$ , $a - 2$", "target": "a + 1 + a - 2 = 0"}, {"rel": "限制性描述", "source": "一个正数的两个平方根分别为 $a + 1$ , $a - 2$", "target": "\\frac { 9 } { 4 }"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "a + 1 + a - 2 = 0"}, {"rel": "代入", "source": "a = \\frac { 1 } { 2 }", "target": "\\frac { 3 } { 2 }"}, {"rel": "被描述", "source": "\\frac { 3 } { 2 }", "target": "\\frac { 9 } { 4 }"}, {"rel": "限制性描述", "source": "这个正数", "target": "\\frac { 9 } { 4 }"}]}}
{"content": "If $\\sqrt { 2 x - 4 }$ and $| y + 2 |$ are opposite in sign, then the value of ${ x } ^ { 2 } - 2 y$ is ____?", "answer": "8", "steps": "Because $\\sqrt { 2 x - 4 }$ is the opposite of $| y + 2 |$, therefore $\\sqrt { 2 x - 4 } + | y + 2 | = 0$, therefore $2 x - 4 = 0$, $y + 2 = 0$, solving for $x = 2$, $y = - 2$. Therefore, $x ^ { 2 } - 2 y = 2 ^ { 2 } - 2 * ( - 2 ) = 8$.", "expr_cands": ["\\sqrt { 2 x - 4 }", "x", "| y + 2 |", "y", "{ x } ^ { 2 } - 2 y", "\\sqrt { 2 x - 4 } + | y + 2 | = 0", "2 x - 4 = 0", "x = 2", "y + 2 = 0", "y = - 2", "x ^ { 2 } - 2 y", "8"], "exprs": ["\\sqrt { 2 x - 4 } + | y + 2 | = 0", "2 x - 4 = 0", "y + 2 = 0", "x = 2", "y = - 2", "8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { 2 x - 4 }"}, {"id": "\\sqrt { 2 x - 4 } + | y + 2 | = 0"}, {"id": "| y + 2 |"}, {"id": "$\\sqrt { 2 x - 4 }$ 与 $| y + 2 |$ 互为相反数"}, {"id": "2 x - 4 = 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "y + 2 = 0"}, {"id": "绝对值恒大于等于0"}, {"id": "x = 2"}, {"id": "y = - 2"}, {"id": "x ^ { 2 } - 2 y"}, {"id": "8"}], "links": [{"rel": "被描述", "source": "\\sqrt { 2 x - 4 }", "target": "\\sqrt { 2 x - 4 } + | y + 2 | = 0"}, {"rel": "被描述", "source": "\\sqrt { 2 x - 4 } + | y + 2 | = 0", "target": "2 x - 4 = 0"}, {"rel": "被描述", "source": "\\sqrt { 2 x - 4 } + | y + 2 | = 0", "target": "y + 2 = 0"}, {"rel": "被描述", "source": "| y + 2 |", "target": "\\sqrt { 2 x - 4 } + | y + 2 | = 0"}, {"rel": "限制性描述", "source": "$\\sqrt { 2 x - 4 }$ 与 $| y + 2 |$ 互为相反数", "target": "\\sqrt { 2 x - 4 } + | y + 2 | = 0"}, {"rel": "等式方程求解", "source": "2 x - 4 = 0", "target": "x = 2"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "2 x - 4 = 0"}, {"rel": "等式方程求解", "source": "y + 2 = 0", "target": "y = - 2"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "y + 2 = 0"}, {"rel": "代入", "source": "x = 2", "target": "8"}, {"rel": "代入", "source": "y = - 2", "target": "8"}, {"rel": "被代入", "source": "x ^ { 2 } - 2 y", "target": "8"}]}}
{"content": "If the monomial $3 x ^ { 2 n - 1 } y ^ { m + 1 }$ is a like term with $- 5 x ^ { n + 1 } y ^ { 2 }$, then the sum of these two monomials is ____?", "answer": "- 2 x ^ { 3 } y ^ { 2 }", "steps": "$\\because$ The monomials $3 x ^ { 2 n - 1 } y ^ { m + 1 }$ and $- 5 x ^ { n + 1 } y ^ { 2 }$ are like terms, $\\therefore$ $2 n - 1 = n + 1$, $m + 1 = 2$, $\\therefore$ $m = 1$, $n = 2$. Substituting into $3 x ^ { 2 n - 1 } y ^ { m + 1 } + ( - 5 x ^ { n + 1 } y ^ { 2 } ) = 3 x ^ { 3 } y ^ { 2 } + ( - 5 x ^ { 3 } y ^ { 2 } ) = - 2 x ^ { 3 } y ^ { 2 }$.", "expr_cands": ["3 x ^ { 2 n - 1 } y ^ { m + 1 }", "y", "n", "x", "m", "- 5 x ^ { n + 1 } y ^ { 2 }", "2 n - 1 = n + 1", "n = 2", "m + 1 = 2", "m = 1", "3 x ^ { 2 n - 1 } y ^ { m + 1 } + ( - 5 x ^ { n + 1 } y ^ { 2 } ) = - 2 x ^ { 3 } y ^ { 2 }", "3 x ^ { 2 n - 1 } y ^ { m + 1 } + ( - 5 x ^ { n + 1 } y ^ { 2 } )", "- 2 x ^ { 3 } y ^ { 2 }"], "exprs": ["2 n - 1 = n + 1", "m + 1 = 2", "n = 2", "m = 1", "- 2 x ^ { 3 } y ^ { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 x ^ { 2 n - 1 } y ^ { m + 1 }"}, {"id": "2 n - 1 = n + 1"}, {"id": "- 5 x ^ { n + 1 } y ^ { 2 }"}, {"id": "单项式 $3 x ^ { 2 n - 1 } y ^ { m + 1 }$ 与 $- 5 x ^ { n + 1 } y ^ { 2 }$ 是同类项"}, {"id": "m + 1 = 2"}, {"id": "m = 1"}, {"id": "n = 2"}, {"id": "- 2 x ^ { 3 } y ^ { 2 }"}, {"id": "这两个单项式的和"}], "links": [{"rel": "被描述", "source": "3 x ^ { 2 n - 1 } y ^ { m + 1 }", "target": "2 n - 1 = n + 1"}, {"rel": "被描述", "source": "3 x ^ { 2 n - 1 } y ^ { m + 1 }", "target": "m + 1 = 2"}, {"rel": "被描述", "source": "3 x ^ { 2 n - 1 } y ^ { m + 1 }", "target": "- 2 x ^ { 3 } y ^ { 2 }"}, {"rel": "等式方程求解", "source": "2 n - 1 = n + 1", "target": "n = 2"}, {"rel": "被描述", "source": "- 5 x ^ { n + 1 } y ^ { 2 }", "target": "2 n - 1 = n + 1"}, {"rel": "被描述", "source": "- 5 x ^ { n + 1 } y ^ { 2 }", "target": "m + 1 = 2"}, {"rel": "被描述", "source": "- 5 x ^ { n + 1 } y ^ { 2 }", "target": "- 2 x ^ { 3 } y ^ { 2 }"}, {"rel": "限制性描述", "source": "单项式 $3 x ^ { 2 n - 1 } y ^ { m + 1 }$ 与 $- 5 x ^ { n + 1 } y ^ { 2 }$ 是同类项", "target": "2 n - 1 = n + 1"}, {"rel": "限制性描述", "source": "单项式 $3 x ^ { 2 n - 1 } y ^ { m + 1 }$ 与 $- 5 x ^ { n + 1 } y ^ { 2 }$ 是同类项", "target": "m + 1 = 2"}, {"rel": "等式方程求解", "source": "m + 1 = 2", "target": "m = 1"}, {"rel": "被描述", "source": "m = 1", "target": "- 2 x ^ { 3 } y ^ { 2 }"}, {"rel": "被描述", "source": "n = 2", "target": "- 2 x ^ { 3 } y ^ { 2 }"}, {"rel": "限制性描述", "source": "这两个单项式的和", "target": "- 2 x ^ { 3 } y ^ { 2 }"}]}}
{"content": "If $- 1$ is a root of the equation $x ^ 2 + mx - 1 = 0$, then the value of $m$ is ____?", "answer": "0", "steps": "Substituting $x = - 1$ into the equation $x ^ 2 + mx - 1 = 0$, we get $( - 1 ) ^ { 2 } + ( - 1 ) m - 1 = 0$, which yields $m = 0$.", "expr_cands": ["- 1", "x ^ { 2 } + mx - 1 = 0", "m", "x", "x = - 1", "- m - 1 + 1 = 0", "( - 1 ) ^ { 2 } + ( - 1 ) m - 1 = 0", "m = 0"], "exprs": ["x = - 1", "( - 1 ) ^ { 2 } + ( - 1 ) m - 1 = 0", "m = 0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 1"}, {"id": "x = - 1"}, {"id": "x ^ { 2 } + mx - 1 = 0"}, {"id": "$- 1$ 是方程 $x ^ { 2 } + mx - 1 = 0$ 的一个根"}, {"id": "( - 1 ) ^ { 2 } + ( - 1 ) m - 1 = 0"}, {"id": "m = 0"}], "links": [{"rel": "被描述", "source": "- 1", "target": "x = - 1"}, {"rel": "代入", "source": "x = - 1", "target": "( - 1 ) ^ { 2 } + ( - 1 ) m - 1 = 0"}, {"rel": "被描述", "source": "x ^ { 2 } + mx - 1 = 0", "target": "x = - 1"}, {"rel": "被代入", "source": "x ^ { 2 } + mx - 1 = 0", "target": "( - 1 ) ^ { 2 } + ( - 1 ) m - 1 = 0"}, {"rel": "限制性描述", "source": "$- 1$ 是方程 $x ^ { 2 } + mx - 1 = 0$ 的一个根", "target": "x = - 1"}, {"rel": "等式方程求解", "source": "( - 1 ) ^ { 2 } + ( - 1 ) m - 1 = 0", "target": "m = 0"}]}}
{"content": "The parabola $y = - 3 x ^ 2$ after being translated $2$ units to the left becomes ____?", "answer": "y = - 3 ( x + 2 ) ^ { 2 }", "steps": "The parabola $y = - 3 x ^ { 2 }$ is translated $2$ units to the left to obtain the parabola $y = - 3 ( x + 2 ) ^ { 2 }$.", "expr_cands": ["y = - 3 x ^ { 2 }", "y", "x", "2", "y = - 3 ( x + 2 ) ^ { 2 }", "- 3 x ^ { 2 } = - 3 ( x + 2 ) ^ { 2 }", "- 3 x ^ { 2 }"], "exprs": ["y = - 3 ( x + 2 ) ^ { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2"}, {"id": "y = - 3 ( x + 2 ) ^ { 2 }"}, {"id": "y = - 3 x ^ { 2 }"}, {"id": "抛物线 $y = - 3 x ^ { 2 }$ 向左平移 $2$ 个单位后得到的抛物线"}], "links": [{"rel": "被描述", "source": "2", "target": "y = - 3 ( x + 2 ) ^ { 2 }"}, {"rel": "被描述", "source": "y = - 3 x ^ { 2 }", "target": "y = - 3 ( x + 2 ) ^ { 2 }"}, {"rel": "限制性描述", "source": "抛物线 $y = - 3 x ^ { 2 }$ 向左平移 $2$ 个单位后得到的抛物线", "target": "y = - 3 ( x + 2 ) ^ { 2 }"}]}}
{"content": "If the value of $\\frac { x ^ 2 - 4 } { x - 2 }$ is $0$, then the condition that $x$ satisfies is ____?", "answer": "x = - 2", "steps": "According to the problem, we have $x ^ 2 - 4 = 0$, and $x - 2 \\neq 0$. Solving for $x$, we get $x = - 2$.", "expr_cands": ["\\frac { x ^ { 2 } - 4 } { x - 2 }", "x", "0", "x ^ { 2 } - 4 = 0", "x = - 2", "x = 2", "x - 2 \\neq 0", "x \\neq 2"], "exprs": ["x ^ { 2 } - 4 = 0", "x - 2 \\neq 0", "x = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x ^ { 2 } - 4 } { x - 2 }"}, {"id": "x ^ { 2 } - 4 = 0"}, {"id": "$\\frac { x ^ { 2 } - 4 } { x - 2 }$ 的值为 $0$"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "x - 2 \\neq 0"}, {"id": "x = - 2"}], "links": [{"rel": "被描述", "source": "\\frac { x ^ { 2 } - 4 } { x - 2 }", "target": "x ^ { 2 } - 4 = 0"}, {"rel": "被描述", "source": "\\frac { x ^ { 2 } - 4 } { x - 2 }", "target": "x - 2 \\neq 0"}, {"rel": "联立", "source": "x ^ { 2 } - 4 = 0", "target": "x = - 2"}, {"rel": "限制性描述", "source": "$\\frac { x ^ { 2 } - 4 } { x - 2 }$ 的值为 $0$", "target": "x ^ { 2 } - 4 = 0"}, {"rel": "限制性描述", "source": "$\\frac { x ^ { 2 } - 4 } { x - 2 }$ 的值为 $0$", "target": "x - 2 \\neq 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x ^ { 2 } - 4 = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x - 2 \\neq 0"}, {"rel": "联立", "source": "x - 2 \\neq 0", "target": "x = - 2"}]}}
{"content": "When $x$ = ____ ?, the value of the fraction $\\frac { x - 5 } { 2 x + 3 }$ is zero.", "answer": "5", "steps": "According to the problem, we can obtain that $x - 5 = 0$ and $2 x + 3 \\neq 0$. Therefore, when $x = 5$, the value of the fraction $\\frac { x - 5 } { 2 x + 3 }$ is zero.", "expr_cands": ["x", "\\frac { x - 5 } { 2 x + 3 }", "x - 5 = 0", "x = 5", "2 x + 3 \\neq 0", "x \\neq - \\frac { 3 } { 2 }"], "exprs": ["x - 5 = 0", "2 x + 3 \\neq 0", "x = 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x - 5 } { 2 x + 3 }"}, {"id": "x - 5 = 0"}, {"id": "分式 $\\frac { x - 5 } { 2 x + 3 }$ 的值为零"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "2 x + 3 \\neq 0"}, {"id": "x = 5"}], "links": [{"rel": "被描述", "source": "\\frac { x - 5 } { 2 x + 3 }", "target": "x - 5 = 0"}, {"rel": "被描述", "source": "\\frac { x - 5 } { 2 x + 3 }", "target": "2 x + 3 \\neq 0"}, {"rel": "联立", "source": "x - 5 = 0", "target": "x = 5"}, {"rel": "限制性描述", "source": "分式 $\\frac { x - 5 } { 2 x + 3 }$ 的值为零", "target": "x - 5 = 0"}, {"rel": "限制性描述", "source": "分式 $\\frac { x - 5 } { 2 x + 3 }$ 的值为零", "target": "2 x + 3 \\neq 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x - 5 = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "2 x + 3 \\neq 0"}, {"rel": "联立", "source": "2 x + 3 \\neq 0", "target": "x = 5"}]}}
{"content": "The algebraic expression $\\sqrt { x + 3 }$ is meaningful, then the range of $x$ is ____?", "answer": "x \\ge - 3", "steps": "$\\because$ The algebraic expression $\\sqrt { x + 3 }$ is meaningful, $\\therefore$ $x + 3 \\geq 0$, $\\therefore$ $x \\geq - 3$.", "expr_cands": ["\\sqrt { x + 3 }", "x", "x + 3 \\ge 0", "- 3 \\le x", "x \\ge - 3"], "exprs": ["x + 3 \\ge 0", "x \\ge - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x + 3 }"}, {"id": "x + 3 \\ge 0"}, {"id": "代数式 $\\sqrt { x + 3 }$ 有意义"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "$x$ 的范围"}, {"id": "x \\ge - 3"}], "links": [{"rel": "被描述", "source": "\\sqrt { x + 3 }", "target": "x + 3 \\ge 0"}, {"rel": "不等式方程求解", "source": "x + 3 \\ge 0", "target": "x \\ge - 3"}, {"rel": "限制性描述", "source": "代数式 $\\sqrt { x + 3 }$ 有意义", "target": "x + 3 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x + 3 \\ge 0"}, {"rel": "限制性描述", "source": "$x$ 的范围", "target": "x + 3 \\ge 0"}]}}
{"content": "The value of $x$ that makes the expression $\\sqrt { x - 1 }$ meaningful is ____ ?", "answer": "x \\ge 1", "steps": "From the given condition, we have $x - 1 \\ge 0$, which implies $x \\ge 1$.", "expr_cands": ["\\sqrt { x - 1 }", "x", "x - 1 \\ge 0", "1 \\le x", "x \\ge 1"], "exprs": ["x - 1 \\ge 0", "x \\ge 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x - 1 }"}, {"id": "x - 1 \\ge 0"}, {"id": "使式子 $\\sqrt { x - 1 }$ 有意义的 $x$ 的值"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x \\ge 1"}], "links": [{"rel": "被描述", "source": "\\sqrt { x - 1 }", "target": "x - 1 \\ge 0"}, {"rel": "不等式方程求解", "source": "x - 1 \\ge 0", "target": "x \\ge 1"}, {"rel": "限制性描述", "source": "使式子 $\\sqrt { x - 1 }$ 有意义的 $x$ 的值", "target": "x - 1 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 1 \\ge 0"}]}}
{"content": "If the value of $x ^ 2 + 2 x$ is $8$, then the value of $4 x ^ 2 - 5 + 8 x$ is ____?", "answer": "27", "steps": "Because $x ^ 2 + 2 x = 8$, therefore $4 x ^ 2 - 5 + 8 x = 4 ( x ^ 2 + 2 x ) - 5 = 4 * 8 - 5 = 32 - 5 = 27$.", "expr_cands": ["x ^ { 2 } + 2 x", "x", "8", "4 x ^ { 2 } - 5 + 8 x", "x ^ { 2 } + 2 x = 8", "x = - 4", "x = 2", "4 ( x ^ { 2 } + 2 x ) - 5", "27"], "exprs": ["x ^ { 2 } + 2 x = 8", "4 ( x ^ { 2 } + 2 x ) - 5", "27"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } + 2 x"}, {"id": "x ^ { 2 } + 2 x = 8"}, {"id": "8"}, {"id": "$x ^ { 2 } + 2 x$ 的值是 $8$"}, {"id": "4 ( x ^ { 2 } + 2 x ) - 5"}, {"id": "4 x ^ { 2 } - 5 + 8 x"}, {"id": "27"}], "links": [{"rel": "被描述", "source": "x ^ { 2 } + 2 x", "target": "x ^ { 2 } + 2 x = 8"}, {"rel": "提取因式参考", "source": "x ^ { 2 } + 2 x", "target": "4 ( x ^ { 2 } + 2 x ) - 5"}, {"rel": "代入", "source": "x ^ { 2 } + 2 x = 8", "target": "27"}, {"rel": "被描述", "source": "8", "target": "x ^ { 2 } + 2 x = 8"}, {"rel": "限制性描述", "source": "$x ^ { 2 } + 2 x$ 的值是 $8$", "target": "x ^ { 2 } + 2 x = 8"}, {"rel": "被代入", "source": "4 ( x ^ { 2 } + 2 x ) - 5", "target": "27"}, {"rel": "提取因式", "source": "4 x ^ { 2 } - 5 + 8 x", "target": "4 ( x ^ { 2 } + 2 x ) - 5"}]}}
{"content": "$x = - 2$ is a solution of the equation $2 a + 3 x = 4$, then the value of $a$ is:", "answer": "5", "steps": "Substituting $x = - 2$ into the equation, we get $2 a + 3 * ( - 2 ) = 4$, which gives us $a = 5$.", "expr_cands": ["x = - 2", "x", "2 a + 3 x = 4", "a", "2 a + 3 * ( - 2 ) = 4", "a = 5"], "exprs": ["2 a + 3 * ( - 2 ) = 4", "a = 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x = - 2"}, {"id": "2 a + 3 * ( - 2 ) = 4"}, {"id": "2 a + 3 x = 4"}, {"id": "a = 5"}], "links": [{"rel": "代入", "source": "x = - 2", "target": "2 a + 3 * ( - 2 ) = 4"}, {"rel": "等式方程求解", "source": "2 a + 3 * ( - 2 ) = 4", "target": "a = 5"}, {"rel": "被代入", "source": "2 a + 3 x = 4", "target": "2 a + 3 * ( - 2 ) = 4"}]}}
{"content": "If $x - y = 5$ and $y - z = 5$, what is the value of $z - x$?", "answer": "- 10", "steps": "Since $x - y = 5$ and $y - z = 5$, we can add these two equations to get $x - y + y - z = 5 + 5$. Simplifying this gives us $x - z = 10$. Therefore, we can also say that $z - x = - 10$.", "expr_cands": ["x - y = 5", "y", "x", "y - z = 5", "z", "z - x", "x - y + y - z", "10", "x - z = 10", "- 10"], "exprs": ["x - z = 10", "- 10"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x - y = 5"}, {"id": "x - z = 10"}, {"id": "y - z = 5"}, {"id": "z - x"}, {"id": "- 10"}], "links": [{"rel": "联立", "source": "x - y = 5", "target": "x - z = 10"}, {"rel": "代入", "source": "x - z = 10", "target": "- 10"}, {"rel": "联立", "source": "y - z = 5", "target": "x - z = 10"}, {"rel": "被代入", "source": "z - x", "target": "- 10"}]}}
{"content": "The value of $m$ if the solution of the equation $2 x - m = 3$ is also a solution of the equation $3 x = 12$ is ____?", "answer": "5", "steps": "From $3 x = 12$, we get $x = 4$. Substituting $x = 4$ into $2 x - m = 3$, we get $8 - m = 3$. Solving for $m$, we get $m = 5$.", "expr_cands": ["x", "2 x - m = 3", "m", "3 x = 12", "x = 4", "8 - m = 3", "m = 5"], "exprs": ["x = 4", "8 - m = 3", "m = 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 x = 12"}, {"id": "x = 4"}, {"id": "2 x - m = 3"}, {"id": "8 - m = 3"}, {"id": "m = 5"}], "links": [{"rel": "等式方程求解", "source": "3 x = 12", "target": "x = 4"}, {"rel": "代入", "source": "x = 4", "target": "8 - m = 3"}, {"rel": "被代入", "source": "2 x - m = 3", "target": "8 - m = 3"}, {"rel": "等式方程求解", "source": "8 - m = 3", "target": "m = 5"}]}}
{"content": "If the solution set of the inequality $2 x + 4 < a$ is $x < 2$, then $a$ must satisfy ____?", "answer": "a = 8", "steps": "Moving terms, we get $2 x < a - 4$. Dividing both sides by 2, we get $x < \\frac { a - 4 } { 2 }$. Since the solution set of the inequality $2 x + 4 < a$ is $x < 2$, we have $\\frac { a - 4 } { 2 } = 2$. Solving for $a$, we get $a = 8$.", "expr_cands": ["2 x + 4 < a", "a", "x", "x < 2", "2 x < a - 4", "1", "x < \\frac { a - 4 } { 2 }", "\\frac { a - 4 } { 2 } = 2", "a = 8"], "exprs": ["x < \\frac { a - 4 } { 2 }", "\\frac { a - 4 } { 2 } = 2", "a = 8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x + 4 < a"}, {"id": "x < \\frac { a - 4 } { 2 }"}, {"id": "x < 2"}, {"id": "\\frac { a - 4 } { 2 } = 2"}, {"id": "不等式 $2 x + 4 < a$ 的解集是 $x < 2$"}, {"id": "a = 8"}], "links": [{"rel": "不等式方程部分求解", "source": "2 x + 4 < a", "target": "x < \\frac { a - 4 } { 2 }"}, {"rel": "被描述", "source": "x < \\frac { a - 4 } { 2 }", "target": "\\frac { a - 4 } { 2 } = 2"}, {"rel": "被描述", "source": "x < 2", "target": "\\frac { a - 4 } { 2 } = 2"}, {"rel": "等式方程求解", "source": "\\frac { a - 4 } { 2 } = 2", "target": "a = 8"}, {"rel": "限制性描述", "source": "不等式 $2 x + 4 < a$ 的解集是 $x < 2$", "target": "\\frac { a - 4 } { 2 } = 2"}]}}
{"content": "Given: $x = \\frac { 2 } { \\sqrt { 3 } - 1 }$, find the value of $x ^ { 2 } - 2 x + 2018$.", "answer": "2020", "steps": "Since $x = \\frac { 2 } { \\sqrt { 3 } - 1 } = \\frac { 2 ( \\sqrt { 3 } + 1 )} { 2 } = \\sqrt { 3 } + 1$, therefore the original expression is equal to ${ x } ^ { 2 } - 2 x + 1 + 2017 = ( x - 1 ) ^ { 2 } + 2017 = ( \\sqrt { 3 } + 1 - 1 ) ^ { 2 } + 2017 = 2020$.", "expr_cands": ["x = \\frac { 2 } { \\sqrt { 3 } - 1 }", "x", "x ^ { 2 } - 2 x + 2018", "x = \\sqrt { 3 } + 1", "{ x } ^ { 2 } - 2 x + 1 + 2017", "2020"], "exprs": ["x = \\sqrt { 3 } + 1", "{ x } ^ { 2 } - 2 x + 1 + 2017", "2020"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x = \\frac { 2 } { \\sqrt { 3 } - 1 }"}, {"id": "x = \\sqrt { 3 } + 1"}, {"id": "x ^ { 2 } - 2 x + 2018"}, {"id": "{ x } ^ { 2 } - 2 x + 1 + 2017"}, {"id": "2020"}], "links": [{"rel": "计算", "source": "x = \\frac { 2 } { \\sqrt { 3 } - 1 }", "target": "x = \\sqrt { 3 } + 1"}, {"rel": "代入", "source": "x = \\sqrt { 3 } + 1", "target": "2020"}, {"rel": "展开", "source": "x ^ { 2 } - 2 x + 2018", "target": "{ x } ^ { 2 } - 2 x + 1 + 2017"}, {"rel": "被代入", "source": "{ x } ^ { 2 } - 2 x + 1 + 2017", "target": "2020"}]}}
{"content": "If the value of the fraction $\\frac { | m | - 1 } { m ^ 2 - m }$ is zero, then $m$ is equal to ____?", "answer": "m = - 1", "steps": "$\\because$ The value of the fraction $\\frac { | m | - 1 } {{ m } ^ { 2 } - m }$ is zero, $\\therefore$ $| m | - 1 = 0$ and ${ m } ^ { 2 } - m \\neq 0$. Solving for $m$, we get $m = - 1$.", "expr_cands": ["\\frac { | m | - 1 } { m ^ { 2 } - m }", "m", "\\frac { | m | - 1 } { { m } ^ { 2 } - m }", "| m | - 1 = 0", "m = - 1", "m = 1", "m ^ { 2 } - m \\neq 0", "( 0 < m \\wedge m < 1 )", "1 < m", "m < 0"], "exprs": ["| m | - 1 = 0", "m ^ { 2 } - m \\neq 0", "m = - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { | m | - 1 } { m ^ { 2 } - m }"}, {"id": "| m | - 1 = 0"}, {"id": "分式 $\\frac { | m | - 1 } { m ^ { 2 } - m }$ 的值为零"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "m ^ { 2 } - m \\neq 0"}, {"id": "m = - 1"}], "links": [{"rel": "被描述", "source": "\\frac { | m | - 1 } { m ^ { 2 } - m }", "target": "| m | - 1 = 0"}, {"rel": "被描述", "source": "\\frac { | m | - 1 } { m ^ { 2 } - m }", "target": "m ^ { 2 } - m \\neq 0"}, {"rel": "联立", "source": "| m | - 1 = 0", "target": "m = - 1"}, {"rel": "限制性描述", "source": "分式 $\\frac { | m | - 1 } { m ^ { 2 } - m }$ 的值为零", "target": "| m | - 1 = 0"}, {"rel": "限制性描述", "source": "分式 $\\frac { | m | - 1 } { m ^ { 2 } - m }$ 的值为零", "target": "m ^ { 2 } - m \\neq 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "| m | - 1 = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "m ^ { 2 } - m \\neq 0"}, {"rel": "联立", "source": "m ^ { 2 } - m \\neq 0", "target": "m = - 1"}]}}
{"content": "When $k$ = ____ ?, the algebraic expression ${ x } ^ { 6 } - 5 k { x } ^ { 4 } { y } ^ { 3 } - 4 { x } ^ { 6 } + \\frac { 1 } { 5 } { x } ^ { 4 } { y } ^ { 3 } + 10$ does not contain the term $x ^ { 4 } y ^ { 3 }$.", "answer": "\\frac { 1 } { 25 }", "steps": "The algebraic expression $x ^ 6 - 5 kx ^ 4 y ^ 3 - 4 x ^ 6 + \\frac { 1 } { 5 } x ^ 4 y ^ 3 + 10$ does not contain the term $x ^ 4 y ^ 3$, which means that combining $- 5 kx ^ 4 y ^ 3$ and $\\frac { 1 } { 5 } x ^ 4 y ^ 3$ results in $0$. Therefore, we have $- 5 k + \\frac { 1 } { 5 } = 0$, and solving for $k$ gives $k = \\frac { 1 } { 25 }$.", "expr_cands": ["k", "{ x } ^ { 6 } - 5 k { x } ^ { 4 } { y } ^ { 3 } - 4 { x } ^ { 6 } + \\frac { 1 } { 5 } { x } ^ { 4 } { y } ^ { 3 } + 10", "y", "x", "x ^ { 4 } y ^ { 3 }", "x ^ { 6 } - 5 kx ^ { 4 } y ^ { 3 } - 4 x ^ { 6 } + \\frac { 1 } { 5 } x ^ { 4 } y ^ { 3 } + 10", "- 5 kx ^ { 4 } y ^ { 3 }", "\\frac { 1 } { 5 } x ^ { 4 } y ^ { 3 }", "0", "- 5 k + \\frac { 1 } { 5 } = 0", "k = \\frac { 1 } { 25 }"], "exprs": ["- 5 k + \\frac { 1 } { 5 } = 0", "k = \\frac { 1 } { 25 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "{ x } ^ { 6 } - 5 k { x } ^ { 4 } { y } ^ { 3 } - 4 { x } ^ { 6 } + \\frac { 1 } { 5 } { x } ^ { 4 } { y } ^ { 3 } + 10"}, {"id": "- 5 k + \\frac { 1 } { 5 } = 0"}, {"id": "代数式 ${ x } ^ { 6 } - 5 k { x } ^ { 4 } { y } ^ { 3 } - 4 { x } ^ { 6 } + \\frac { 1 } { 5 } { x } ^ { 4 } { y } ^ { 3 } + 10$ 中不含 $x ^ { 4 } y ^ { 3 }$ 项"}, {"id": "k = \\frac { 1 } { 25 }"}], "links": [{"rel": "被描述", "source": "{ x } ^ { 6 } - 5 k { x } ^ { 4 } { y } ^ { 3 } - 4 { x } ^ { 6 } + \\frac { 1 } { 5 } { x } ^ { 4 } { y } ^ { 3 } + 10", "target": "- 5 k + \\frac { 1 } { 5 } = 0"}, {"rel": "等式方程求解", "source": "- 5 k + \\frac { 1 } { 5 } = 0", "target": "k = \\frac { 1 } { 25 }"}, {"rel": "限制性描述", "source": "代数式 ${ x } ^ { 6 } - 5 k { x } ^ { 4 } { y } ^ { 3 } - 4 { x } ^ { 6 } + \\frac { 1 } { 5 } { x } ^ { 4 } { y } ^ { 3 } + 10$ 中不含 $x ^ { 4 } y ^ { 3 }$ 项", "target": "- 5 k + \\frac { 1 } { 5 } = 0"}]}}
{"content": "If $x = 2$ is a solution of the equation $ax + 3 bx - 10 = 0$, then the value of $3 a + 9 b$ is ____?", "answer": "15", "steps": "Because $x = 2$ is a solution of the equation $ax + 3 bx - 10 = 0$, therefore $2 a + 6 b - 10 = 0$, therefore $2 a + 6 b = 10$, therefore $a + 3 b = 5$, therefore the original expression is equal to $3 ( a + 3 b ) = 3 \\times 5 = 15$.", "expr_cands": ["x = 2", "x", "ax + 3 bx - 10 = 0", "b", "a", "3 a + 9 b", "2 a + 6 b - 10 = 0", "2 a + 6 b = 10", "a + 3 b = 5", "3 ( a + 3 b )", "15"], "exprs": ["2 a + 6 b - 10 = 0", "2 a + 6 b = 10", "a + 3 b = 5", "3 ( a + 3 b )", "15"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "ax + 3 bx - 10 = 0"}, {"id": "2 a + 6 b - 10 = 0"}, {"id": "x = 2"}, {"id": "$x = 2$ 是方程 $ax + 3 bx - 10 = 0$ 的解"}, {"id": "2 a + 6 b = 10"}, {"id": "a + 3 b = 5"}, {"id": "3 a + 9 b"}, {"id": "3 ( a + 3 b )"}, {"id": "15"}], "links": [{"rel": "被描述", "source": "ax + 3 bx - 10 = 0", "target": "2 a + 6 b - 10 = 0"}, {"rel": "移项", "source": "2 a + 6 b - 10 = 0", "target": "2 a + 6 b = 10"}, {"rel": "被描述", "source": "x = 2", "target": "2 a + 6 b - 10 = 0"}, {"rel": "限制性描述", "source": "$x = 2$ 是方程 $ax + 3 bx - 10 = 0$ 的解", "target": "2 a + 6 b - 10 = 0"}, {"rel": "同乘除", "source": "2 a + 6 b = 10", "target": "a + 3 b = 5"}, {"rel": "提取因式参考", "source": "a + 3 b = 5", "target": "3 ( a + 3 b )"}, {"rel": "代入", "source": "a + 3 b = 5", "target": "15"}, {"rel": "提取因式", "source": "3 a + 9 b", "target": "3 ( a + 3 b )"}, {"rel": "被代入", "source": "3 ( a + 3 b )", "target": "15"}]}}
{"content": "If $( 3 m - 2 ) x ^ { 2 } y ^ { n - 1 }$ is a sixth-degree monomial with coefficient 1 in terms of $x$ and $y$, then $m - n ^ { 2 }$ = ____?", "answer": "- 24", "steps": "From the given information, we can obtain: $3 m - 2 = 1$, $2 + n - 1 = 6$. Solving for $m$ and $n$, we get $m = 1$, $n = 5$. Therefore, $m - n ^ 2 = 1 - 5 ^ 2 = - 24$.", "expr_cands": ["( 3 m - 2 ) x ^ { 2 } y ^ { n - 1 }", "m", "y", "x", "n", "1", "m - n ^ { 2 }", "3 m - 2 = 1", "m = 1", "2 + n - 1 = 6", "n = 5", "m - { n } ^ { 2 }", "- 24"], "exprs": ["3 m - 2 = 1", "2 + n - 1 = 6", "m = 1", "n = 5", "- 24"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( 3 m - 2 ) x ^ { 2 } y ^ { n - 1 }"}, {"id": "3 m - 2 = 1"}, {"id": "$( 3 m - 2 ) x ^ { 2 } y ^ { n - 1 }$ 是关于 $x$ , $y$ 的系数为 $1$ 的六次单项式"}, {"id": "2 + n - 1 = 6"}, {"id": "m = 1"}, {"id": "n = 5"}, {"id": "m - { n } ^ { 2 }"}, {"id": "- 24"}], "links": [{"rel": "被描述", "source": "( 3 m - 2 ) x ^ { 2 } y ^ { n - 1 }", "target": "3 m - 2 = 1"}, {"rel": "被描述", "source": "( 3 m - 2 ) x ^ { 2 } y ^ { n - 1 }", "target": "2 + n - 1 = 6"}, {"rel": "等式方程求解", "source": "3 m - 2 = 1", "target": "m = 1"}, {"rel": "限制性描述", "source": "$( 3 m - 2 ) x ^ { 2 } y ^ { n - 1 }$ 是关于 $x$ , $y$ 的系数为 $1$ 的六次单项式", "target": "3 m - 2 = 1"}, {"rel": "限制性描述", "source": "$( 3 m - 2 ) x ^ { 2 } y ^ { n - 1 }$ 是关于 $x$ , $y$ 的系数为 $1$ 的六次单项式", "target": "2 + n - 1 = 6"}, {"rel": "等式方程求解", "source": "2 + n - 1 = 6", "target": "n = 5"}, {"rel": "代入", "source": "m = 1", "target": "- 24"}, {"rel": "代入", "source": "n = 5", "target": "- 24"}, {"rel": "被代入", "source": "m - { n } ^ { 2 }", "target": "- 24"}]}}
{"content": "The equation $x ^ { 2 } + 4 x + k = 0$ has a root of $2$. What is the value of $k$?", "answer": "- 12", "steps": "According to the problem, we have $2 ^ { 2 } + 4 \\times 2 + k = 0$, which gives us $k = - 12$ when solved.", "expr_cands": ["x ^ { 2 } + 4 x + k = 0", "x", "k", "2", "2 ^ { 2 } + 4 * 2 + k = 0", "k = - 12"], "exprs": ["2 ^ { 2 } + 4 * 2 + k = 0", "k = - 12"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2"}, {"id": "2 ^ { 2 } + 4 * 2 + k = 0"}, {"id": "x ^ { 2 } + 4 x + k = 0"}, {"id": "x"}, {"id": "方程 $x ^ { 2 } + 4 x + k = 0$ 的一个根是 $2$"}, {"id": "k = - 12"}], "links": [{"rel": "被描述", "source": "2", "target": "2 ^ { 2 } + 4 * 2 + k = 0"}, {"rel": "等式方程求解", "source": "2 ^ { 2 } + 4 * 2 + k = 0", "target": "k = - 12"}, {"rel": "被描述", "source": "x ^ { 2 } + 4 x + k = 0", "target": "2 ^ { 2 } + 4 * 2 + k = 0"}, {"rel": "被描述", "source": "x", "target": "2 ^ { 2 } + 4 * 2 + k = 0"}, {"rel": "限制性描述", "source": "方程 $x ^ { 2 } + 4 x + k = 0$ 的一个根是 $2$", "target": "2 ^ { 2 } + 4 * 2 + k = 0"}]}}
{"content": "If $| x - 1 | + | y + 3 | = 0$, then $( x + 1 ) ( y + 1 )$ is equal to ____?", "answer": "- 4", "steps": "$\\because | x - 1 | + | y + 3 | = 0$, $\\therefore x - 1 = 0$, $y + 3 = 0$, solving for $x = 1$, $y = - 3$, $\\therefore$ the original expression $= ( 1 + 1 ) \\times ( - 3 + 1 ) = - 4$.", "expr_cands": ["| x - 1 | + | y + 3 | = 0", "y", "x", "( x + 1 ) ( y + 1 )", "x - 1 = 0", "x = 1", "y + 3 = 0", "y = - 3", "( 1 + 1 ) * ( - 3 + 1 )", "- 4"], "exprs": ["x - 1 = 0", "y + 3 = 0", "x = 1", "y = - 3", "( 1 + 1 ) * ( - 3 + 1 )", "- 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| x - 1 | + | y + 3 | = 0"}, {"id": "x - 1 = 0"}, {"id": "绝对值恒大于等于0"}, {"id": "y + 3 = 0"}, {"id": "x = 1"}, {"id": "y = - 3"}, {"id": "( x + 1 ) ( y + 1 )"}, {"id": "( 1 + 1 ) * ( - 3 + 1 )"}, {"id": "- 4"}], "links": [{"rel": "被描述", "source": "| x - 1 | + | y + 3 | = 0", "target": "x - 1 = 0"}, {"rel": "被描述", "source": "| x - 1 | + | y + 3 | = 0", "target": "y + 3 = 0"}, {"rel": "等式方程求解", "source": "x - 1 = 0", "target": "x = 1"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "x - 1 = 0"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "y + 3 = 0"}, {"rel": "等式方程求解", "source": "y + 3 = 0", "target": "y = - 3"}, {"rel": "代入", "source": "x = 1", "target": "( 1 + 1 ) * ( - 3 + 1 )"}, {"rel": "代入", "source": "y = - 3", "target": "( 1 + 1 ) * ( - 3 + 1 )"}, {"rel": "被代入", "source": "( x + 1 ) ( y + 1 )", "target": "( 1 + 1 ) * ( - 3 + 1 )"}, {"rel": "计算", "source": "( 1 + 1 ) * ( - 3 + 1 )", "target": "- 4"}]}}
{"content": "Given that $x = 3$ is a solution to the equation $ax - 6 = a + 10$, then $a$ = ____ ?", "answer": "8", "steps": "$\\because x = 3$ is a solution to the equation $ax - 6 = a + 10$, $\\therefore x = 3$ satisfies the equation $ax - 6 = a + 10$, $\\therefore 3 a - 6 = a + 10$, solving for $a$ yields $a = 8$.", "expr_cands": ["x = 3", "x", "ax - 6 = a + 10", "a", "3 a - 6 = a + 10", "ax - 6", "a + 10", "a = 8"], "exprs": ["3 a - 6 = a + 10", "a = 8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "ax - 6 = a + 10"}, {"id": "3 a - 6 = a + 10"}, {"id": "x = 3"}, {"id": "a = 8"}], "links": [{"rel": "被代入", "source": "ax - 6 = a + 10", "target": "3 a - 6 = a + 10"}, {"rel": "等式方程求解", "source": "3 a - 6 = a + 10", "target": "a = 8"}, {"rel": "代入", "source": "x = 3", "target": "3 a - 6 = a + 10"}]}}
{"content": "If $x$ is a rational number, what is the maximum value of the expression $2018 - | x + 2 |$?", "answer": "2018", "steps": "$\\because x$ is a rational number, the expression $2018 - | x + 2 |$ has a maximum value. $\\therefore$ When $| x + 2 | = 0$, $2018 - | x + 2 |$ is maximum at $2018$.", "expr_cands": ["x", "2018 - | x + 2 |", "| x + 2 | = 0", "x = - 2", "2018"], "exprs": ["| x + 2 | = 0", "2018"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2018 - | x + 2 |"}, {"id": "| x + 2 | = 0"}, {"id": "$x$ 为有理数"}, {"id": "式子 $2018 - | x + 2 |$ 存在最大值"}, {"id": "绝对值恒大于等于0"}, {"id": "2018"}], "links": [{"rel": "被描述", "source": "2018 - | x + 2 |", "target": "| x + 2 | = 0"}, {"rel": "被代入", "source": "2018 - | x + 2 |", "target": "2018"}, {"rel": "代入", "source": "| x + 2 | = 0", "target": "2018"}, {"rel": "限制性描述", "source": "$x$ 为有理数", "target": "| x + 2 | = 0"}, {"rel": "限制性描述", "source": "式子 $2018 - | x + 2 |$ 存在最大值", "target": "| x + 2 | = 0"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "| x + 2 | = 0"}]}}
{"content": "If the equation $2 x + a = 0$ has a solution of $x = 2$, then the value of $a$ is ____?", "answer": "- 4", "steps": "Because $x = 2$ is a solution of the equation $2 x + a = 0$, therefore $4 + a = 0$. Solving for $a$, we get $a = - 4$.", "expr_cands": ["x", "2 x + a = 0", "a", "x = 2", "a + 4 = 0", "4 + a = 0", "a = - 4"], "exprs": ["4 + a = 0", "a = - 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x + a = 0"}, {"id": "4 + a = 0"}, {"id": "x = 2"}, {"id": "a = - 4"}], "links": [{"rel": "被代入", "source": "2 x + a = 0", "target": "4 + a = 0"}, {"rel": "等式方程求解", "source": "4 + a = 0", "target": "a = - 4"}, {"rel": "代入", "source": "x = 2", "target": "4 + a = 0"}]}}
{"content": "If the equation $\\frac { x } { x - 3 } + 5 = \\frac { m - 3 } { x - 3 }$ has no solution for $x$, then the value of $m$ is ____?", "answer": "6", "steps": "$\\frac { x } { x - 3 } + 5 = \\frac { m - 3 } { x - 3 }$, $x + 5 ( x - 3 ) = m - 3 x + 5 x - 15 = m - 3$ $\\therefore$ $x = \\frac { 1 } { 6 } m + 2$, when $\\frac { 1 } { 6 } m + 2 - 3 = 0$, the equation has no solution, so $m = 6$ is obtained.", "expr_cands": ["x", "\\frac { x } { x - 3 } + 5 = \\frac { m - 3 } { x - 3 }", "m", "x + 5 ( x - 3 ) = m - 3", "x = \\frac { 1 } { 6 } m + 2", "\\frac { 1 } { 6 } m + 2 - 3 = 0", "m = 6"], "exprs": ["x + 5 ( x - 3 ) = m - 3", "x = \\frac { 1 } { 6 } m + 2", "\\frac { 1 } { 6 } m + 2 - 3 = 0", "m = 6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x } { x - 3 } + 5 = \\frac { m - 3 } { x - 3 }"}, {"id": "x + 5 ( x - 3 ) = m - 3"}, {"id": "x = \\frac { 1 } { 6 } m + 2"}, {"id": "\\frac { 1 } { 6 } m + 2 - 3 = 0"}, {"id": "关于 $x$ 的方程 $\\frac { x } { x - 3 } + 5 = \\frac { m - 3 } { x - 3 }$ 无解"}, {"id": "分式方程无解，则分母为0"}, {"id": "m = 6"}], "links": [{"rel": "同乘除", "source": "\\frac { x } { x - 3 } + 5 = \\frac { m - 3 } { x - 3 }", "target": "x + 5 ( x - 3 ) = m - 3"}, {"rel": "被描述", "source": "\\frac { x } { x - 3 } + 5 = \\frac { m - 3 } { x - 3 }", "target": "\\frac { 1 } { 6 } m + 2 - 3 = 0"}, {"rel": "等式方程部分求解", "source": "x + 5 ( x - 3 ) = m - 3", "target": "x = \\frac { 1 } { 6 } m + 2"}, {"rel": "被描述", "source": "x = \\frac { 1 } { 6 } m + 2", "target": "\\frac { 1 } { 6 } m + 2 - 3 = 0"}, {"rel": "等式方程求解", "source": "\\frac { 1 } { 6 } m + 2 - 3 = 0", "target": "m = 6"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $\\frac { x } { x - 3 } + 5 = \\frac { m - 3 } { x - 3 }$ 无解", "target": "\\frac { 1 } { 6 } m + 2 - 3 = 0"}, {"rel": "属性描述", "source": "分式方程无解，则分母为0", "target": "\\frac { 1 } { 6 } m + 2 - 3 = 0"}]}}
{"content": "The straight line $y = 3 x + 4$ is translated down $5$ units to obtain the line $l$. The corresponding function expression for the line $l$ is ____?", "answer": "y = 3 x - 1", "steps": "The equation of the line $y = 3 x + 4$ after being translated down $5$ units is $y = 3 x + 4 - 5$, which simplifies to $y = 3 x - 1$.", "expr_cands": ["y = 3 x + 4", "y", "x", "5", "l", "y = 3 x + 4 - 5", "3 x + 4 = 3 x + 4 - 5", "3 x - 1"], "exprs": ["y = 3 x + 4 - 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = 3 x + 4"}, {"id": "y = 3 x + 4 - 5"}, {"id": "5"}, {"id": "将直线 $y = 3 x + 4$ 向下平移 $5$ 个单位得到直线 $l$"}, {"id": "直线对应的函数表达式"}], "links": [{"rel": "被描述", "source": "y = 3 x + 4", "target": "y = 3 x + 4 - 5"}, {"rel": "被描述", "source": "5", "target": "y = 3 x + 4 - 5"}, {"rel": "限制性描述", "source": "将直线 $y = 3 x + 4$ 向下平移 $5$ 个单位得到直线 $l$", "target": "y = 3 x + 4 - 5"}, {"rel": "限制性描述", "source": "直线对应的函数表达式", "target": "y = 3 x + 4 - 5"}]}}
{"content": "If $a = - 2018$, then the value of the expression $| a ^ 2 + 2017 a + 1 | + | a ^ 2 + 2019 a - 1 |$ is ____?", "answer": "4038", "steps": "\\because $a = - 2018$, \\therefore the original expression $= | a ( a + 2017 ) + 1 | + | a ( a + 2019 ) - 1 | = | - 2018 ( - 2018 + 2017 ) + 1 | + | - 2018 ( - 2018 + 2019 ) - 1 | = | 2018 + 1 | + | - 2018 - 1 | = 2019 + 2019 = 4038$.", "expr_cands": ["a = - 2018", "a", "| a ^ { 2 } + 2017 a + 1 | + | a ^ { 2 } + 2019 a - 1 |", "| a ( a + 2017 ) + 1 | + | a ( a + 2019 ) - 1 |", "4038"], "exprs": ["4038"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a = - 2018"}, {"id": "4038"}, {"id": "| a ^ { 2 } + 2017 a + 1 | + | a ^ { 2 } + 2019 a - 1 |"}], "links": [{"rel": "代入", "source": "a = - 2018", "target": "4038"}, {"rel": "被代入", "source": "| a ^ { 2 } + 2017 a + 1 | + | a ^ { 2 } + 2019 a - 1 |", "target": "4038"}]}}
{"content": "If the monomial $3 { a } ^ { n } { b } ^ { 2 } c$ is a fifth degree monomial, then $n$ = ____ ?", "answer": "2", "steps": "Since the monomial $3 { a } ^ { n } { b } ^ { 2 } c$ is a fifth degree monomial, we have $n + 2 + 1 = 5$. Solving for $n$, we get $n = 2$.", "expr_cands": ["3 { a } ^ { n } { b } ^ { 2 } c", "a", "b", "n", "c", "n + 2 + 1 = 5", "n = 2"], "exprs": ["n + 2 + 1 = 5", "n = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 { a } ^ { n } { b } ^ { 2 } c"}, {"id": "n + 2 + 1 = 5"}, {"id": "单项式 $3 { a } ^ { n } { b } ^ { 2 } c$ 是五次单项式"}, {"id": "n = 2"}], "links": [{"rel": "被描述", "source": "3 { a } ^ { n } { b } ^ { 2 } c", "target": "n + 2 + 1 = 5"}, {"rel": "等式方程求解", "source": "n + 2 + 1 = 5", "target": "n = 2"}, {"rel": "限制性描述", "source": "单项式 $3 { a } ^ { n } { b } ^ { 2 } c$ 是五次单项式", "target": "n + 2 + 1 = 5"}]}}
{"content": "The positive integer solution of the inequality $- 3 x + 6 > 0$ is ____?", "answer": "1", "steps": "Moving terms yields: $- 3 x > - 6$. Dividing both sides by $- 3$ gives: $x < 2$. Therefore, the positive integer solution is: $1$.", "expr_cands": ["- 3 x + 6 > 0", "x", "- 3 x > - 6", "x < 2", "1"], "exprs": ["x < 2", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 3 x + 6 > 0"}, {"id": "x < 2"}, {"id": "1"}, {"id": "不等式 $- 3 x + 6 > 0$ 的正整数解"}], "links": [{"rel": "不等式方程求解", "source": "- 3 x + 6 > 0", "target": "x < 2"}, {"rel": "被描述", "source": "x < 2", "target": "1"}, {"rel": "限制性描述", "source": "不等式 $- 3 x + 6 > 0$ 的正整数解", "target": "1"}]}}
{"content": "Given that $a$ and $b$ are opposite numbers, $c$ and $d$ are reciprocal numbers, and $e$ is the number with the smallest absolute value, what is the value of the algebraic expression $5 ( a + b ) ^ 2 + \\frac { 1 } { 2 } cd - | e - 1 |$?", "answer": "- \\frac { 1 } { 2 }", "steps": "Since $a$ and $b$ are opposite numbers, and $c$ and $d$ are reciprocals, and $e$ is the number with the smallest absolute value, therefore $a + b = 0$, $cd = 1$, $e = 0$. Therefore, $5 ( a + b ) ^ 2 + \\frac { 1 } { 2 } cd - | e - 1 | = 5 * 0 + \\frac { 1 } { 2 } * 1 - 1 = - \\frac { 1 } { 2 }$.", "expr_cands": ["a", "b", "c", "d", "e", "5 ( a + b ) ^ { 2 } + \\frac { 1 } { 2 } cd - | e - 1 |", "a + b = 0", "cd = 1", "e = 0", "- \\frac { 1 } { 2 }"], "exprs": ["a + b = 0", "cd = 1", "e = 0", "- \\frac { 1 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a + b = 0"}, {"id": "b"}, {"id": "$a$ , $b$ 互为相反数"}, {"id": "c"}, {"id": "cd = 1"}, {"id": "d"}, {"id": "$c$ , $d$ 互为倒数"}, {"id": "e"}, {"id": "e = 0"}, {"id": "$e$ 是绝对值最小的数"}, {"id": "5 ( a + b ) ^ { 2 } + \\frac { 1 } { 2 } cd - | e - 1 |"}, {"id": "- \\frac { 1 } { 2 }"}], "links": [{"rel": "被描述", "source": "a", "target": "a + b = 0"}, {"rel": "代入", "source": "a + b = 0", "target": "- \\frac { 1 } { 2 }"}, {"rel": "被描述", "source": "b", "target": "a + b = 0"}, {"rel": "限制性描述", "source": "$a$ , $b$ 互为相反数", "target": "a + b = 0"}, {"rel": "被描述", "source": "c", "target": "cd = 1"}, {"rel": "代入", "source": "cd = 1", "target": "- \\frac { 1 } { 2 }"}, {"rel": "被描述", "source": "d", "target": "cd = 1"}, {"rel": "限制性描述", "source": "$c$ , $d$ 互为倒数", "target": "cd = 1"}, {"rel": "被描述", "source": "e", "target": "e = 0"}, {"rel": "代入", "source": "e = 0", "target": "- \\frac { 1 } { 2 }"}, {"rel": "限制性描述", "source": "$e$ 是绝对值最小的数", "target": "e = 0"}, {"rel": "被代入", "source": "5 ( a + b ) ^ { 2 } + \\frac { 1 } { 2 } cd - | e - 1 |", "target": "- \\frac { 1 } { 2 }"}]}}
{"content": "When $x$ = ____ ?, the value of the fraction $\\frac { x } { x + 1 }$ is $1$ greater than the value of $\\frac { 6 } { x ^ 2 - 1 }$. ", "answer": "- 5", "steps": "According to the problem, we have $\\frac { x } { x + 1 } - \\frac { 6 } { x ^ 2 - 1 } = 1$. Simplifying the equation by getting rid of the denominators, we get $x ( x - 1 ) - 6 = x ^ 2 - 1$. Solving for $x$, we get $x = - 5$. Checking our solution, we see that $x = - 5$ satisfies the equation $\\frac { x } { x + 1 } - \\frac { 6 } { x ^ 2 - 1 } = 1$.", "expr_cands": ["x", "\\frac { x } { x + 1 }", "\\frac { 6 } { x ^ { 2 } - 1 }", "1", "\\frac { x } { x + 1 } - \\frac { 6 } { x ^ { 2 } - 1 } = 1", "x = - 5", "x ( x - 1 ) - 6 = x ^ { 2 } - 1"], "exprs": ["\\frac { x } { x + 1 } - \\frac { 6 } { x ^ { 2 } - 1 } = 1", "x = - 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x } { x + 1 }"}, {"id": "\\frac { x } { x + 1 } - \\frac { 6 } { x ^ { 2 } - 1 } = 1"}, {"id": "\\frac { 6 } { x ^ { 2 } - 1 }"}, {"id": "1"}, {"id": "分式 $\\frac { x } { x + 1 }$ 比 $\\frac { 6 } { x ^ { 2 } - 1 }$ 的值大 $1$"}, {"id": "x = - 5"}], "links": [{"rel": "被描述", "source": "\\frac { x } { x + 1 }", "target": "\\frac { x } { x + 1 } - \\frac { 6 } { x ^ { 2 } - 1 } = 1"}, {"rel": "等式方程求解", "source": "\\frac { x } { x + 1 } - \\frac { 6 } { x ^ { 2 } - 1 } = 1", "target": "x = - 5"}, {"rel": "被描述", "source": "\\frac { 6 } { x ^ { 2 } - 1 }", "target": "\\frac { x } { x + 1 } - \\frac { 6 } { x ^ { 2 } - 1 } = 1"}, {"rel": "被描述", "source": "1", "target": "\\frac { x } { x + 1 } - \\frac { 6 } { x ^ { 2 } - 1 } = 1"}, {"rel": "限制性描述", "source": "分式 $\\frac { x } { x + 1 }$ 比 $\\frac { 6 } { x ^ { 2 } - 1 }$ 的值大 $1$", "target": "\\frac { x } { x + 1 } - \\frac { 6 } { x ^ { 2 } - 1 } = 1"}]}}
{"content": "The product of the two roots of the quadratic equation $x ^ 2 + x - 2 = 0$ is _____.", "answer": "- 2", "steps": "The quadratic equation $x ^ 2 + x - 2 = 0$ has two roots denoted by $\\alpha$ and $\\beta$. Therefore, $\\alpha \\beta = - 2$. Hence, the product of the two roots of the quadratic equation $x ^ 2 + x - 2 = 0$ is $- 2$.", "expr_cands": ["x ^ { 2 } + x - 2 = 0", "x", "x = - 2", "x = 1", "\\alpha \\beta = - 2", "\\alpha", "\\beta", "- 2"], "exprs": ["- 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } + x - 2 = 0"}, {"id": "- 2"}, {"id": "一元二次方程 $x ^ { 2 } + x - 2 = 0$ 的两根之积"}, {"id": "一元二次方程根与系数关系，两根之积"}], "links": [{"rel": "被描述", "source": "x ^ { 2 } + x - 2 = 0", "target": "- 2"}, {"rel": "限制性描述", "source": "一元二次方程 $x ^ { 2 } + x - 2 = 0$ 的两根之积", "target": "- 2"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之积", "target": "- 2"}]}}
{"content": "If $( x + y ) : y = 6 : 1$, then $x : y$ = ", "answer": "5 : 1", "steps": "Since $( x + y ) : y = 6 : 1$, it follows that $x + y = 6 y$. Therefore, $x = 5 y$. Hence, $x : y = 5 : 1$.", "expr_cands": ["( x + y ) : y = 6 : 1", "y", "x", "x : y", "x + y = 6 y", "x = 5 y", "5"], "exprs": ["x + y = 6 y", "x = 5 y", "5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( x + y ) : y = 6 : 1"}, {"id": "x + y = 6 y"}, {"id": "x = 5 y"}, {"id": "x : y"}, {"id": "5"}], "links": [{"rel": "同乘除", "source": "( x + y ) : y = 6 : 1", "target": "x + y = 6 y"}, {"rel": "移项", "source": "x + y = 6 y", "target": "x = 5 y"}, {"rel": "代入", "source": "x = 5 y", "target": "5"}, {"rel": "被代入", "source": "x : y", "target": "5"}]}}
{"content": "If $a$ and $b$ are reciprocals, then $- 5 ab$ is equal to ____?", "answer": "- 5", "steps": "Since $a$ and $b$ are reciprocals, we have $ab = 1$. Therefore, the original expression is equal to $- 5 \\times 1 = - 5$.", "expr_cands": ["a", "b", "- 5 ab", "ab = 1", "- 5 * 1", "- 5"], "exprs": ["ab = 1", "- 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "ab = 1"}, {"id": "b"}, {"id": "$a$ , $b$ 互为倒数"}, {"id": "- 5 ab"}, {"id": "- 5"}], "links": [{"rel": "被描述", "source": "a", "target": "ab = 1"}, {"rel": "代入", "source": "ab = 1", "target": "- 5"}, {"rel": "被描述", "source": "b", "target": "ab = 1"}, {"rel": "限制性描述", "source": "$a$ , $b$ 互为倒数", "target": "ab = 1"}, {"rel": "被代入", "source": "- 5 ab", "target": "- 5"}]}}
{"content": "If $a ^ 2 + b + 5 = 0$, then the value of the algebraic expression $3 a ^ 2 + 3 b + 10$ is ____?", "answer": "- 5", "steps": "Since $a ^ 2 + b + 5 = 0$, it follows that $a ^ 2 + b = - 5$. Therefore, $3 a ^ 2 + 3 b + 10 = 3 ( a ^ 2 + b ) + 10 = 3 * ( - 5 ) + 10 = - 5$.", "expr_cands": ["a ^ { 2 } + b + 5 = 0", "a", "b", "3 a ^ { 2 } + 3 b + 10", "a ^ { 2 } + b = - 5", "3 ( a ^ { 2 } + b ) + 10", "- 5"], "exprs": ["a ^ { 2 } + b = - 5", "3 ( a ^ { 2 } + b ) + 10", "- 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a ^ { 2 } + b + 5 = 0"}, {"id": "a ^ { 2 } + b = - 5"}, {"id": "3 a ^ { 2 } + 3 b + 10"}, {"id": "3 ( a ^ { 2 } + b ) + 10"}, {"id": "- 5"}], "links": [{"rel": "移项", "source": "a ^ { 2 } + b + 5 = 0", "target": "a ^ { 2 } + b = - 5"}, {"rel": "提取因式参考", "source": "a ^ { 2 } + b = - 5", "target": "3 ( a ^ { 2 } + b ) + 10"}, {"rel": "代入", "source": "a ^ { 2 } + b = - 5", "target": "- 5"}, {"rel": "提取因式", "source": "3 a ^ { 2 } + 3 b + 10", "target": "3 ( a ^ { 2 } + b ) + 10"}, {"rel": "被代入", "source": "3 ( a ^ { 2 } + b ) + 10", "target": "- 5"}]}}
{"content": "If $x = 2$ is a solution to the one-variable linear equation $mx - n = 3$, then the value of $2 - 6 m + 3 n$ is ____?", "answer": "- 7", "steps": "Substituting $x = 2$ into the linear equation $mx - n = 3$ yields $2 m - n = 3$. Therefore, $2 - 6 m + 3 n = 2 - 3 ( 2 m - n ) = 2 - 3 * 3 = - 7$.", "expr_cands": ["x = 2", "x", "mx - n = 3", "m", "n", "2 - 6 m + 3 n", "2 m - n = 3", "2 - 3 ( 2 m - n )", "- 7"], "exprs": ["2 m - n = 3", "2 - 3 ( 2 m - n )", "- 7"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "mx - n = 3"}, {"id": "2 m - n = 3"}, {"id": "x = 2"}, {"id": "2 - 6 m + 3 n"}, {"id": "2 - 3 ( 2 m - n )"}, {"id": "- 7"}], "links": [{"rel": "被代入", "source": "mx - n = 3", "target": "2 m - n = 3"}, {"rel": "提取因式参考", "source": "2 m - n = 3", "target": "2 - 3 ( 2 m - n )"}, {"rel": "代入", "source": "2 m - n = 3", "target": "- 7"}, {"rel": "代入", "source": "x = 2", "target": "2 m - n = 3"}, {"rel": "提取因式", "source": "2 - 6 m + 3 n", "target": "2 - 3 ( 2 m - n )"}, {"rel": "被代入", "source": "2 - 3 ( 2 m - n )", "target": "- 7"}]}}
{"content": "The equation of the line $y = 2 x + 3$ after translating it $5$ units downwards is _____.", "answer": "y = 2 x - 2", "steps": "The line $y = 2 x + 3$ is translated down $5$ units to become the line $y = 2 x + 3 - 5$, which is equivalent to $y = 2 x - 2$.", "expr_cands": ["y = 2 x + 3", "x", "y", "5", "y = 2 x + 3 - 5", "2 x + 3 = 2 x + 3 - 5", "2 x - 2"], "exprs": ["y = 2 x + 3 - 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = 2 x + 3"}, {"id": "y = 2 x + 3 - 5"}, {"id": "5"}, {"id": "将直线 $y = 2 x + 3$ 向下平移 $5$ 个单位长度后"}], "links": [{"rel": "被描述", "source": "y = 2 x + 3", "target": "y = 2 x + 3 - 5"}, {"rel": "被描述", "source": "5", "target": "y = 2 x + 3 - 5"}, {"rel": "限制性描述", "source": "将直线 $y = 2 x + 3$ 向下平移 $5$ 个单位长度后", "target": "y = 2 x + 3 - 5"}]}}
{"content": "If $2 a - 3$ is a factor of the polynomial $4 a ^ 2 + ma - 9$, then the value of $m$ is ____?", "answer": "0", "steps": "Since $2 a - 3$ is a factor of the polynomial $4 a ^ 2 + ma - 9$, therefore when $2 a - 3 = 0$, we have $4 a ^ 2 + ma - 9 = 0$. That is, when $a = \\frac { 3 } { 2 }$, we have $4 a ^ 2 + ma - 9 = 0$. Thus, substituting $a = \\frac { 3 } { 2 }$ into the equation gives $9 + \\frac { 3 } { 2 } m - 9 = 0$. Therefore, $m = 0$.", "expr_cands": ["2 a - 3", "a", "4 a ^ { 2 } + ma - 9", "m", "2 a - 3 = 0", "a = \\frac { 3 } { 2 }", "4 a ^ { 2 } + ma - 9 = 0", "9 + \\frac { 3 } { 2 } m - 9 = 0", "m = 0"], "exprs": ["2 a - 3 = 0", "a = \\frac { 3 } { 2 }", "9 + \\frac { 3 } { 2 } m - 9 = 0", "m = 0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 a - 3"}, {"id": "2 a - 3 = 0"}, {"id": "4 a ^ { 2 } + ma - 9"}, {"id": "$2 a - 3$ 是多项式 $4 a ^ { 2 } + ma - 9$ 的一个因式"}, {"id": "当 $2 a - 3 = 0$ 时"}, {"id": "a = \\frac { 3 } { 2 }"}, {"id": "9 + \\frac { 3 } { 2 } m - 9 = 0"}, {"id": "4 a ^ { 2 } + ma - 9 = 0"}, {"id": "m = 0"}], "links": [{"rel": "被描述", "source": "2 a - 3", "target": "2 a - 3 = 0"}, {"rel": "等式方程求解", "source": "2 a - 3 = 0", "target": "a = \\frac { 3 } { 2 }"}, {"rel": "被描述", "source": "4 a ^ { 2 } + ma - 9", "target": "2 a - 3 = 0"}, {"rel": "限制性描述", "source": "$2 a - 3$ 是多项式 $4 a ^ { 2 } + ma - 9$ 的一个因式", "target": "2 a - 3 = 0"}, {"rel": "限制性描述", "source": "当 $2 a - 3 = 0$ 时", "target": "2 a - 3 = 0"}, {"rel": "代入", "source": "a = \\frac { 3 } { 2 }", "target": "9 + \\frac { 3 } { 2 } m - 9 = 0"}, {"rel": "等式方程求解", "source": "9 + \\frac { 3 } { 2 } m - 9 = 0", "target": "m = 0"}, {"rel": "被代入", "source": "4 a ^ { 2 } + ma - 9 = 0", "target": "9 + \\frac { 3 } { 2 } m - 9 = 0"}]}}
{"content": "Given $x$ satisfies the inequality $x - a > 0$, $y$ satisfies the inequality $2 - y \\geq 1$, and if $x > y$ always holds, then the range of values for $a$ is ____?", "answer": "a \\ge 1", "steps": "Solve the inequality $x - a > 0$ to get $x > a$, and solve the inequality $2 - y \\ge 1$ to get $y \\le 1$. Since $x > y$ always holds, the maximum value of $y$ is $1$, which is less than all values of $x$. Therefore, $1 \\le a$, which means the range of values for $a$ is $a \\ge 1$.", "expr_cands": ["x", "x - a > 0", "a", "y", "2 - y \\ge 1", "x > y", "x > a", "y \\le 1", "1", "1 \\le a", "a \\ge 1"], "exprs": ["x > a", "y \\le 1", "1 \\le a", "a \\ge 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x - a > 0"}, {"id": "x > a"}, {"id": "2 - y \\ge 1"}, {"id": "y \\le 1"}, {"id": "x > y"}, {"id": "1 \\le a"}, {"id": "$x$ 满足不等式 $x - a > 0$ , $y$ 满足不等式 $2 - y \\ge 1$"}, {"id": "$x > y$ 恒成立"}, {"id": "a \\ge 1"}], "links": [{"rel": "不等式方程部分求解", "source": "x - a > 0", "target": "x > a"}, {"rel": "被描述", "source": "x > a", "target": "1 \\le a"}, {"rel": "不等式方程求解", "source": "2 - y \\ge 1", "target": "y \\le 1"}, {"rel": "被描述", "source": "y \\le 1", "target": "1 \\le a"}, {"rel": "被描述", "source": "x > y", "target": "1 \\le a"}, {"rel": "不等式方程求解", "source": "1 \\le a", "target": "a \\ge 1"}, {"rel": "限制性描述", "source": "$x$ 满足不等式 $x - a > 0$ , $y$ 满足不等式 $2 - y \\ge 1$", "target": "1 \\le a"}, {"rel": "限制性描述", "source": "$x > y$ 恒成立", "target": "1 \\le a"}]}}
{"content": "Given that the equation $x ^ 2 + kx + 3 = 0$ has a root of $x = 3$, what is the value of $k$?", "answer": "- 4", "steps": "According to the problem, we have $3 ^ 2 + 3 k + 3 = 0$, which gives us $k = - 4$ as the solution.", "expr_cands": ["x", "x ^ { 2 } + kx + 3 = 0", "k", "x = 3", "3 ^ { 2 } + 3 k + 3 = 0", "k = - 4"], "exprs": ["3 ^ { 2 } + 3 k + 3 = 0", "k = - 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } + kx + 3 = 0"}, {"id": "3 ^ { 2 } + 3 k + 3 = 0"}, {"id": "x = 3"}, {"id": "k = - 4"}], "links": [{"rel": "被代入", "source": "x ^ { 2 } + kx + 3 = 0", "target": "3 ^ { 2 } + 3 k + 3 = 0"}, {"rel": "等式方程求解", "source": "3 ^ { 2 } + 3 k + 3 = 0", "target": "k = - 4"}, {"rel": "代入", "source": "x = 3", "target": "3 ^ { 2 } + 3 k + 3 = 0"}]}}
{"content": "Given the equation $x ^ 2 - 4 x + m = 0$, if one root is $x = 3$, what is the other root?", "answer": "x = 1", "steps": "Assuming the other root of the equation is $x _ 2$, according to the given condition, we have $3 + x _ 2 = 4$. Solving for $x _ 2$, we get $x _ 2 = 1$.", "expr_cands": ["x ^ { 2 } - 4 x + m = 0", "m", "x", "x = 3", "x _ { 2 }", "3 + x _ { 2 } = 4", "x_{2} = 1", "x _ { 2 } = 1"], "exprs": ["x _ { 2 }", "3 + x _ { 2 } = 4", "x _ { 2 } = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "设方程的另一个根为 $x _ { 2 }$"}, {"id": "x _ { 2 }"}, {"id": "x ^ { 2 } - 4 x + m = 0"}, {"id": "3 + x _ { 2 } = 4"}, {"id": "x = 3"}, {"id": "方程 $x ^ { 2 } - 4 x + m = 0$ 一个根是 $x = 3$"}, {"id": "一元二次方程根与系数关系，两根之和"}, {"id": "x _ { 2 } = 1"}], "links": [{"rel": "假设描述", "source": "设方程的另一个根为 $x _ { 2 }$", "target": "x _ { 2 }"}, {"rel": "限制性描述", "source": "设方程的另一个根为 $x _ { 2 }$", "target": "3 + x _ { 2 } = 4"}, {"rel": "被描述", "source": "x _ { 2 }", "target": "3 + x _ { 2 } = 4"}, {"rel": "被描述", "source": "x ^ { 2 } - 4 x + m = 0", "target": "3 + x _ { 2 } = 4"}, {"rel": "等式方程求解", "source": "3 + x _ { 2 } = 4", "target": "x _ { 2 } = 1"}, {"rel": "被描述", "source": "x = 3", "target": "3 + x _ { 2 } = 4"}, {"rel": "限制性描述", "source": "方程 $x ^ { 2 } - 4 x + m = 0$ 一个根是 $x = 3$", "target": "3 + x _ { 2 } = 4"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "3 + x _ { 2 } = 4"}]}}
{"content": "Calculate given: $3 * 9 ^ { m } * 27 ^ { m } = 3 ^ { 21 }$, then the value of $m$ is ____?", "answer": "4", "steps": "$\\because$ $3 * 9 ^ { m } * 27 ^ { m } = 3 * 3 ^ { 2 m } * 3 ^ { 3 m } = 3 ^ { 5 m + 1 } = 3 ^ { 21 }$, $\\therefore$ $5 m + 1 = 21$, so we get $m = 4$.", "expr_cands": ["3 * 9 ^ { m } * 27 ^ { m } = 3 ^ { 21 }", "m", "3 ^ { 5 m + 1 } = 3 ^ { 21 }", "m = 4", "5 m + 1 = 21"], "exprs": ["m = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 * 9 ^ { m } * 27 ^ { m } = 3 ^ { 21 }"}, {"id": "m = 4"}], "links": [{"rel": "等式方程求解", "source": "3 * 9 ^ { m } * 27 ^ { m } = 3 ^ { 21 }", "target": "m = 4"}]}}
{"content": "Given the line $y = ( m - 1 ) x + m ^ 2 + 2$ passes through the first, second, and fourth quadrants and intersects the line $y = x + 6$ at the same point on the $y$-axis, find the equation of this line.", "answer": "y = - 3 x + 6", "steps": "$\\because$ The line $y = ( m - 1 ) x + m ^ 2 + 2$ passes through the first, second, and fourth quadrants and intersects the line $y = x + 6$ at the same point on the $y$-axis. $\\therefore$ $m ^ 2 + 2 = 6$, $\\therefore$ $m = - 2$, $\\therefore$ the equation of this line is $y = - 3 x + 6$.", "expr_cands": ["y = ( m - 1 ) x + m ^ { 2 } + 2", "x", "m", "y", "y = x + 6", "m ^ { 2 } + x ( m - 1 ) + 2 = x + 6", "m ^ { 2 } + 2 = 6", "m = - 2", "m = 2", "y = - 3 x + 6", "x + 6 = - 3 x + 6", "x + 6"], "exprs": ["m ^ { 2 } + 2 = 6", "m = - 2", "y = - 3 x + 6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = ( m - 1 ) x + m ^ { 2 } + 2"}, {"id": "m ^ { 2 } + 2 = 6"}, {"id": "y = x + 6"}, {"id": "y"}, {"id": "直线 $y = ( m - 1 ) x + m ^ { 2 } + 2$ 过第一"}, {"id": "四象限且与直线 $y = x + 6$ 交于 $y$ 轴同一点"}, {"id": "m = - 2"}, {"id": "y = - 3 x + 6"}], "links": [{"rel": "被描述", "source": "y = ( m - 1 ) x + m ^ { 2 } + 2", "target": "m ^ { 2 } + 2 = 6"}, {"rel": "联立", "source": "y = ( m - 1 ) x + m ^ { 2 } + 2", "target": "y = - 3 x + 6"}, {"rel": "等式方程求解", "source": "m ^ { 2 } + 2 = 6", "target": "m = - 2"}, {"rel": "被描述", "source": "y = x + 6", "target": "m ^ { 2 } + 2 = 6"}, {"rel": "被描述", "source": "y", "target": "m ^ { 2 } + 2 = 6"}, {"rel": "限制性描述", "source": "直线 $y = ( m - 1 ) x + m ^ { 2 } + 2$ 过第一", "target": "m ^ { 2 } + 2 = 6"}, {"rel": "限制性描述", "source": "四象限且与直线 $y = x + 6$ 交于 $y$ 轴同一点", "target": "m ^ { 2 } + 2 = 6"}, {"rel": "联立", "source": "m = - 2", "target": "y = - 3 x + 6"}]}}
{"content": "Given the inequality $4 x - 3 a > - 1$ and the inequality $2 ( x - 1 ) + 3 > 5$ have the same solution set, then $a$ = ____ ?", "answer": "3", "steps": "Solve the inequality $4 x - 3 a > - 1$ to get $x > \\frac { 3 a - 1 } { 4 }$. Solve the inequality $2 ( x - 1 ) + 3 > 5$ to get $x > 2$. Since the solution sets of the two inequalities are the same, we have $\\frac { 3 a - 1 } { 4 } = 2$. Solving for $a$, we get $a = 3$.", "expr_cands": ["4 x - 3 a > - 1", "x", "a", "2 ( x - 1 ) + 3 > 5", "x > \\frac { 3 a - 1 } { 4 }", "2 < x", "x > 2", "\\frac { 3 a - 1 } { 4 } = 2", "a = 3"], "exprs": ["x > \\frac { 3 a - 1 } { 4 }", "x > 2", "\\frac { 3 a - 1 } { 4 } = 2", "a = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 x - 3 a > - 1"}, {"id": "x > \\frac { 3 a - 1 } { 4 }"}, {"id": "2 ( x - 1 ) + 3 > 5"}, {"id": "x > 2"}, {"id": "\\frac { 3 a - 1 } { 4 } = 2"}, {"id": "不等式 $4 x - 3 a > - 1$ 与不等式 $2 ( x - 1 ) + 3 > 5$ 的解集相同"}, {"id": "a = 3"}], "links": [{"rel": "不等式方程部分求解", "source": "4 x - 3 a > - 1", "target": "x > \\frac { 3 a - 1 } { 4 }"}, {"rel": "被描述", "source": "x > \\frac { 3 a - 1 } { 4 }", "target": "\\frac { 3 a - 1 } { 4 } = 2"}, {"rel": "不等式方程求解", "source": "2 ( x - 1 ) + 3 > 5", "target": "x > 2"}, {"rel": "被描述", "source": "x > 2", "target": "\\frac { 3 a - 1 } { 4 } = 2"}, {"rel": "等式方程求解", "source": "\\frac { 3 a - 1 } { 4 } = 2", "target": "a = 3"}, {"rel": "限制性描述", "source": "不等式 $4 x - 3 a > - 1$ 与不等式 $2 ( x - 1 ) + 3 > 5$ 的解集相同", "target": "\\frac { 3 a - 1 } { 4 } = 2"}]}}
{"content": "Given that the equation $x ^ 2 - 3 x - k = 0$ has a root of $2$, the value of $k$ is ____?", "answer": "- 2", "steps": "Substituting $x = 2$ into the equation $x ^ 2 - 3 x - k = 0$ yields $4 - 6 - k = 0$, which gives the solution $k = - 2$.", "expr_cands": ["x ^ { 2 } - 3 x - k = 0", "x", "k", "2", "x = 2", "- k - 2 = 0", "4 - 6 - k = 0", "k = - 2"], "exprs": ["x = 2", "4 - 6 - k = 0", "k = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2"}, {"id": "x = 2"}, {"id": "x ^ { 2 } - 3 x - k = 0"}, {"id": "x"}, {"id": "方程 $x ^ { 2 } - 3 x - k = 0$ 有一根是 $2$"}, {"id": "4 - 6 - k = 0"}, {"id": "k = - 2"}], "links": [{"rel": "被描述", "source": "2", "target": "x = 2"}, {"rel": "代入", "source": "x = 2", "target": "4 - 6 - k = 0"}, {"rel": "被描述", "source": "x ^ { 2 } - 3 x - k = 0", "target": "x = 2"}, {"rel": "被代入", "source": "x ^ { 2 } - 3 x - k = 0", "target": "4 - 6 - k = 0"}, {"rel": "被描述", "source": "x", "target": "x = 2"}, {"rel": "限制性描述", "source": "方程 $x ^ { 2 } - 3 x - k = 0$ 有一根是 $2$", "target": "x = 2"}, {"rel": "等式方程求解", "source": "4 - 6 - k = 0", "target": "k = - 2"}]}}
{"content": "If the line $y = - 2 x + 3 b + 2$ passes through the first, second, and fourth quadrants, then the range of values for $b$ is ____?", "answer": "b > - \\frac { 2 } { 3 }", "steps": "Since the line $y = - 2 x + 3 b + 2$ passes through the first, second, and fourth quadrants, it follows that $3 b + 2 > 0$. Therefore, $b > - \\frac { 2 } { 3 }$.", "expr_cands": ["y = - 2 x + 3 b + 2", "b", "y", "x", "3 b + 2 > 0", "- \\frac { 2 } { 3 } < b", "b > - \\frac { 2 } { 3 }"], "exprs": ["3 b + 2 > 0", "b > - \\frac { 2 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = - 2 x + 3 b + 2"}, {"id": "3 b + 2 > 0"}, {"id": "直线 $y = - 2 x + 3 b + 2$ 经过第一"}, {"id": "二、四象限"}, {"id": "b > - \\frac { 2 } { 3 }"}], "links": [{"rel": "被描述", "source": "y = - 2 x + 3 b + 2", "target": "3 b + 2 > 0"}, {"rel": "不等式方程求解", "source": "3 b + 2 > 0", "target": "b > - \\frac { 2 } { 3 }"}, {"rel": "限制性描述", "source": "直线 $y = - 2 x + 3 b + 2$ 经过第一", "target": "3 b + 2 > 0"}, {"rel": "限制性描述", "source": "二、四象限", "target": "3 b + 2 > 0"}]}}
{"content": "Given a reciprocal function $y = \\frac { m - 1 } { x } ( x > 0 )$, where $y$ increases as $x$ increases, the possible range of values for $m$ is ____?", "answer": "m < 1", "steps": "$\\because$ The inverse proportion function is $y = \\frac { m - 1 } { x }$ ($x > 0$), and $y$ increases as $x$ increases. $\\therefore$ $m - 1 < 0$, which means $m < 1$.", "expr_cands": ["y = \\frac { m - 1 } { x } ( x > 0 )", "y", "m", "x", "m - 1 < 0", "m < 1"], "exprs": ["m - 1 < 0", "m < 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = \\frac { m - 1 } { x } ( x > 0 )"}, {"id": "m - 1 < 0"}, {"id": "y"}, {"id": "x"}, {"id": "己知反比例函数 $y = \\frac { m - 1 } { x } ( x > 0 )$ , $y$ 随 $x$ 的增大而增大"}, {"id": "m < 1"}], "links": [{"rel": "被描述", "source": "y = \\frac { m - 1 } { x } ( x > 0 )", "target": "m - 1 < 0"}, {"rel": "不等式方程求解", "source": "m - 1 < 0", "target": "m < 1"}, {"rel": "被描述", "source": "y", "target": "m - 1 < 0"}, {"rel": "被描述", "source": "x", "target": "m - 1 < 0"}, {"rel": "限制性描述", "source": "己知反比例函数 $y = \\frac { m - 1 } { x } ( x > 0 )$ , $y$ 随 $x$ 的增大而增大", "target": "m - 1 < 0"}]}}
{"content": "If $x = 1$ is a solution of the equation $2 x + a = 6$, then the value of $a$ is ____?", "answer": "4", "steps": "Substituting $x = 1$ into the equation $2 x + a = 6$, we get $2 + a = 6$. Solving for $a$, we get $a = 4$.", "expr_cands": ["x = 1", "x", "2 x + a = 6", "a", "a + 2 = 6", "2 + a = 6", "a = 4"], "exprs": ["2 + a = 6", "a = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x + a = 6"}, {"id": "2 + a = 6"}, {"id": "x = 1"}, {"id": "$x = 1$ 是方程 $2 x + a = 6$ 的解"}, {"id": "a = 4"}], "links": [{"rel": "被描述", "source": "2 x + a = 6", "target": "2 + a = 6"}, {"rel": "等式方程求解", "source": "2 + a = 6", "target": "a = 4"}, {"rel": "被描述", "source": "x = 1", "target": "2 + a = 6"}, {"rel": "限制性描述", "source": "$x = 1$ 是方程 $2 x + a = 6$ 的解", "target": "2 + a = 6"}]}}
{"content": "Given that $x = 4$ is a solution to the equation $ax - 2 = a + 10$, what is the value of $a$?", "answer": "4", "steps": "Substituting $x = 4$ into the equation gives: $4 a - 2 = a + 10$, solving for $a$ gives: $a = 4$.", "expr_cands": ["x = 4", "x", "ax - 2 = a + 10", "a", "4 a - 2 = a + 10", "a = 4"], "exprs": ["4 a - 2 = a + 10", "a = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "ax - 2 = a + 10"}, {"id": "4 a - 2 = a + 10"}, {"id": "x = 4"}, {"id": "$x = 4$ 是方程式 $ax - 2 = a + 10$ 的解"}, {"id": "a = 4"}], "links": [{"rel": "被描述", "source": "ax - 2 = a + 10", "target": "4 a - 2 = a + 10"}, {"rel": "等式方程求解", "source": "4 a - 2 = a + 10", "target": "a = 4"}, {"rel": "被描述", "source": "x = 4", "target": "4 a - 2 = a + 10"}, {"rel": "限制性描述", "source": "$x = 4$ 是方程式 $ax - 2 = a + 10$ 的解", "target": "4 a - 2 = a + 10"}]}}
{"content": "To make the expression $\\sqrt { x - 3 }$ meaningful, what is the possible value of $x$?", "answer": "x \\ge 3", "steps": "From the given condition, we have $x - 3 \\ge 0$. Solving for $x$, we get $x \\ge 3$.", "expr_cands": ["\\sqrt { x - 3 }", "x", "x - 3 \\ge 0", "3 \\le x", "x \\ge 3"], "exprs": ["x - 3 \\ge 0", "x \\ge 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x - 3 }"}, {"id": "x - 3 \\ge 0"}, {"id": "要使式子 $\\sqrt { x - 3 }$ 有意义"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x \\ge 3"}], "links": [{"rel": "被描述", "source": "\\sqrt { x - 3 }", "target": "x - 3 \\ge 0"}, {"rel": "不等式方程求解", "source": "x - 3 \\ge 0", "target": "x \\ge 3"}, {"rel": "限制性描述", "source": "要使式子 $\\sqrt { x - 3 }$ 有意义", "target": "x - 3 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 3 \\ge 0"}]}}
{"content": "If the value of $\\frac { 2 } { 3 } x - 4$ is not greater than $6$, what is the range of possible values for $x$?", "answer": "x \\le 15", "steps": "$\\frac { 2 } { 3 } x - 4 \\le 6$, moving terms yields $\\frac { 2 } { 3 } x \\le 10$, solving gives $x \\le 15$.", "expr_cands": ["\\frac { 2 } { 3 } x - 4", "x", "6", "\\frac { 2 } { 3 } x - 4 \\le 6", "x \\le 15", "\\frac { 2 } { 3 } x \\le 10"], "exprs": ["\\frac { 2 } { 3 } x - 4 \\le 6", "x \\le 15"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 2 } { 3 } x - 4"}, {"id": "\\frac { 2 } { 3 } x - 4 \\le 6"}, {"id": "6"}, {"id": "$\\frac { 2 } { 3 } x - 4$ 的值不大于 $6$"}, {"id": "x \\le 15"}], "links": [{"rel": "被描述", "source": "\\frac { 2 } { 3 } x - 4", "target": "\\frac { 2 } { 3 } x - 4 \\le 6"}, {"rel": "不等式方程求解", "source": "\\frac { 2 } { 3 } x - 4 \\le 6", "target": "x \\le 15"}, {"rel": "被描述", "source": "6", "target": "\\frac { 2 } { 3 } x - 4 \\le 6"}, {"rel": "限制性描述", "source": "$\\frac { 2 } { 3 } x - 4$ 的值不大于 $6$", "target": "\\frac { 2 } { 3 } x - 4 \\le 6"}]}}
{"content": "If ${ ( x - \\frac { 1 } { 2 } ) } ^ { 0 }$ is undefined, then the value of ${ x } ^ { - 2 }$ is ____?", "answer": "4", "steps": "From the given information, we have $x = \\frac { 1 } { 2 }$. When $x = \\frac { 1 } { 2 }$, ${ x } ^ { - 2 } = \\frac { 1 } { x ^ 2 } = \\frac { 1 } {( \\frac { 1 } { 2 }) ^ 2 } = 4$.", "expr_cands": ["{ ( x - \\frac { 1 } { 2 } ) } ^ { 0 }", "x", "{ x } ^ { - 2 }", "x = \\frac { 1 } { 2 }", "4"], "exprs": ["x = \\frac { 1 } { 2 }", "4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "{ ( x - \\frac { 1 } { 2 } ) } ^ { 0 }"}, {"id": "x = \\frac { 1 } { 2 }"}, {"id": "${ ( x - \\frac { 1 } { 2 } ) } ^ { 0 }$ 没有意义"}, {"id": "多项式零次方项，若在实数范围内有意义，则底数不为0，无意义则底数为0"}, {"id": "{ x } ^ { - 2 }"}, {"id": "4"}], "links": [{"rel": "被描述", "source": "{ ( x - \\frac { 1 } { 2 } ) } ^ { 0 }", "target": "x = \\frac { 1 } { 2 }"}, {"rel": "代入", "source": "x = \\frac { 1 } { 2 }", "target": "4"}, {"rel": "限制性描述", "source": "${ ( x - \\frac { 1 } { 2 } ) } ^ { 0 }$ 没有意义", "target": "x = \\frac { 1 } { 2 }"}, {"rel": "属性描述", "source": "多项式零次方项，若在实数范围内有意义，则底数不为0，无意义则底数为0", "target": "x = \\frac { 1 } { 2 }"}, {"rel": "被代入", "source": "{ x } ^ { - 2 }", "target": "4"}]}}
{"content": "Given $\\sqrt { x + 3 } + | 3 x + 2 y - 15 | = 0$, what is the square root of the arithmetic mean of $x$ and $y$?", "answer": "\\sqrt { 3 }", "steps": "From the given information, we have $x + 3 = 0$ and $3 x + 2 y - 15 = 0$. Solving these equations, we get $x = - 3$ and $y = 12$. Therefore, $\\sqrt { x + y } = \\sqrt { - 3 + 12 } = 3$. Hence, the arithmetic square root of $\\sqrt { x + y }$ is $\\sqrt { 3 }$.", "expr_cands": ["\\sqrt { x + 3 } + | 3 x + 2 y - 15 | = 0", "x", "y", "\\sqrt { x + y }", "x + 3 = 0", "x = - 3", "3 x + 2 y - 15 = 0", "y = 12", "3", "\\sqrt { 3 }"], "exprs": ["x + 3 = 0", "3 x + 2 y - 15 = 0", "x = - 3", "y = 12", "\\sqrt { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x + 3 } + | 3 x + 2 y - 15 | = 0"}, {"id": "x + 3 = 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x = - 3"}, {"id": "3 x + 2 y - 15 = 0"}, {"id": "绝对值恒大于等于0"}, {"id": "y = 12"}, {"id": "\\sqrt { x + y }"}, {"id": "\\sqrt { 3 }"}, {"id": "$\\sqrt { x + y }$ 的算术平方根"}], "links": [{"rel": "被描述", "source": "\\sqrt { x + 3 } + | 3 x + 2 y - 15 | = 0", "target": "x + 3 = 0"}, {"rel": "被描述", "source": "\\sqrt { x + 3 } + | 3 x + 2 y - 15 | = 0", "target": "3 x + 2 y - 15 = 0"}, {"rel": "等式方程求解", "source": "x + 3 = 0", "target": "x = - 3"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x + 3 = 0"}, {"rel": "联立", "source": "x = - 3", "target": "y = 12"}, {"rel": "被描述", "source": "x = - 3", "target": "\\sqrt { 3 }"}, {"rel": "联立", "source": "3 x + 2 y - 15 = 0", "target": "y = 12"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "3 x + 2 y - 15 = 0"}, {"rel": "被描述", "source": "y = 12", "target": "\\sqrt { 3 }"}, {"rel": "被描述", "source": "\\sqrt { x + y }", "target": "\\sqrt { 3 }"}, {"rel": "限制性描述", "source": "$\\sqrt { x + y }$ 的算术平方根", "target": "\\sqrt { 3 }"}]}}
{"content": "Regarding the quadratic equation in one variable $x$, $x ^ 2 + mx + 2 = 0$, if one of its roots is $1$, then the value of $m$ is ____?", "answer": "m = - 3", "steps": "Substituting $x = 1$, we get $1 + m + 2 = 0$, which gives $m = - 3$.", "expr_cands": ["x", "{ x } ^ { 2 } + mx + 2 = 0", "m", "1", "x = 1", "1 + m + 2 = 0", "m = - 3"], "exprs": ["x = 1", "1 + m + 2 = 0", "m = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "1"}, {"id": "x = 1"}, {"id": "x"}, {"id": "{ x } ^ { 2 } + mx + 2 = 0"}, {"id": "关于 $x$ 的一元二次方程 ${ x } ^ { 2 } + mx + 2 = 0$ 的一个根为 $1$"}, {"id": "1 + m + 2 = 0"}, {"id": "m = - 3"}], "links": [{"rel": "被描述", "source": "1", "target": "x = 1"}, {"rel": "代入", "source": "x = 1", "target": "1 + m + 2 = 0"}, {"rel": "被描述", "source": "x", "target": "x = 1"}, {"rel": "被描述", "source": "{ x } ^ { 2 } + mx + 2 = 0", "target": "x = 1"}, {"rel": "被代入", "source": "{ x } ^ { 2 } + mx + 2 = 0", "target": "1 + m + 2 = 0"}, {"rel": "限制性描述", "source": "关于 $x$ 的一元二次方程 ${ x } ^ { 2 } + mx + 2 = 0$ 的一个根为 $1$", "target": "x = 1"}, {"rel": "等式方程求解", "source": "1 + m + 2 = 0", "target": "m = - 3"}]}}
{"content": "Given that $x = 5$ is a solution to the equation $mx - 8 = 20 + m$, then $m$ = ____ ?", "answer": "7", "steps": "Substituting $x = 5$ into the equation, we get $5 m - 8 = 20 + m$. Solving for $m$, we get $m = 7$.", "expr_cands": ["x = 5", "x", "mx - 8 = 20 + m", "m", "5 m - 8 = 20 + m", "m = 7"], "exprs": ["5 m - 8 = 20 + m", "m = 7"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "mx - 8 = 20 + m"}, {"id": "5 m - 8 = 20 + m"}, {"id": "x = 5"}, {"id": "m = 7"}], "links": [{"rel": "被代入", "source": "mx - 8 = 20 + m", "target": "5 m - 8 = 20 + m"}, {"rel": "等式方程求解", "source": "5 m - 8 = 20 + m", "target": "m = 7"}, {"rel": "代入", "source": "x = 5", "target": "5 m - 8 = 20 + m"}]}}
{"content": "The range of values of $a$ that make the square root expression $\\sqrt { 2 a + 1 }$ meaningful is _____.", "answer": "a \\ge - \\frac { 1 } { 2 }", "steps": "From the given condition, we have $2 a + 1 \\geq 0$. Solving for $a$, we get $a \\geq - \\frac { 1 } { 2 }$.", "expr_cands": ["\\sqrt { 2 a + 1 }", "a", "2 a + 1 \\ge 0", "- \\frac { 1 } { 2 } \\le a", "a \\ge - \\frac { 1 } { 2 }"], "exprs": ["2 a + 1 \\ge 0", "a \\ge - \\frac { 1 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { 2 a + 1 }"}, {"id": "2 a + 1 \\ge 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "使二次根式 $\\sqrt { 2 a + 1 }$ 有意义的 $a$ 的取值范围"}, {"id": "a \\ge - \\frac { 1 } { 2 }"}], "links": [{"rel": "被描述", "source": "\\sqrt { 2 a + 1 }", "target": "2 a + 1 \\ge 0"}, {"rel": "不等式方程求解", "source": "2 a + 1 \\ge 0", "target": "a \\ge - \\frac { 1 } { 2 }"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "2 a + 1 \\ge 0"}, {"rel": "限制性描述", "source": "使二次根式 $\\sqrt { 2 a + 1 }$ 有意义的 $a$ 的取值范围", "target": "2 a + 1 \\ge 0"}]}}
{"content": "In the equation $2 x + 3 y = 16$, if $x = 2$, then $y$ = ____ ?", "answer": "4", "steps": "The equation $2 x + 3 y = 16$ is given. Substituting $x = 2$, we get $4 + 3 y = 16$, which can be solved to obtain $y = 4$.", "expr_cands": ["2 x + 3 y = 16", "y", "x", "x = 2", "4 + 3 y = 16", "y = 4"], "exprs": ["4 + 3 y = 16", "y = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x + 3 y = 16"}, {"id": "4 + 3 y = 16"}, {"id": "x = 2"}, {"id": "y = 4"}], "links": [{"rel": "被代入", "source": "2 x + 3 y = 16", "target": "4 + 3 y = 16"}, {"rel": "等式方程求解", "source": "4 + 3 y = 16", "target": "y = 4"}, {"rel": "代入", "source": "x = 2", "target": "4 + 3 y = 16"}]}}
{"content": "If $5 x ^ { 2 } y ^ { n }$ and $4 x ^ { m - 1 } y$ are similar terms, then the value of the algebraic expression $m - n$ is ____?", "answer": "2", "steps": "From the given information, we have $m - 1 = 2$ and $n = 1$. Solving for $m$ and $n$, we get $m = 3$ and $n = 1$. Therefore, $m - n = 3 - 1 = 2$.", "expr_cands": ["5 x ^ { 2 } y ^ { n }", "y", "n", "x", "4 x ^ { m - 1 } y", "m", "m - n", "m - 1 = 2", "m = 3", "n = 1", "2"], "exprs": ["m - 1 = 2", "n = 1", "m = 3", "2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 x ^ { m - 1 } y"}, {"id": "m - 1 = 2"}, {"id": "5 x ^ { 2 } y ^ { n }"}, {"id": "$5 x ^ { 2 } y ^ { n }$ 与 $4 x ^ { m - 1 } y$ 是同类项"}, {"id": "m = 3"}, {"id": "n = 1"}, {"id": "m - n"}, {"id": "2"}], "links": [{"rel": "被描述", "source": "4 x ^ { m - 1 } y", "target": "m - 1 = 2"}, {"rel": "被描述", "source": "4 x ^ { m - 1 } y", "target": "n = 1"}, {"rel": "等式方程求解", "source": "m - 1 = 2", "target": "m = 3"}, {"rel": "被描述", "source": "5 x ^ { 2 } y ^ { n }", "target": "m - 1 = 2"}, {"rel": "被描述", "source": "5 x ^ { 2 } y ^ { n }", "target": "n = 1"}, {"rel": "限制性描述", "source": "$5 x ^ { 2 } y ^ { n }$ 与 $4 x ^ { m - 1 } y$ 是同类项", "target": "m - 1 = 2"}, {"rel": "限制性描述", "source": "$5 x ^ { 2 } y ^ { n }$ 与 $4 x ^ { m - 1 } y$ 是同类项", "target": "n = 1"}, {"rel": "代入", "source": "m = 3", "target": "2"}, {"rel": "代入", "source": "n = 1", "target": "2"}, {"rel": "被代入", "source": "m - n", "target": "2"}]}}
{"content": "When $a = \\sqrt { 3 } + \\sqrt { 2 }$ and $b = \\sqrt { 3 } - \\sqrt { 2 }$, what is the value of the algebraic expression $a ^ { 2 } - b ^ { 2 }$?", "answer": "4 \\sqrt { 6 }", "steps": "Since $a = \\sqrt { 3 } + \\sqrt { 2 }$ and $b = \\sqrt { 3 } - \\sqrt { 2 }$, we have $a + b = 2 \\sqrt { 3 }$ and $a - b = 2 \\sqrt { 2 }$. Therefore, $a ^ 2 - b ^ 2 = ( a + b ) ( a - b ) = 2 \\sqrt { 3 } \\cdot 2 \\sqrt { 2 } = 4 \\sqrt { 6 }$.", "expr_cands": ["a = \\sqrt { 3 } + \\sqrt { 2 }", "a", "b = \\sqrt { 3 } - \\sqrt { 2 }", "b", "a ^ { 2 } - b ^ { 2 }", "a + b", "2 \\sqrt { 3 }", "a - b", "2 \\sqrt { 2 }", "4 \\sqrt { 6 }"], "exprs": ["4 \\sqrt { 6 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a ^ { 2 } - b ^ { 2 }"}, {"id": "4 \\sqrt { 6 }"}, {"id": "a = \\sqrt { 3 } + \\sqrt { 2 }"}, {"id": "b = \\sqrt { 3 } - \\sqrt { 2 }"}], "links": [{"rel": "被代入", "source": "a ^ { 2 } - b ^ { 2 }", "target": "4 \\sqrt { 6 }"}, {"rel": "代入", "source": "a = \\sqrt { 3 } + \\sqrt { 2 }", "target": "4 \\sqrt { 6 }"}, {"rel": "代入", "source": "b = \\sqrt { 3 } - \\sqrt { 2 }", "target": "4 \\sqrt { 6 }"}]}}
{"content": "If the fractional equation $\\frac { 3 } { x + 2 } + \\frac { a } { x + 2 } = 1$ has a positive root, then the value of $a$ is ____?", "answer": "- 3", "steps": "Eliminating the denominator in the fractional equation, we get $3 + a = x + 2$. Since the fractional equation has an extraneous root, we have $x + 2 = 0$, which means $x = - 2$. Substituting $x = - 2$ into the polynomial equation, we get $a = - 3$.", "expr_cands": ["\\frac { 3 } { x + 2 } + \\frac { a } { x + 2 } = 1", "x", "a", "3 + a = x + 2", "x + 2 = 0", "x = - 2", "a = - 3"], "exprs": ["3 + a = x + 2", "x + 2 = 0", "x = - 2", "a = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 3 } { x + 2 } + \\frac { a } { x + 2 } = 1"}, {"id": "3 + a = x + 2"}, {"id": "x + 2 = 0"}, {"id": "分式方程 $\\frac { 3 } { x + 2 } + \\frac { a } { x + 2 } = 1$ 有增根"}, {"id": "分式有增根，则分母为0"}, {"id": "x = - 2"}, {"id": "a = - 3"}], "links": [{"rel": "同乘除", "source": "\\frac { 3 } { x + 2 } + \\frac { a } { x + 2 } = 1", "target": "3 + a = x + 2"}, {"rel": "被描述", "source": "\\frac { 3 } { x + 2 } + \\frac { a } { x + 2 } = 1", "target": "x + 2 = 0"}, {"rel": "联立", "source": "3 + a = x + 2", "target": "a = - 3"}, {"rel": "等式方程求解", "source": "x + 2 = 0", "target": "x = - 2"}, {"rel": "限制性描述", "source": "分式方程 $\\frac { 3 } { x + 2 } + \\frac { a } { x + 2 } = 1$ 有增根", "target": "x + 2 = 0"}, {"rel": "属性描述", "source": "分式有增根，则分母为0", "target": "x + 2 = 0"}, {"rel": "联立", "source": "x = - 2", "target": "a = - 3"}]}}
{"content": "Given $27 ^ b = 9 * 3 ^ { a + 3 }$ and $16 = 4 * 2 ^ { 2 b - 2 }$, what is the value of $a + b$?", "answer": "3", "steps": "Because $27 ^ { b } = 3 ^ { 3 b } = 9 * 3 ^ { a + 3 } = 3 ^ { a + 5 }$, $16 = 2 ^ { 4 } = 4 * 2 ^ { 2 b - 2 } = 2 ^ { 2 b }$, therefore $a + 5 = 3 b$, $2 b = 4$, which leads to $b = 2$, $a = 1$, thus $a + b = 1 + 2 = 3$.", "expr_cands": ["27 ^ { b } = 9 * 3 ^ { a + 3 }", "a", "b", "16 = 4 * 2 ^ { 2 b - 2 }", "a + b", "27 ^ { b } = 3 ^ { a + 5 }", "16 = 2 ^ { 2 b }", "b = 2", "a + 5 = 3 b", "2 b = 4", "a = 1", "3"], "exprs": ["27 ^ { b } = 3 ^ { a + 5 }", "16 = 2 ^ { 2 b }", "b = 2", "a = 1", "3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "27 ^ { b } = 9 * 3 ^ { a + 3 }"}, {"id": "27 ^ { b } = 3 ^ { a + 5 }"}, {"id": "16 = 4 * 2 ^ { 2 b - 2 }"}, {"id": "16 = 2 ^ { 2 b }"}, {"id": "b = 2"}, {"id": "a = 1"}, {"id": "a + b"}, {"id": "3"}], "links": [{"rel": "计算", "source": "27 ^ { b } = 9 * 3 ^ { a + 3 }", "target": "27 ^ { b } = 3 ^ { a + 5 }"}, {"rel": "联立", "source": "27 ^ { b } = 3 ^ { a + 5 }", "target": "a = 1"}, {"rel": "计算", "source": "16 = 4 * 2 ^ { 2 b - 2 }", "target": "16 = 2 ^ { 2 b }"}, {"rel": "等式方程求解", "source": "16 = 2 ^ { 2 b }", "target": "b = 2"}, {"rel": "联立", "source": "b = 2", "target": "a = 1"}, {"rel": "代入", "source": "b = 2", "target": "3"}, {"rel": "代入", "source": "a = 1", "target": "3"}, {"rel": "被代入", "source": "a + b", "target": "3"}]}}
{"content": "If $- 2 a ^ { 2 } b \\cdot ma ^ { 2 } b ^ { 2 } = 6 a ^ { 4 } b ^ { 3 }$, then the value of $m$ is ____?", "answer": "- 3", "steps": "$- 2 a ^ { 2 } b \\cdot ma ^ { 2 } b ^ { 2 } = - 2 ma ^ { 4 } b ^ { 3 }$, because $- 2 a ^ { 2 } b \\cdot ma ^ { 2 } b ^ { 2 } = 6 a ^ { 4 } b ^ { 3 }$, therefore $- 2 m = 6$, therefore $m = - 3$.", "expr_cands": ["- 2 a ^ { 2 } b \\cdot ma ^ { 2 } b ^ { 2 } = 6 a ^ { 4 } b ^ { 3 }", "m", "b", "a", "- 2 a ^ { 2 } b \\cdot ma ^ { 2 } b ^ { 2 }", "- 2 ma ^ { 4 } b ^ { 3 }", "- 2 m = 6", "m = - 3"], "exprs": ["- 2 m = 6", "m = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 2 a ^ { 2 } b \\cdot ma ^ { 2 } b ^ { 2 } = 6 a ^ { 4 } b ^ { 3 }"}, {"id": "- 2 m = 6"}, {"id": "m = - 3"}], "links": [{"rel": "同乘除", "source": "- 2 a ^ { 2 } b \\cdot ma ^ { 2 } b ^ { 2 } = 6 a ^ { 4 } b ^ { 3 }", "target": "- 2 m = 6"}, {"rel": "等式方程求解", "source": "- 2 m = 6", "target": "m = - 3"}]}}
{"content": "Given the equation about $x$: $x ^ 2 - ( m + 2 ) x + ( 2 m - 1 ) = 0$, if one root of this equation is $1$, then the other root of the equation is ____?", "answer": "3", "steps": "$\\because$ One root of the equation is $1$, $\\therefore$ $1 ^ 2 - ( m + 2 ) + 2 m - 1 = 0$, solving for $m$ gives $m = 2$, $\\therefore$ the original equation is $x ^ 2 - 4 x + 3 = 0$. Let the other root be $x _ 2$, then $x _ 2 + 1 = 4$. $\\therefore$ $x _ 2 = 3$. Hence, the other root of the equation is $3$.", "expr_cands": ["x", "x ^ { 2 } - ( m + 2 ) x + ( 2 m - 1 ) = 0", "m", "1", "1 ^ { 2 } - ( m + 2 ) + 2 m - 1 = 0", "m = 2", "x ^ { 2 } - 4 x + 3 = 0", "x = 1", "x = 3", "x _ { 2 }", "x _ { 2 } + 1 = 4", "x_{2} = 3", "x _ { 2 } = 3", "3"], "exprs": ["1 ^ { 2 } - ( m + 2 ) + 2 m - 1 = 0", "x _ { 2 }", "m = 2", "x ^ { 2 } - 4 x + 3 = 0", "x _ { 2 } + 1 = 4", "x _ { 2 } = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "1"}, {"id": "1 ^ { 2 } - ( m + 2 ) + 2 m - 1 = 0"}, {"id": "x ^ { 2 } - ( m + 2 ) x + ( 2 m - 1 ) = 0"}, {"id": "x"}, {"id": "关于 $x$ 的方程 $x ^ { 2 } - ( m + 2 ) x + ( 2 m - 1 ) = 0$"}, {"id": "此方程的一个根是 $1$"}, {"id": "m = 2"}, {"id": "x ^ { 2 } - 4 x + 3 = 0"}, {"id": "设另一根为 $x _ { 2 }$"}, {"id": "x _ { 2 }"}, {"id": "x _ { 2 } + 1 = 4"}, {"id": "一元二次方程根与系数关系，两根之和"}, {"id": "x _ { 2 } = 3"}], "links": [{"rel": "被描述", "source": "1", "target": "1 ^ { 2 } - ( m + 2 ) + 2 m - 1 = 0"}, {"rel": "被描述", "source": "1", "target": "x _ { 2 } + 1 = 4"}, {"rel": "等式方程求解", "source": "1 ^ { 2 } - ( m + 2 ) + 2 m - 1 = 0", "target": "m = 2"}, {"rel": "被描述", "source": "x ^ { 2 } - ( m + 2 ) x + ( 2 m - 1 ) = 0", "target": "1 ^ { 2 } - ( m + 2 ) + 2 m - 1 = 0"}, {"rel": "被代入", "source": "x ^ { 2 } - ( m + 2 ) x + ( 2 m - 1 ) = 0", "target": "x ^ { 2 } - 4 x + 3 = 0"}, {"rel": "被描述", "source": "x", "target": "1 ^ { 2 } - ( m + 2 ) + 2 m - 1 = 0"}, {"rel": "被描述", "source": "x", "target": "x _ { 2 } + 1 = 4"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $x ^ { 2 } - ( m + 2 ) x + ( 2 m - 1 ) = 0$", "target": "1 ^ { 2 } - ( m + 2 ) + 2 m - 1 = 0"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $x ^ { 2 } - ( m + 2 ) x + ( 2 m - 1 ) = 0$", "target": "x _ { 2 } + 1 = 4"}, {"rel": "限制性描述", "source": "此方程的一个根是 $1$", "target": "1 ^ { 2 } - ( m + 2 ) + 2 m - 1 = 0"}, {"rel": "限制性描述", "source": "此方程的一个根是 $1$", "target": "x _ { 2 } + 1 = 4"}, {"rel": "代入", "source": "m = 2", "target": "x ^ { 2 } - 4 x + 3 = 0"}, {"rel": "被描述", "source": "x ^ { 2 } - 4 x + 3 = 0", "target": "x _ { 2 } + 1 = 4"}, {"rel": "假设描述", "source": "设另一根为 $x _ { 2 }$", "target": "x _ { 2 }"}, {"rel": "限制性描述", "source": "设另一根为 $x _ { 2 }$", "target": "x _ { 2 } + 1 = 4"}, {"rel": "被描述", "source": "x _ { 2 }", "target": "x _ { 2 } + 1 = 4"}, {"rel": "等式方程求解", "source": "x _ { 2 } + 1 = 4", "target": "x _ { 2 } = 3"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "x _ { 2 } + 1 = 4"}]}}
{"content": "Given that $a$ and $b$ are opposite numbers, $m$ and $n$ are reciprocal, and $x$ is the largest negative integer, what is the value of $- 2 mn + \\frac { a + b } { m - n } + x ^ 2$?", "answer": "- 1", "steps": "According to the problem, we know that $a + b = 0$, $mn = 1$, and $x = - 1$. The original expression is equal to $- 2 \\times 1 + \\frac { 0 } { m - n } + ( - 1 ) ^ 2 = - 2 + 0 + 1 = - 1$.", "expr_cands": ["a", "b", "m", "n", "x", "- 2 mn + \\frac { a + b } { m - n } + x ^ { 2 }", "a + b = 0", "mn = 1", "x = - 1", "- 2 * 1 + \\frac { 0 } { m - n } + ( - 1 ) ^ { 2 }", "- 1"], "exprs": ["a + b = 0", "mn = 1", "x = - 1", "- 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a + b = 0"}, {"id": "b"}, {"id": "$a$ , $b$ 互为相反数"}, {"id": "m"}, {"id": "mn = 1"}, {"id": "n"}, {"id": "$m$ , $n$ 互为倒数"}, {"id": "x"}, {"id": "x = - 1"}, {"id": "$x$ 是最大的负整数"}, {"id": "- 2 mn + \\frac { a + b } { m - n } + x ^ { 2 }"}, {"id": "- 1"}], "links": [{"rel": "被描述", "source": "a", "target": "a + b = 0"}, {"rel": "代入", "source": "a + b = 0", "target": "- 1"}, {"rel": "被描述", "source": "b", "target": "a + b = 0"}, {"rel": "限制性描述", "source": "$a$ , $b$ 互为相反数", "target": "a + b = 0"}, {"rel": "被描述", "source": "m", "target": "mn = 1"}, {"rel": "代入", "source": "mn = 1", "target": "- 1"}, {"rel": "被描述", "source": "n", "target": "mn = 1"}, {"rel": "限制性描述", "source": "$m$ , $n$ 互为倒数", "target": "mn = 1"}, {"rel": "被描述", "source": "x", "target": "x = - 1"}, {"rel": "代入", "source": "x = - 1", "target": "- 1"}, {"rel": "限制性描述", "source": "$x$ 是最大的负整数", "target": "x = - 1"}, {"rel": "被代入", "source": "- 2 mn + \\frac { a + b } { m - n } + x ^ { 2 }", "target": "- 1"}]}}
{"content": "If a linear function is $y = kx - 5$, and when $x = - 2$, $y = 7$, then the value of $k$ is ____?", "answer": "- 6", "steps": "\\because $y = kx - 5$, when $x = - 2$, $y = 7$, \\therefore substituting gives: $7 = - 2 k - 5$, solving for $k$, we get: $k = - 6$.", "expr_cands": ["y = kx - 5", "x", "k", "y", "x = - 2", "y = 7", "7 = - 2 k - 5", "k = - 6"], "exprs": ["7 = - 2 k - 5", "k = - 6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = kx - 5"}, {"id": "7 = - 2 k - 5"}, {"id": "x = - 2"}, {"id": "y = 7"}, {"id": "k = - 6"}], "links": [{"rel": "被代入", "source": "y = kx - 5", "target": "7 = - 2 k - 5"}, {"rel": "等式方程求解", "source": "7 = - 2 k - 5", "target": "k = - 6"}, {"rel": "代入", "source": "x = - 2", "target": "7 = - 2 k - 5"}, {"rel": "代入", "source": "y = 7", "target": "7 = - 2 k - 5"}]}}
{"content": "If $y = ( a - 2 ) x + { a } ^ { 2 } - 4$ is a proportional function, then the value of $a$ is ____?", "answer": "- 2", "steps": "From the given condition, we have $a ^ 2 - 4 = 0$ and $a - 2 \\neq 0$. Solving for $a$, we get $a = - 2$.", "expr_cands": ["y = ( a - 2 ) x + { a } ^ { 2 } - 4", "a", "y", "x", "a ^ { 2 } - 4 = 0", "a = - 2", "a = 2", "a - 2 \\neq 0", "a \\neq 2"], "exprs": ["a ^ { 2 } - 4 = 0", "a - 2 \\neq 0", "a = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = ( a - 2 ) x + { a } ^ { 2 } - 4"}, {"id": "a ^ { 2 } - 4 = 0"}, {"id": "$y = ( a - 2 ) x + { a } ^ { 2 } - 4$ 为正比例函数"}, {"id": "a - 2 \\neq 0"}, {"id": "a = - 2"}], "links": [{"rel": "被描述", "source": "y = ( a - 2 ) x + { a } ^ { 2 } - 4", "target": "a ^ { 2 } - 4 = 0"}, {"rel": "被描述", "source": "y = ( a - 2 ) x + { a } ^ { 2 } - 4", "target": "a - 2 \\neq 0"}, {"rel": "联立", "source": "a ^ { 2 } - 4 = 0", "target": "a = - 2"}, {"rel": "限制性描述", "source": "$y = ( a - 2 ) x + { a } ^ { 2 } - 4$ 为正比例函数", "target": "a ^ { 2 } - 4 = 0"}, {"rel": "限制性描述", "source": "$y = ( a - 2 ) x + { a } ^ { 2 } - 4$ 为正比例函数", "target": "a - 2 \\neq 0"}, {"rel": "联立", "source": "a - 2 \\neq 0", "target": "a = - 2"}]}}
{"content": "Given $9 ^ { m } * 27 ^ { n } = 81$, what is the value of $6 - 4 m - 6 n$?", "answer": "- 2", "steps": "Since $9 ^ m \\times 27 ^ n = 81$, it follows that $3 ^ { 2 m } \\times 3 ^ { 3 n } = 3 ^ 4$. Therefore, $2 m + 3 n = 4$. This implies that $6 - 4 m - 6 n = 6 - 2 ( 2 m + 3 n ) = 6 - 2 \\times 4 = 6 - 8 = - 2$.", "expr_cands": ["9 ^ { m } * 27 ^ { n } = 81", "m", "n", "6 - 4 m - 6 n", "3 ^ { 2 m } \\times 3 ^ { 3 n }", "81", "2 m + 3 n = 4", "6 - 2 ( 2 m + 3 n )", "- 2"], "exprs": ["2 m + 3 n = 4", "6 - 2 ( 2 m + 3 n )", "- 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "9 ^ { m } * 27 ^ { n } = 81"}, {"id": "2 m + 3 n = 4"}, {"id": "6 - 4 m - 6 n"}, {"id": "6 - 2 ( 2 m + 3 n )"}, {"id": "- 2"}], "links": [{"rel": "同取对数", "source": "9 ^ { m } * 27 ^ { n } = 81", "target": "2 m + 3 n = 4"}, {"rel": "提取因式参考", "source": "2 m + 3 n = 4", "target": "6 - 2 ( 2 m + 3 n )"}, {"rel": "代入", "source": "2 m + 3 n = 4", "target": "- 2"}, {"rel": "提取因式", "source": "6 - 4 m - 6 n", "target": "6 - 2 ( 2 m + 3 n )"}, {"rel": "被代入", "source": "6 - 2 ( 2 m + 3 n )", "target": "- 2"}]}}
{"content": "If the value of the expression $x - 1$ is $- 2$, then the value of $x$ is ____?", "answer": "- 1", "steps": "According to the problem, we have $x - 1 = - 2$. Moving the constant term to the other side, we get $x = - 2 + 1$. Combining like terms, we have $x = - 1$.", "expr_cands": ["x - 1", "x", "- 2", "x - 1 = - 2", "x = - 1", "x = - 2 + 1"], "exprs": ["x - 1 = - 2", "x = - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x - 1"}, {"id": "x - 1 = - 2"}, {"id": "- 2"}, {"id": "式子 $x - 1$ 的值是 $- 2$"}, {"id": "x = - 1"}], "links": [{"rel": "被描述", "source": "x - 1", "target": "x - 1 = - 2"}, {"rel": "等式方程求解", "source": "x - 1 = - 2", "target": "x = - 1"}, {"rel": "被描述", "source": "- 2", "target": "x - 1 = - 2"}, {"rel": "限制性描述", "source": "式子 $x - 1$ 的值是 $- 2$", "target": "x - 1 = - 2"}]}}
{"content": "If $A : B = \\frac { 1 } { 9 }$, then $( A * 9 ) : ( B * 9 )$ = ____?", "answer": "\\frac { 1 } { 9 }", "steps": "Because $A : B = \\frac { 1 } { 9 }$, therefore $( A * 9 ) : ( B * 9 ) = 1 : 9$.", "expr_cands": ["A : B = \\frac { 1 } { 9 }", "A", "B", "( A * 9 ) : ( B * 9 )", "\\frac { 1 } { 9 }"], "exprs": ["\\frac { 1 } { 9 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( A * 9 ) : ( B * 9 )"}, {"id": "\\frac { 1 } { 9 }"}, {"id": "A : B = \\frac { 1 } { 9 }"}], "links": [{"rel": "被代入", "source": "( A * 9 ) : ( B * 9 )", "target": "\\frac { 1 } { 9 }"}, {"rel": "代入", "source": "A : B = \\frac { 1 } { 9 }", "target": "\\frac { 1 } { 9 }"}]}}
{"content": "The equation $\\sqrt { x - 1 } \\cdot \\sqrt { x + 1 } = \\sqrt { { x } ^ { 2 } - 1 }$ holds true when ____ ?", "answer": "x \\ge 1", "steps": "$\\because \\sqrt { x + 1 } \\times \\sqrt { x - 1 } = \\sqrt { { x } ^ { 2 } - 1 }$ holds, $\\therefore x + 1 \\ge 0$, $x - 1 \\ge 0$. Solving, we get $x \\ge 1$.", "expr_cands": ["\\sqrt { x - 1 } \\cdot \\sqrt { x + 1 } = \\sqrt { { x } ^ { 2 } - 1 }", "x", "\\sqrt { x + 1 } \\times \\sqrt { x - 1 } = \\sqrt { { x } ^ { 2 } - 1 }", "x + 1 \\ge 0", "- 1 \\le x", "x - 1 \\ge 0", "1 \\le x", "x \\ge 1"], "exprs": ["x + 1 \\ge 0", "x - 1 \\ge 0", "x \\ge 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x - 1 } \\cdot \\sqrt { x + 1 } = \\sqrt { { x } ^ { 2 } - 1 }"}, {"id": "x + 1 \\ge 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "式 $\\sqrt { x - 1 } \\cdot \\sqrt { x + 1 } = \\sqrt { { x } ^ { 2 } - 1 }$ 成立的条件"}, {"id": "x - 1 \\ge 0"}, {"id": "x \\ge 1"}], "links": [{"rel": "被描述", "source": "\\sqrt { x - 1 } \\cdot \\sqrt { x + 1 } = \\sqrt { { x } ^ { 2 } - 1 }", "target": "x + 1 \\ge 0"}, {"rel": "被描述", "source": "\\sqrt { x - 1 } \\cdot \\sqrt { x + 1 } = \\sqrt { { x } ^ { 2 } - 1 }", "target": "x - 1 \\ge 0"}, {"rel": "联立", "source": "x + 1 \\ge 0", "target": "x \\ge 1"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x + 1 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 1 \\ge 0"}, {"rel": "限制性描述", "source": "式 $\\sqrt { x - 1 } \\cdot \\sqrt { x + 1 } = \\sqrt { { x } ^ { 2 } - 1 }$ 成立的条件", "target": "x + 1 \\ge 0"}, {"rel": "限制性描述", "source": "式 $\\sqrt { x - 1 } \\cdot \\sqrt { x + 1 } = \\sqrt { { x } ^ { 2 } - 1 }$ 成立的条件", "target": "x - 1 \\ge 0"}, {"rel": "联立", "source": "x - 1 \\ge 0", "target": "x \\ge 1"}]}}
{"content": "Given $2 : x = 3 : 9$, then $x$ = ____ ?", "answer": "6", "steps": "Since $2 : x = 3 : 9$, therefore $3 x = 18$, therefore $x = 6$.", "expr_cands": ["2 : x = 3 : 9", "x", "x = 6", "3 x = 18"], "exprs": ["x = 6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 : x = 3 : 9"}, {"id": "x = 6"}], "links": [{"rel": "等式方程求解", "source": "2 : x = 3 : 9", "target": "x = 6"}]}}
{"content": "If the equation $( m - 5 ) x = 0$ holds for any $x$, what is the possible value(s) of $m$?", "answer": "5", "steps": "According to the problem, we have $m - 5 = 0$, which implies $m = 5$.", "expr_cands": ["x", "( m - 5 ) x = 0", "m", "m - 5 = 0", "m = 5"], "exprs": ["m - 5 = 0", "m = 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( m - 5 ) x = 0"}, {"id": "m - 5 = 0"}, {"id": "关于 $x$ 的方程 $( m - 5 ) x = 0$ 对任意 $x$ 都成立"}, {"id": "m = 5"}], "links": [{"rel": "被描述", "source": "( m - 5 ) x = 0", "target": "m - 5 = 0"}, {"rel": "等式方程求解", "source": "m - 5 = 0", "target": "m = 5"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $( m - 5 ) x = 0$ 对任意 $x$ 都成立", "target": "m - 5 = 0"}]}}
{"content": "If the quadratic equation $x ^ 2 + 2 x + a = 0$ has no real roots, then the possible values of $a$ are ____?", "answer": "a > 1", "steps": "$\\because$ The quadratic equation $x ^ 2 + 2 x + a = 0$ has no real roots. $\\therefore$ $\\Delta = 2 ^ 2 - 4 a < 0$, which implies $a > 1$.", "expr_cands": ["x ^ { 2 } + 2 x + a = 0", "x", "a", "2 ^ { 2 } - 4 a < 0", "1 < a", "a > 1"], "exprs": ["2 ^ { 2 } - 4 a < 0", "a > 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } + 2 x + a = 0"}, {"id": "2 ^ { 2 } - 4 a < 0"}, {"id": "一元二次方程 $x ^ { 2 } + 2 x + a = 0$ 无实根"}, {"id": "一元二次方程根与系数关系，方程无解，则根的判别式小于0"}, {"id": "a > 1"}], "links": [{"rel": "被描述", "source": "x ^ { 2 } + 2 x + a = 0", "target": "2 ^ { 2 } - 4 a < 0"}, {"rel": "不等式方程求解", "source": "2 ^ { 2 } - 4 a < 0", "target": "a > 1"}, {"rel": "限制性描述", "source": "一元二次方程 $x ^ { 2 } + 2 x + a = 0$ 无实根", "target": "2 ^ { 2 } - 4 a < 0"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，方程无解，则根的判别式小于0", "target": "2 ^ { 2 } - 4 a < 0"}]}}
{"content": "Given a parabola $y = { x } ^ { 2 } - 2 x - m$ that intersects the $x$-axis at only one point, the value of the algebraic expression ${ m } ^ { 2 } - 2 m$ is ____?", "answer": "3", "steps": "$\\because$ The parabola $y = { x } ^ { 2 } - 2 x - m$ has only one intersection with the $x$-axis. $\\therefore \\Delta = ( - 2 ) ^ 2 + 4 m = 0$, which solves for $m = - 1$. $\\therefore { m } ^ { 2 } - 2 m = 1 + 2 = 3$.", "expr_cands": ["y = { x } ^ { 2 } - 2 x - m", "m", "y", "x", "{ m } ^ { 2 } - 2 m", "( - 2 ) ^ { 2 } + 4 m = 0", "m = - 1", "3"], "exprs": ["( - 2 ) ^ { 2 } + 4 m = 0", "m = - 1", "3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = { x } ^ { 2 } - 2 x - m"}, {"id": "( - 2 ) ^ { 2 } + 4 m = 0"}, {"id": "抛物线 $y = { x } ^ { 2 } - 2 x - m$ 与 $x$ 轴只有一个交点"}, {"id": "一元二次方程根与系数关系，方程有解，则根的判别式大于等于0"}, {"id": "m = - 1"}, {"id": "{ m } ^ { 2 } - 2 m"}, {"id": "3"}], "links": [{"rel": "被描述", "source": "y = { x } ^ { 2 } - 2 x - m", "target": "( - 2 ) ^ { 2 } + 4 m = 0"}, {"rel": "等式方程求解", "source": "( - 2 ) ^ { 2 } + 4 m = 0", "target": "m = - 1"}, {"rel": "限制性描述", "source": "抛物线 $y = { x } ^ { 2 } - 2 x - m$ 与 $x$ 轴只有一个交点", "target": "( - 2 ) ^ { 2 } + 4 m = 0"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，方程有解，则根的判别式大于等于0", "target": "( - 2 ) ^ { 2 } + 4 m = 0"}, {"rel": "代入", "source": "m = - 1", "target": "3"}, {"rel": "被代入", "source": "{ m } ^ { 2 } - 2 m", "target": "3"}]}}
{"content": "Calculate: If $a = 3 - \\sqrt { 10 }$, then the algebraic expression $a ^ 2 - 6 a - 2$ = ____?", "answer": "- 1", "steps": "Since $a = 3 - \\sqrt { 10 }$, it follows that $a ^ 2 - 6 a - 2 = ( a - 3 ) ^ 2 - 11 = ( 3 - \\sqrt { 10 } - 3 ) ^ 2 - 11 = 10 - 11 = - 1$.", "expr_cands": ["a = 3 - \\sqrt { 10 }", "a", "a ^ { 2 } - 6 a - 2", "- 1"], "exprs": ["- 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a ^ { 2 } - 6 a - 2"}, {"id": "- 1"}, {"id": "a = 3 - \\sqrt { 10 }"}], "links": [{"rel": "被代入", "source": "a ^ { 2 } - 6 a - 2", "target": "- 1"}, {"rel": "代入", "source": "a = 3 - \\sqrt { 10 }", "target": "- 1"}]}}
{"content": "Given that the value of the polynomial $4 x ^ 2 - 3 mx + 2 + m$ is independent of the value of $m$, what is the value of $x$?", "answer": "\\frac { 1 } { 3 }", "steps": "$\\because$ The value of the polynomial $4 x ^ 2 - 3 mx + 2 + m$ is independent of the size of $m$, $\\therefore$ we have $4 x ^ 2 - 3 mx + 2 + m = 4 x ^ 2 + 2 + ( - 3 x + 1 ) m$. Thus, $- 3 x + 1 = 0$, and we solve for $x$ to get $x = \\frac { 1 } { 3 }$.", "expr_cands": ["4 x ^ { 2 } - 3 mx + 2 + m", "x", "m", "4 x ^ { 2 } + 2 + ( - 3 x + 1 ) m", "- 3 x + 1 = 0", "x = \\frac { 1 } { 3 }"], "exprs": ["4 x ^ { 2 } + 2 + ( - 3 x + 1 ) m", "- 3 x + 1 = 0", "x = \\frac { 1 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 x ^ { 2 } - 3 mx + 2 + m"}, {"id": "4 x ^ { 2 } + 2 + ( - 3 x + 1 ) m"}, {"id": "m"}, {"id": "- 3 x + 1 = 0"}, {"id": "多项式 $4 x ^ { 2 } - 3 mx + 2 + m$ 的值与 $m$ 的大小无关"}, {"id": "x = \\frac { 1 } { 3 }"}], "links": [{"rel": "提取因式", "source": "4 x ^ { 2 } - 3 mx + 2 + m", "target": "4 x ^ { 2 } + 2 + ( - 3 x + 1 ) m"}, {"rel": "被描述", "source": "4 x ^ { 2 } + 2 + ( - 3 x + 1 ) m", "target": "- 3 x + 1 = 0"}, {"rel": "提取因式参考", "source": "m", "target": "4 x ^ { 2 } + 2 + ( - 3 x + 1 ) m"}, {"rel": "等式方程求解", "source": "- 3 x + 1 = 0", "target": "x = \\frac { 1 } { 3 }"}, {"rel": "限制性描述", "source": "多项式 $4 x ^ { 2 } - 3 mx + 2 + m$ 的值与 $m$ 的大小无关", "target": "- 3 x + 1 = 0"}]}}
{"content": "If $\\frac { 1 } { a } - \\frac { 1 } { b } = 3$, then the value of $\\frac { 2 ab } { a - b }$ is ____?", "answer": "- \\frac { 2 } { 3 }", "steps": "Since $\\frac { 1 } { a } - \\frac { 1 } { b } = 3$, therefore $\\frac { b - a } { ab } = 3$, therefore $a - b = - 3 ab$, and the original expression is $\\frac { 2 ab } { - 3 ab } = - \\frac { 2 } { 3 }$.", "expr_cands": ["\\frac { { 1 } } { a } - \\frac { 1 } { b } = 3", "b", "a", "\\frac { 2 ab } { a - b }", "\\frac { 1 } { a } - \\frac { 1 } { b } = 3", "\\frac { b - a } { ab } = 3", "a - b = - 3 ab", "\\frac { 2 ab } { - 3 ab }", "- \\frac { 2 } { 3 }"], "exprs": ["a - b = - 3 ab", "- \\frac { 2 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { { 1 } } { a } - \\frac { 1 } { b } = 3"}, {"id": "a - b = - 3 ab"}, {"id": "\\frac { 2 ab } { a - b }"}, {"id": "- \\frac { 2 } { 3 }"}], "links": [{"rel": "同乘除", "source": "\\frac { { 1 } } { a } - \\frac { 1 } { b } = 3", "target": "a - b = - 3 ab"}, {"rel": "代入", "source": "a - b = - 3 ab", "target": "- \\frac { 2 } { 3 }"}, {"rel": "被代入", "source": "\\frac { 2 ab } { a - b }", "target": "- \\frac { 2 } { 3 }"}]}}
{"content": "If $3 ^ n = 27$, then $2 ^ { - n }$ is equal to ____?", "answer": "\\frac { 1 } { 8 }", "steps": "Because $3 ^ n = 27$, therefore $n = 3$, therefore $2 ^ { - n } = \\frac { 1 } { 2 } ^ n = \\frac { 1 } { 2 } ^ 3 = \\frac { 1 } { 8 }$.", "expr_cands": ["3 ^ { n } = 27", "n", "2 ^ { - n }", "n = 3", "\\frac { 1 } { 8 }"], "exprs": ["n = 3", "\\frac { 1 } { 8 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 ^ { n } = 27"}, {"id": "n = 3"}, {"id": "2 ^ { - n }"}, {"id": "\\frac { 1 } { 8 }"}], "links": [{"rel": "同取对数", "source": "3 ^ { n } = 27", "target": "n = 3"}, {"rel": "代入", "source": "n = 3", "target": "\\frac { 1 } { 8 }"}, {"rel": "被代入", "source": "2 ^ { - n }", "target": "\\frac { 1 } { 8 }"}]}}
{"content": "If the difference between the monomials $2 a ^ { 4 } b ^ { 3 m }$ and $3 a ^ { 2 n } b ^ { 6 }$ is still a monomial, then $m - n$ = ____ ?", "answer": "0", "steps": "$\\because$ The difference between the monomials $2 a ^ { 4 } b ^ { 3 m }$ and $3 a ^ { 2 n } b ^ { 6 }$ is still a monomial, $\\therefore$ the monomials $2 a ^ { 4 } b ^ { 3 m }$ and $3 a ^ { 2 n } b ^ { 6 }$ are like terms, $\\therefore$ $2 n = 4$, $3 m = 6$, solving for $m$ and $n$, we get: $m = 2$, $n = 2$, so $m - n = 2 - 2 = 0$.", "expr_cands": ["2 a ^ { 4 } b ^ { 3 m }", "b", "a", "m", "3 a ^ { 2 n } b ^ { 6 }", "n", "m - n", "2 n = 4", "n = 2", "3 m = 6", "m = 2", "0"], "exprs": ["2 n = 4", "3 m = 6", "n = 2", "m = 2", "0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 a ^ { 2 n } b ^ { 6 }"}, {"id": "2 n = 4"}, {"id": "2 a ^ { 4 } b ^ { 3 m }"}, {"id": "单项式 $2 a ^ { 4 } b ^ { 3 m }$ 与 $3 a ^ { 2 n } b ^ { 6 }$ 的差仍是单项式"}, {"id": "单项式 $2 a ^ { 4 } b ^ { 3 m }$ 与 $3 a ^ { 2 n } b ^ { 6 }$ 是同类项"}, {"id": "3 m = 6"}, {"id": "m = 2"}, {"id": "n = 2"}, {"id": "m - n"}, {"id": "0"}], "links": [{"rel": "被描述", "source": "3 a ^ { 2 n } b ^ { 6 }", "target": "2 n = 4"}, {"rel": "被描述", "source": "3 a ^ { 2 n } b ^ { 6 }", "target": "3 m = 6"}, {"rel": "等式方程求解", "source": "2 n = 4", "target": "n = 2"}, {"rel": "被描述", "source": "2 a ^ { 4 } b ^ { 3 m }", "target": "2 n = 4"}, {"rel": "被描述", "source": "2 a ^ { 4 } b ^ { 3 m }", "target": "3 m = 6"}, {"rel": "限制性描述", "source": "单项式 $2 a ^ { 4 } b ^ { 3 m }$ 与 $3 a ^ { 2 n } b ^ { 6 }$ 的差仍是单项式", "target": "2 n = 4"}, {"rel": "限制性描述", "source": "单项式 $2 a ^ { 4 } b ^ { 3 m }$ 与 $3 a ^ { 2 n } b ^ { 6 }$ 的差仍是单项式", "target": "3 m = 6"}, {"rel": "限制性描述", "source": "单项式 $2 a ^ { 4 } b ^ { 3 m }$ 与 $3 a ^ { 2 n } b ^ { 6 }$ 是同类项", "target": "2 n = 4"}, {"rel": "限制性描述", "source": "单项式 $2 a ^ { 4 } b ^ { 3 m }$ 与 $3 a ^ { 2 n } b ^ { 6 }$ 是同类项", "target": "3 m = 6"}, {"rel": "等式方程求解", "source": "3 m = 6", "target": "m = 2"}, {"rel": "代入", "source": "m = 2", "target": "0"}, {"rel": "代入", "source": "n = 2", "target": "0"}, {"rel": "被代入", "source": "m - n", "target": "0"}]}}
{"content": "The solution to the equation $( x + 6 ) ( x - 6 ) - x ( x - 9 ) = 0$ is ____?", "answer": "x = 4", "steps": "Because $( x + 6 ) ( x - 6 ) - x ( x - 9 ) = x ^ { 2 } - 36 - ( x ^ { 2 } - 9 x ) = x ^ { 2 } - 36 - x ^ { 2 } + 9 x = 9 x - 36$, therefore the equation can be simplified to $9 x - 36 = 0$, and the solution is $x = 4$.", "expr_cands": ["( x + 6 ) ( x - 6 ) - x ( x - 9 ) = 0", "x", "( x + 6 ) ( x - 6 ) - x ( x - 9 )", "9 x - 36", "9 x - 36 = 0", "x = 4"], "exprs": ["x = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( x + 6 ) ( x - 6 ) - x ( x - 9 ) = 0"}, {"id": "x = 4"}], "links": [{"rel": "等式方程求解", "source": "( x + 6 ) ( x - 6 ) - x ( x - 9 ) = 0", "target": "x = 4"}]}}
{"content": "The solution to the equation $\\frac { 4 } { 5 } x - 3 = 13$ is ____ ?", "answer": "x = 20", "steps": "The equation is rearranged and combined to get: $\\frac { 4 } { 5 } x = 16$, which is solved to get: $x = 20$.", "expr_cands": ["\\frac { 4 } { 5 } x - 3 = 13", "x", "\\frac { 4 } { 5 } x = 16", "x = 20"], "exprs": ["x = 20"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 4 } { 5 } x - 3 = 13"}, {"id": "x = 20"}], "links": [{"rel": "等式方程求解", "source": "\\frac { 4 } { 5 } x - 3 = 13", "target": "x = 20"}]}}
{"content": "Regarding the quadratic equation in one variable $x$, $x ^ 2 + nx - 3 n = 0$, if one of its roots is $x = 1$, then $n$ = ____?", "answer": "\\frac { 1 } { 2 }", "steps": "$\\because$ One root of the quadratic equation $x ^ 2 + nx - 3 n = 0$ with respect to $x$ is $x = 1$, $\\therefore$ $1 + n - 3 n = 0$, solving for $n$, we get $n = \\frac { 1 } { 2 }$.", "expr_cands": ["x", "x ^ { 2 } + nx - 3 n = 0", "n", "x = 1", "1 + n - 3 n = 0", "n = \\frac { 1 } { 2 }"], "exprs": ["1 + n - 3 n = 0", "n = \\frac { 1 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } + nx - 3 n = 0"}, {"id": "1 + n - 3 n = 0"}, {"id": "x = 1"}, {"id": "n = \\frac { 1 } { 2 }"}], "links": [{"rel": "被代入", "source": "x ^ { 2 } + nx - 3 n = 0", "target": "1 + n - 3 n = 0"}, {"rel": "等式方程求解", "source": "1 + n - 3 n = 0", "target": "n = \\frac { 1 } { 2 }"}, {"rel": "代入", "source": "x = 1", "target": "1 + n - 3 n = 0"}]}}
{"content": "Given that the two square roots of $x$ are $2 a - 1$ and $a - 5$, find the value of $x$.", "answer": "9", "steps": "$\\because$ $x$ has two square roots, which are $2 a - 1$ and $a - 5$, $\\therefore$ $2 a - 1 + a - 5 = 0$, solving for $a$, we get $a = 2$, then $2 a - 1 = 3$, so $x = 9$.", "expr_cands": ["x", "2 a - 1", "a", "a - 5", "2 a - 1 + a - 5 = 0", "a = 2", "3", "x = 9"], "exprs": ["2 a - 1 + a - 5 = 0", "a = 2", "3", "x = 9"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 a - 1"}, {"id": "2 a - 1 + a - 5 = 0"}, {"id": "a - 5"}, {"id": "$x$ 的两个平方根分别是 $2 a - 1$ 和 $a - 5$"}, {"id": "平方根互为相反数"}, {"id": "a = 2"}, {"id": "3"}, {"id": "x"}, {"id": "x = 9"}, {"id": "平方"}], "links": [{"rel": "被描述", "source": "2 a - 1", "target": "2 a - 1 + a - 5 = 0"}, {"rel": "被代入", "source": "2 a - 1", "target": "3"}, {"rel": "等式方程求解", "source": "2 a - 1 + a - 5 = 0", "target": "a = 2"}, {"rel": "被描述", "source": "a - 5", "target": "2 a - 1 + a - 5 = 0"}, {"rel": "限制性描述", "source": "$x$ 的两个平方根分别是 $2 a - 1$ 和 $a - 5$", "target": "2 a - 1 + a - 5 = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "2 a - 1 + a - 5 = 0"}, {"rel": "代入", "source": "a = 2", "target": "3"}, {"rel": "被描述", "source": "3", "target": "x = 9"}, {"rel": "被描述", "source": "x", "target": "x = 9"}, {"rel": "限制性描述", "source": "平方", "target": "x = 9"}]}}
{"content": "If $a$ and $b$ are both integers, $a + b = - 2$, and $a \\ge 2 b$, then the maximum value of $\\frac { a } { b }$ is _____.", "answer": "1", "steps": "$a = - 2 - b$, $- 2 - b \\ge 2 b$, solving gives: $b \\le - \\frac { 2 } { 3 }$, $b = - 2 - a$, $a \\ge - 4 - 2 a$, solving gives: $a \\ge - \\frac { 4 } { 3 }$, since $a$ and $b$ are both integers, therefore when $a = - 1$, $b = - 1$, $\\frac { a } { b }$ has a maximum value of $1$.", "expr_cands": ["a", "b", "a + b = - 2", "a \\ge 2 b", "\\frac { a } { b }", "a = - 2 - b", "- 2 - b \\ge 2 b", "b \\le - \\frac { 2 } { 3 }", "b = - 2 - a", "b = b", "a \\ge - 4 - 2 a", "- \\frac { 4 } { 3 } \\le a", "a \\ge - \\frac { 4 } { 3 }", "a = - 1", "b = - 1", "1"], "exprs": ["a = - 2 - b", "b = - 2 - a", "a = - 1", "- 2 - b \\ge 2 b", "a \\ge - 4 - 2 a", "b \\le - \\frac { 2 } { 3 }", "a \\ge - \\frac { 4 } { 3 }", "b = - 1", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a + b = - 2"}, {"id": "a = - 2 - b"}, {"id": "a \\ge 2 b"}, {"id": "- 2 - b \\ge 2 b"}, {"id": "b \\le - \\frac { 2 } { 3 }"}, {"id": "b = - 2 - a"}, {"id": "a \\ge - 4 - 2 a"}, {"id": "a \\ge - \\frac { 4 } { 3 }"}, {"id": "a"}, {"id": "a = - 1"}, {"id": "- \\frac { 4 } { 3 } \\le a"}, {"id": "$a$ , $b$ 均为整数"}, {"id": "$\\frac { a } { b }$ 有最大值"}, {"id": "当 $a = - 1$ , $b = - 1$ 时"}, {"id": "b"}, {"id": "b = - 1"}, {"id": "\\frac { a } { b }"}, {"id": "1"}], "links": [{"rel": "等式方程部分求解", "source": "a + b = - 2", "target": "a = - 2 - b"}, {"rel": "等式方程部分求解", "source": "a + b = - 2", "target": "b = - 2 - a"}, {"rel": "代入", "source": "a = - 2 - b", "target": "- 2 - b \\ge 2 b"}, {"rel": "被代入", "source": "a \\ge 2 b", "target": "- 2 - b \\ge 2 b"}, {"rel": "被代入", "source": "a \\ge 2 b", "target": "a \\ge - 4 - 2 a"}, {"rel": "不等式方程求解", "source": "- 2 - b \\ge 2 b", "target": "b \\le - \\frac { 2 } { 3 }"}, {"rel": "被描述", "source": "b \\le - \\frac { 2 } { 3 }", "target": "b = - 1"}, {"rel": "代入", "source": "b = - 2 - a", "target": "a \\ge - 4 - 2 a"}, {"rel": "不等式方程求解", "source": "a \\ge - 4 - 2 a", "target": "a \\ge - \\frac { 4 } { 3 }"}, {"rel": "被描述", "source": "a", "target": "a = - 1"}, {"rel": "代入", "source": "a = - 1", "target": "1"}, {"rel": "被描述", "source": "- \\frac { 4 } { 3 } \\le a", "target": "a = - 1"}, {"rel": "限制性描述", "source": "$a$ , $b$ 均为整数", "target": "a = - 1"}, {"rel": "限制性描述", "source": "$a$ , $b$ 均为整数", "target": "b = - 1"}, {"rel": "限制性描述", "source": "$\\frac { a } { b }$ 有最大值", "target": "a = - 1"}, {"rel": "限制性描述", "source": "$\\frac { a } { b }$ 有最大值", "target": "b = - 1"}, {"rel": "限制性描述", "source": "当 $a = - 1$ , $b = - 1$ 时", "target": "a = - 1"}, {"rel": "限制性描述", "source": "当 $a = - 1$ , $b = - 1$ 时", "target": "b = - 1"}, {"rel": "被描述", "source": "b", "target": "b = - 1"}, {"rel": "代入", "source": "b = - 1", "target": "1"}, {"rel": "被代入", "source": "\\frac { a } { b }", "target": "1"}]}}
{"content": "If the linear term of $( mx + 8 ) ( 2 - 3 x )$ is absent, then the value of $m$ is ____?", "answer": "12", "steps": "$( mx + 8 ) ( 2 - 3 x ) = - 3 mx ^ { 2 } - 24 x + 2 mx + 16 = - 3 mx ^ { 2 } + ( 2 m - 24 ) x + 16$ . From the given information, we have $2 m - 24 = 0$, which implies $m = 12$.", "expr_cands": ["( mx + 8 ) ( 2 - 3 x )", "m", "x", "- 3 mx ^ { 2 } + ( 2 m - 24 ) x + 16", "2 m - 24 = 0", "m = 12"], "exprs": ["- 3 mx ^ { 2 } + ( 2 m - 24 ) x + 16", "2 m - 24 = 0", "m = 12"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( mx + 8 ) ( 2 - 3 x )"}, {"id": "- 3 mx ^ { 2 } + ( 2 m - 24 ) x + 16"}, {"id": "2 m - 24 = 0"}, {"id": "$( mx + 8 ) ( 2 - 3 x )$ 中不含 $x$ 的一次项"}, {"id": "m = 12"}], "links": [{"rel": "展开", "source": "( mx + 8 ) ( 2 - 3 x )", "target": "- 3 mx ^ { 2 } + ( 2 m - 24 ) x + 16"}, {"rel": "被描述", "source": "- 3 mx ^ { 2 } + ( 2 m - 24 ) x + 16", "target": "2 m - 24 = 0"}, {"rel": "等式方程求解", "source": "2 m - 24 = 0", "target": "m = 12"}, {"rel": "限制性描述", "source": "$( mx + 8 ) ( 2 - 3 x )$ 中不含 $x$ 的一次项", "target": "2 m - 24 = 0"}]}}
{"content": "Given that the value of the polynomial $4 { a } ^ 3 - 2 a + 5$ is $7$, what is the value of the polynomial $6 { a } ^ 3 - 3 a + 5$?", "answer": "8", "steps": "\\because $4 a ^ { 3 } - 2 a + 5 = 7$, which means $2 a ^ { 3 } - a = 1$. \\therefore The original expression is equal to $3 ( 2 a ^ { 3 } - a ) + 5 = 3 * 1 + 5 = 8$.", "expr_cands": ["4 { a } ^ { 3 } - 2 a + 5", "a", "7", "6 { a } ^ { 3 } - 3 a + 5", "4 a ^ { 3 } - 2 a + 5 = 7", "a = 1", "2 a ^ { 3 } - a = 1", "3 ( 2 a ^ { 3 } - a ) + 5", "8"], "exprs": ["4 a ^ { 3 } - 2 a + 5 = 7", "2 a ^ { 3 } - a = 1", "3 ( 2 a ^ { 3 } - a ) + 5", "8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 { a } ^ { 3 } - 2 a + 5"}, {"id": "4 a ^ { 3 } - 2 a + 5 = 7"}, {"id": "7"}, {"id": "多项式 $4 { a } ^ { 3 } - 2 a + 5$ 的值是 $7$"}, {"id": "2 a ^ { 3 } - a = 1"}, {"id": "6 { a } ^ { 3 } - 3 a + 5"}, {"id": "3 ( 2 a ^ { 3 } - a ) + 5"}, {"id": "8"}], "links": [{"rel": "被描述", "source": "4 { a } ^ { 3 } - 2 a + 5", "target": "4 a ^ { 3 } - 2 a + 5 = 7"}, {"rel": "移项", "source": "4 a ^ { 3 } - 2 a + 5 = 7", "target": "2 a ^ { 3 } - a = 1"}, {"rel": "被描述", "source": "7", "target": "4 a ^ { 3 } - 2 a + 5 = 7"}, {"rel": "限制性描述", "source": "多项式 $4 { a } ^ { 3 } - 2 a + 5$ 的值是 $7$", "target": "4 a ^ { 3 } - 2 a + 5 = 7"}, {"rel": "提取因式参考", "source": "2 a ^ { 3 } - a = 1", "target": "3 ( 2 a ^ { 3 } - a ) + 5"}, {"rel": "代入", "source": "2 a ^ { 3 } - a = 1", "target": "8"}, {"rel": "提取因式", "source": "6 { a } ^ { 3 } - 3 a + 5", "target": "3 ( 2 a ^ { 3 } - a ) + 5"}, {"rel": "被代入", "source": "3 ( 2 a ^ { 3 } - a ) + 5", "target": "8"}]}}
{"content": "If $\\frac { x } { 2 } = \\frac { y } { 3 }$, then $\\frac { 7 x ^ 2 - y ^ 2 } { x ^ 2 - 2 xy + 3 y ^ 2 }$ = ____ ?", "answer": "1", "steps": "\\because $\\frac { x } { 2 } = \\frac { y } { 3 }$ , \\therefore let $\\frac { x } { 2 } = \\frac { y } { 3 } = k$ , then $x = 2 k$ , $y = 3 k$ , \\therefore the original expression = $\\frac { 28 k ^ { 2 } - 9 k ^ { 2 } } { 4 k ^ { 2 } - 12 k ^ { 2 } + 27 k ^ { 2 } } = 1$ .", "expr_cands": ["\\frac { x } { 2 } = \\frac { y } { 3 }", "y", "x", "\\frac { 7 x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } - 2 xy + 3 y ^ { 2 } }", "\\frac { x } { 2 } = k", "k", "x = 2 k", "y = 3 k", "\\frac { 28 k ^ { 2 } - 9 k ^ { 2 } } { 4 k ^ { 2 } - 12 k ^ { 2 } + 27 k ^ { 2 } }", "1"], "exprs": ["x = 2 k", "y = 3 k", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "令 $\\frac { x } { 2 } = \\frac { y } { 3 } = k$"}, {"id": "x = 2 k"}, {"id": "y = 3 k"}, {"id": "\\frac { 7 x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } - 2 xy + 3 y ^ { 2 } }"}, {"id": "1"}], "links": [{"rel": "假设描述", "source": "令 $\\frac { x } { 2 } = \\frac { y } { 3 } = k$", "target": "x = 2 k"}, {"rel": "假设描述", "source": "令 $\\frac { x } { 2 } = \\frac { y } { 3 } = k$", "target": "y = 3 k"}, {"rel": "代入", "source": "x = 2 k", "target": "1"}, {"rel": "代入", "source": "y = 3 k", "target": "1"}, {"rel": "被代入", "source": "\\frac { 7 x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } - 2 xy + 3 y ^ { 2 } }", "target": "1"}]}}
{"content": "The equation $3 x + a = x - 7$ has a negative root for $x$. The possible values of real number $a$ are _____.", "answer": "a > - 7", "steps": "From $3 x + a = x - 7$, we can solve for $x$ to get $x = \\frac { - 7 - a } { 2 }$. Since the root of the equation $3 x + a = x - 7$ with respect to $x$ is negative, we have $- a - 7 < 0$. Solving for $a$, we get $a > - 7$.", "expr_cands": ["x", "3 x + a = x - 7", "a", "x = \\frac { - 7 - a } { 2 }", "- \\frac { a } { 2 } - \\frac { 21 } { 2 } = - \\frac { a } { 2 } - 7 - \\frac { 7 } { 2 }", "- a - 7 < 0", "- 7 < a", "a > - 7"], "exprs": ["x = \\frac { - 7 - a } { 2 }", "- a - 7 < 0", "a > - 7"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 x + a = x - 7"}, {"id": "x = \\frac { - 7 - a } { 2 }"}, {"id": "- a - 7 < 0"}, {"id": "关于 $x$ 的方程 $3 x + a = x - 7$ 的根是负数"}, {"id": "a > - 7"}], "links": [{"rel": "等式方程部分求解", "source": "3 x + a = x - 7", "target": "x = \\frac { - 7 - a } { 2 }"}, {"rel": "被描述", "source": "x = \\frac { - 7 - a } { 2 }", "target": "- a - 7 < 0"}, {"rel": "不等式方程求解", "source": "- a - 7 < 0", "target": "a > - 7"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $3 x + a = x - 7$ 的根是负数", "target": "- a - 7 < 0"}]}}
{"content": "If $2005 - 200.5 = x - 20.05$, then $x$ is equal to ____?", "answer": "1824.55", "steps": "$2005 - 200.5 = x - 20.05$ is rearranged to x = 2005 - 200.5 + 20.05. Combining like terms, we get x = 1824.55.", "expr_cands": ["2005 - 200.5 = x - 20.05", "x", "x = 1824.55", "x = 2005 - 200.5 + 20.05"], "exprs": ["x = 1824.55"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2005 - 200.5 = x - 20.05"}, {"id": "x = 1824.55"}], "links": [{"rel": "等式方程求解", "source": "2005 - 200.5 = x - 20.05", "target": "x = 1824.55"}]}}
{"content": "Translate the above math content in English, you should keep the content wrapped in $unchanged.What is the new function obtained by translating the line$y=-2x+1$ one unit upward?", "answer": "y = - 2 x + 2", "steps": "According to the principle of adding up and subtracting down, we know that the equation of the line obtained by shifting the line $y = - 2 x + 1$ one unit upward is $y = - 2 x + 1 + 1$, which is $y = - 2 x + 2$.", "expr_cands": ["y = - 2 x + 1", "y", "x", "1", "y = - 2 x + 1 + 1", "1 - 2 x = - 2 x + 1 + 1", "- 2 x + 2"], "exprs": ["y = - 2 x + 1 + 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = - 2 x + 1"}, {"id": "y = - 2 x + 1 + 1"}, {"id": "1"}, {"id": "将直线 $y = - 2 x + 1$ 向上平移 $1$ 个单位"}, {"id": "\" 上加下减 \" 的原可"}], "links": [{"rel": "被描述", "source": "y = - 2 x + 1", "target": "y = - 2 x + 1 + 1"}, {"rel": "被描述", "source": "1", "target": "y = - 2 x + 1 + 1"}, {"rel": "限制性描述", "source": "将直线 $y = - 2 x + 1$ 向上平移 $1$ 个单位", "target": "y = - 2 x + 1 + 1"}, {"rel": "属性描述", "source": "\" 上加下减 \" 的原可", "target": "y = - 2 x + 1 + 1"}]}}
{"content": "Given $a = \\frac { 37 } { 14 }$, $b = \\frac { 7 } { 74 }$, find the value of $( a + b ) ^ 2 - ( a - b ) ^ 2$.", "answer": "1", "steps": "\\because $( a + b ) ^ { 2 } - ( a - b ) ^ { 2 } = ( a ^ { 2 } + 2 ab + b ^ { 2 } ) - ( a ^ { 2 } - 2 ab + b ^ { 2 } ) = 4 ab$ , \\therefore substituting $a = \\frac { 37 } { 14 }$ , $b = \\frac { 7 } { 74 }$ into the above equation yields: the original expression $= 4 ab = 4 * \\frac { 37 } { 14 } * \\frac { 7 } { 74 } = 1$.", "expr_cands": ["a = \\frac { 37 } { 14 }", "a", "b = \\frac { 7 } { 74 }", "b", "( a + b ) ^ { 2 } - ( a - b ) ^ { 2 }", "4 ab", "1"], "exprs": ["4 ab", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( a + b ) ^ { 2 } - ( a - b ) ^ { 2 }"}, {"id": "4 ab"}, {"id": "1"}, {"id": "a = \\frac { 37 } { 14 }"}, {"id": "b = \\frac { 7 } { 74 }"}], "links": [{"rel": "计算", "source": "( a + b ) ^ { 2 } - ( a - b ) ^ { 2 }", "target": "4 ab"}, {"rel": "被代入", "source": "4 ab", "target": "1"}, {"rel": "代入", "source": "a = \\frac { 37 } { 14 }", "target": "1"}, {"rel": "代入", "source": "b = \\frac { 7 } { 74 }", "target": "1"}]}}
{"content": "If the value of $2 a$ is the opposite of the value of $1 - a$, then $a$ is equal to ____?", "answer": "- 1", "steps": "From the given information, we have $2 a + 1 - a = 0$, which can be solved to obtain $a = - 1$.", "expr_cands": ["2 a", "a", "1 - a", "2 a + 1 - a = 0", "a = - 1"], "exprs": ["2 a + 1 - a = 0", "a = - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 a"}, {"id": "2 a + 1 - a = 0"}, {"id": "1 - a"}, {"id": "$2 a$ 的值与 $1 - a$ 的值互为相反数"}, {"id": "a = - 1"}], "links": [{"rel": "被描述", "source": "2 a", "target": "2 a + 1 - a = 0"}, {"rel": "等式方程求解", "source": "2 a + 1 - a = 0", "target": "a = - 1"}, {"rel": "被描述", "source": "1 - a", "target": "2 a + 1 - a = 0"}, {"rel": "限制性描述", "source": "$2 a$ 的值与 $1 - a$ 的值互为相反数", "target": "2 a + 1 - a = 0"}]}}
{"content": "In the inverse proportion function $y = \\frac { 9 } { x }$, when $x = 3$, the function value is ____?", "answer": "3", "steps": "Substituting $x = 3$ into $y = \\frac { 9 } { x }$ gives $y = \\frac { 9 } { 3 } = 3$.", "expr_cands": ["y = \\frac { 9 } { x }", "y", "x", "x = 3", "y = 3"], "exprs": ["y = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = \\frac { 9 } { x }"}, {"id": "y = 3"}, {"id": "x = 3"}], "links": [{"rel": "被代入", "source": "y = \\frac { 9 } { x }", "target": "y = 3"}, {"rel": "代入", "source": "x = 3", "target": "y = 3"}]}}
{"content": "The range of values for $x$ that make the algebraic expression $\\sqrt { x - 2 }$ valid is _____.", "answer": "x \\ge 2", "steps": "Because $\\sqrt { x - 2 }$ is defined, therefore $x - 2 \\ge 0$, which implies $x \\ge 2$.", "expr_cands": ["\\sqrt { x - 2 }", "x", "x - 2 \\ge 0", "2 \\le x", "x \\ge 2"], "exprs": ["x - 2 \\ge 0", "x \\ge 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x - 2 }"}, {"id": "x - 2 \\ge 0"}, {"id": "使代数式 $\\sqrt { x - 2 }$ 成立的 $x$ 的取值范围"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x \\ge 2"}], "links": [{"rel": "被描述", "source": "\\sqrt { x - 2 }", "target": "x - 2 \\ge 0"}, {"rel": "不等式方程求解", "source": "x - 2 \\ge 0", "target": "x \\ge 2"}, {"rel": "限制性描述", "source": "使代数式 $\\sqrt { x - 2 }$ 成立的 $x$ 的取值范围", "target": "x - 2 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 2 \\ge 0"}]}}
{"content": "If $a$ and $b$ are opposite numbers, then $| a + b - 1 |$ = ____ ?", "answer": "1", "steps": "Since $a$ and $b$ are opposite numbers, we have $a + b = 0$. Therefore, $| a + b - 1 | = | 0 - 1 | = | - 1 | = 1$.", "expr_cands": ["a", "b", "| a + b - 1 |", "a + b = 0", "1"], "exprs": ["a + b = 0", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a + b = 0"}, {"id": "b"}, {"id": "$a$ , $b$ 互为相反数"}, {"id": "| a + b - 1 |"}, {"id": "1"}], "links": [{"rel": "被描述", "source": "a", "target": "a + b = 0"}, {"rel": "代入", "source": "a + b = 0", "target": "1"}, {"rel": "被描述", "source": "b", "target": "a + b = 0"}, {"rel": "限制性描述", "source": "$a$ , $b$ 互为相反数", "target": "a + b = 0"}, {"rel": "被代入", "source": "| a + b - 1 |", "target": "1"}]}}
{"content": "Given that $x ^ m = 4$ and $x ^ m = 3$, what is the value of $x ^ { m + n }$?", "answer": "12", "steps": "Since $x ^ m = 4$ and $x ^ n = 3$, therefore $x ^ { m + n } = x ^ m \\cdot x ^ n = 3 \\cdot 4 = 12$.", "expr_cands": ["{ x } ^ { m } = 4", "x", "m", "{ x } ^ { m } = 3", "{ x } ^ { m + n }", "n", "{ x } ^ { n } = 3", "12"], "exprs": ["12"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "{ x } ^ { m + n }"}, {"id": "12"}, {"id": "{ x } ^ { m } = 4"}, {"id": "{ x } ^ { n } = 3"}], "links": [{"rel": "被代入", "source": "{ x } ^ { m + n }", "target": "12"}, {"rel": "代入", "source": "{ x } ^ { m } = 4", "target": "12"}, {"rel": "代入", "source": "{ x } ^ { n } = 3", "target": "12"}]}}
{"content": "The solution to the one-variable linear equation $ax + 4 = 10$ for $x$ is $x = 2$. What is the value of $a$?", "answer": "3", "steps": "Substituting $x = 2$ into the equation gives $2 a + 4 = 10$, solving for $a$ gives $a = 3$.", "expr_cands": ["x", "ax + 4 = 10", "a", "x = 2", "2 a + 4 = 10", "a = 3"], "exprs": ["2 a + 4 = 10", "a = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "ax + 4 = 10"}, {"id": "2 a + 4 = 10"}, {"id": "x = 2"}, {"id": "a = 3"}], "links": [{"rel": "被代入", "source": "ax + 4 = 10", "target": "2 a + 4 = 10"}, {"rel": "等式方程求解", "source": "2 a + 4 = 10", "target": "a = 3"}, {"rel": "代入", "source": "x = 2", "target": "2 a + 4 = 10"}]}}
{"content": "Given the inverse proportion function $y = \\frac { k } { x }$, when $x = 2$, $y = - 3$, then $k$ = ____?", "answer": "- 6", "steps": "Substituting $x = 2$ and $y = - 3$ into $y = \\frac { k } { x }$, we get $\\frac { k } { 2 } = - 3$. Solving for $k$, we get $k = - 6$.", "expr_cands": ["y = \\frac { k } { x }", "y", "k", "x", "x = 2", "y = - 3", "- 3 = \\frac { k } { 2 }", "\\frac { k } { 2 } = - 3", "k = - 6"], "exprs": ["\\frac { k } { 2 } = - 3", "k = - 6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = \\frac { k } { x }"}, {"id": "\\frac { k } { 2 } = - 3"}, {"id": "x = 2"}, {"id": "y = - 3"}, {"id": "k = - 6"}], "links": [{"rel": "被代入", "source": "y = \\frac { k } { x }", "target": "\\frac { k } { 2 } = - 3"}, {"rel": "等式方程求解", "source": "\\frac { k } { 2 } = - 3", "target": "k = - 6"}, {"rel": "代入", "source": "x = 2", "target": "\\frac { k } { 2 } = - 3"}, {"rel": "代入", "source": "y = - 3", "target": "\\frac { k } { 2 } = - 3"}]}}
{"content": "The solution to the equation $3 x + 4 = 11 - 4 x$ is", "answer": "x = 1", "steps": "The equation $3 x + 4 = 11 - 4 x$ is rearranged and combined to get $7 x = 7$. Solving for $x$ gives $x = 1$.", "expr_cands": ["3 x + 4 = 11 - 4 x", "x", "x = 1", "7 x = 7"], "exprs": ["x = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 x + 4 = 11 - 4 x"}, {"id": "x = 1"}], "links": [{"rel": "等式方程求解", "source": "3 x + 4 = 11 - 4 x", "target": "x = 1"}]}}
{"content": "If the fractional equation $\\frac { 5 } { x - 1 } = \\frac { x + 4 } { x ( x - 1 ) }$ has a positive root, then the root is ____?", "answer": "x = 1", "steps": "Multiplying both sides by $x ( x - 1 )$, we get: $5 x = x + 4$. Therefore, $x = 1$. Checking, we see that $x = 1$ is an extraneous root of the rational equation.", "expr_cands": ["\\frac { 5 } { x - 1 } = \\frac { x + 4 } { x ( x - 1 ) }", "x", "x ( x - 1 )", "5 x = x + 4", "x = 1"], "exprs": ["x = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 5 } { x - 1 } = \\frac { x + 4 } { x ( x - 1 ) }"}, {"id": "x = 1"}, {"id": "分式方程 $\\frac { 5 } { x - 1 } = \\frac { x + 4 } { x ( x - 1 ) }$ 有增根"}, {"id": "分式有增根，则分母为0"}], "links": [{"rel": "被描述", "source": "\\frac { 5 } { x - 1 } = \\frac { x + 4 } { x ( x - 1 ) }", "target": "x = 1"}, {"rel": "限制性描述", "source": "分式方程 $\\frac { 5 } { x - 1 } = \\frac { x + 4 } { x ( x - 1 ) }$ 有增根", "target": "x = 1"}, {"rel": "属性描述", "source": "分式有增根，则分母为0", "target": "x = 1"}]}}
{"content": "The range of values of the independent variable $x$ that make the function $y = \\frac { 5 } { \\sqrt { x - 2 } }$ meaningful is ____ ?", "answer": "x > 2", "steps": "From the given condition, we have $x - 2 > 0$, which implies $x > 2$.", "expr_cands": ["y = \\frac { 5 } { \\sqrt { x - 2 } }", "y", "x", "x - 2 > 0", "2 < x", "x > 2"], "exprs": ["x - 2 > 0", "x > 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = \\frac { 5 } { \\sqrt { x - 2 } }"}, {"id": "x - 2 > 0"}, {"id": "使函数 $y = \\frac { 5 } { \\sqrt { x - 2 } }$ 有意义的自变量 $x$ 的取值范围"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "分式有意义，则分母不为0"}, {"id": "x > 2"}], "links": [{"rel": "被描述", "source": "y = \\frac { 5 } { \\sqrt { x - 2 } }", "target": "x - 2 > 0"}, {"rel": "不等式方程求解", "source": "x - 2 > 0", "target": "x > 2"}, {"rel": "限制性描述", "source": "使函数 $y = \\frac { 5 } { \\sqrt { x - 2 } }$ 有意义的自变量 $x$ 的取值范围", "target": "x - 2 > 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 2 > 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "x - 2 > 0"}]}}
{"content": "If the value of $4 m - 5$ is the opposite of the value of $3 m - 9$, then $m$ equals ____?", "answer": "2", "steps": "According to the problem, we have $4 m - 5 + 3 m - 9 = 0$. By rearranging and combining terms, we get $7 m = 14$. Solving for $m$, we get $m = 2$.", "expr_cands": ["4 m - 5", "m", "3 m - 9", "4 m - 5 + 3 m - 9 = 0", "m = 2", "7 m = 14"], "exprs": ["4 m - 5 + 3 m - 9 = 0", "m = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 m - 5"}, {"id": "4 m - 5 + 3 m - 9 = 0"}, {"id": "3 m - 9"}, {"id": "$4 m - 5$ 的值与 $3 m - 9$ 的值互为相反数"}, {"id": "m = 2"}], "links": [{"rel": "被描述", "source": "4 m - 5", "target": "4 m - 5 + 3 m - 9 = 0"}, {"rel": "等式方程求解", "source": "4 m - 5 + 3 m - 9 = 0", "target": "m = 2"}, {"rel": "被描述", "source": "3 m - 9", "target": "4 m - 5 + 3 m - 9 = 0"}, {"rel": "限制性描述", "source": "$4 m - 5$ 的值与 $3 m - 9$ 的值互为相反数", "target": "4 m - 5 + 3 m - 9 = 0"}]}}
{"content": "Translate the above math content in English, you should keep the content wrapped in $ unchanged. The equation of the line y = 3x + 1 is translated 2 units to the left and 4 units down. The equation of the new line is ____?", "answer": "y = 3 x + 3", "steps": "The line $y = 3 x + 1$ is translated $2$ units to the left and $4$ units down, resulting in the equation $y = 3 ( x + 2 ) + 1 - 4$, which simplifies to $y = 3 x + 3$.", "expr_cands": ["y = 3 x + 1", "y", "x", "2", "4", "y = 3 ( x + 2 ) + 1 - 4", "3 x + 1 = 3 ( x + 2 ) + 1 - 4", "3 x + 3"], "exprs": ["y = 3 ( x + 2 ) + 1 - 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = 3 x + 1"}, {"id": "y = 3 ( x + 2 ) + 1 - 4"}, {"id": "2"}, {"id": "4"}, {"id": "将直线 $y = 3 x + 1$ 向左平移 $2$ 个单位并向下平移 $4$ 个单位"}, {"id": "平移后所得直线的解析式"}], "links": [{"rel": "被描述", "source": "y = 3 x + 1", "target": "y = 3 ( x + 2 ) + 1 - 4"}, {"rel": "被描述", "source": "2", "target": "y = 3 ( x + 2 ) + 1 - 4"}, {"rel": "被描述", "source": "4", "target": "y = 3 ( x + 2 ) + 1 - 4"}, {"rel": "限制性描述", "source": "将直线 $y = 3 x + 1$ 向左平移 $2$ 个单位并向下平移 $4$ 个单位", "target": "y = 3 ( x + 2 ) + 1 - 4"}, {"rel": "限制性描述", "source": "平移后所得直线的解析式", "target": "y = 3 ( x + 2 ) + 1 - 4"}]}}
{"content": "The coefficient of the monomial $- \\frac { 2 a { b } ^ 2 } { 5 }$ is $m$, and the degree of the polynomial $2 { a } ^ 2 { b } ^ 3 + 3 { b } ^ 2 { c } ^ 2 - 1$ is $n$. Find $m + n$.", "answer": "\\frac { 23 } { 5 }", "steps": "Because the coefficient of the monomial $- \\frac { 2 a { b } ^ 2 } { 5 }$ is $m$, and the degree of the polynomial $2 a ^ 2 b ^ 3 + 3 b ^ 2 c ^ 2 - 1$ is $n$, therefore $m = - \\frac { 2 } { 5 }$ and $n = 5$. Thus, $m + n = \\frac { 23 } { 5 }$.", "expr_cands": ["- \\frac { 2 a { b } ^ { 2 } } { 5 }", "b", "a", "m", "2 { a } ^ { 2 } { b } ^ { 3 } + 3 { b } ^ { 2 } { c } ^ { 2 } - 1", "c", "n", "m + n", "2 a ^ { 2 } b ^ { 3 } + 3 b ^ { 2 } c ^ { 2 } - 1", "m = - \\frac { 2 } { 5 }", "n = 5", "\\frac { 23 } { 5 }"], "exprs": ["m = - \\frac { 2 } { 5 }", "n = 5", "\\frac { 23 } { 5 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- \\frac { 2 a { b } ^ { 2 } } { 5 }"}, {"id": "m = - \\frac { 2 } { 5 }"}, {"id": "m"}, {"id": "单项式 $- \\frac { 2 a { b } ^ { 2 } } { 5 }$ 的系数是 $m$"}, {"id": "2 { a } ^ { 2 } { b } ^ { 3 } + 3 { b } ^ { 2 } { c } ^ { 2 } - 1"}, {"id": "n = 5"}, {"id": "n"}, {"id": "多项式 $2 { a } ^ { 2 } { b } ^ { 3 } + 3 { b } ^ { 2 } { c } ^ { 2 } - 1$ 的次数是 $n$"}, {"id": "m + n"}, {"id": "\\frac { 23 } { 5 }"}], "links": [{"rel": "被描述", "source": "- \\frac { 2 a { b } ^ { 2 } } { 5 }", "target": "m = - \\frac { 2 } { 5 }"}, {"rel": "代入", "source": "m = - \\frac { 2 } { 5 }", "target": "\\frac { 23 } { 5 }"}, {"rel": "被描述", "source": "m", "target": "m = - \\frac { 2 } { 5 }"}, {"rel": "限制性描述", "source": "单项式 $- \\frac { 2 a { b } ^ { 2 } } { 5 }$ 的系数是 $m$", "target": "m = - \\frac { 2 } { 5 }"}, {"rel": "被描述", "source": "2 { a } ^ { 2 } { b } ^ { 3 } + 3 { b } ^ { 2 } { c } ^ { 2 } - 1", "target": "n = 5"}, {"rel": "代入", "source": "n = 5", "target": "\\frac { 23 } { 5 }"}, {"rel": "被描述", "source": "n", "target": "n = 5"}, {"rel": "限制性描述", "source": "多项式 $2 { a } ^ { 2 } { b } ^ { 3 } + 3 { b } ^ { 2 } { c } ^ { 2 } - 1$ 的次数是 $n$", "target": "n = 5"}, {"rel": "被代入", "source": "m + n", "target": "\\frac { 23 } { 5 }"}]}}
{"content": "When $x = 5$, what is the value of the quadratic radical $\\sqrt { 2 x - 1 }$?", "answer": "3", "steps": "Substituting $x = 5$ into $\\sqrt { 2 x - 1 }$ yields $\\sqrt { 2 \\times 5 - 1 } = \\sqrt { 9 } = 3$.", "expr_cands": ["x = 5", "x", "\\sqrt { 2 x - 1 }", "\\sqrt { 2 \\times 5 - 1 }", "3"], "exprs": ["3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x = 5"}, {"id": "3"}, {"id": "\\sqrt { 2 x - 1 }"}], "links": [{"rel": "代入", "source": "x = 5", "target": "3"}, {"rel": "被代入", "source": "\\sqrt { 2 x - 1 }", "target": "3"}]}}
{"content": "When $m$ takes what value, the fraction $\\frac { 2 m } { 3 m + 2 }$ is meaningful ____?", "answer": "m \\neq - \\frac { 2 } { 3 }", "steps": "From the given condition, we have $3 m + 2 \\neq 0$. Solving for $m$, we get $m \\neq - \\frac { 2 } { 3 }$.", "expr_cands": ["m", "\\frac { 2 m } { 3 m + 2 }", "3 m + 2 \\neq 0", "m \\neq - \\frac { 2 } { 3 }"], "exprs": ["3 m + 2 \\neq 0", "m \\neq - \\frac { 2 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 2 m } { 3 m + 2 }"}, {"id": "3 m + 2 \\neq 0"}, {"id": "分式 $\\frac { 2 m } { 3 m + 2 }$ 有意义"}, {"id": "分式有意义，则分母不为0"}, {"id": "m \\neq - \\frac { 2 } { 3 }"}], "links": [{"rel": "被描述", "source": "\\frac { 2 m } { 3 m + 2 }", "target": "3 m + 2 \\neq 0"}, {"rel": "不等式方程求解", "source": "3 m + 2 \\neq 0", "target": "m \\neq - \\frac { 2 } { 3 }"}, {"rel": "限制性描述", "source": "分式 $\\frac { 2 m } { 3 m + 2 }$ 有意义", "target": "3 m + 2 \\neq 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "3 m + 2 \\neq 0"}]}}
{"content": "If the square root of a positive number is $a + 8$ and $3 a - 4$, then the positive number is ____?", "answer": "49", "steps": "$\\because$ The square root of a positive number is $a + 8$ and $3 a - 4$, $\\therefore$ $a + 8 + 3 a - 4 = 0$, $a = - 1$, $a + 8 = 7$, $\\therefore$ this positive number is $7 ^ 2 = 49$.", "expr_cands": ["a + 8", "a", "3 a - 4", "a + 8 + 3 a - 4 = 0", "a = - 1", "7", "7 ^ { 2 }", "49"], "exprs": ["a + 8 + 3 a - 4 = 0", "a = - 1", "7", "49"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a + 8"}, {"id": "a + 8 + 3 a - 4 = 0"}, {"id": "3 a - 4"}, {"id": "一个正数的平方根是 $a + 8$ 和 $3 a - 4$"}, {"id": "平方根互为相反数"}, {"id": "a = - 1"}, {"id": "7"}, {"id": "49"}, {"id": "平方"}], "links": [{"rel": "被描述", "source": "a + 8", "target": "a + 8 + 3 a - 4 = 0"}, {"rel": "被代入", "source": "a + 8", "target": "7"}, {"rel": "等式方程求解", "source": "a + 8 + 3 a - 4 = 0", "target": "a = - 1"}, {"rel": "被描述", "source": "3 a - 4", "target": "a + 8 + 3 a - 4 = 0"}, {"rel": "限制性描述", "source": "一个正数的平方根是 $a + 8$ 和 $3 a - 4$", "target": "a + 8 + 3 a - 4 = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "a + 8 + 3 a - 4 = 0"}, {"rel": "代入", "source": "a = - 1", "target": "7"}, {"rel": "被描述", "source": "7", "target": "49"}, {"rel": "限制性描述", "source": "平方", "target": "49"}]}}
{"content": "If the product of $( { x } ^ { 2 } + bx + 8 ) ( { x } ^ { 2 } - 3 x + c )$ does not contain terms of $x ^ { 2 }$ and $x ^ { 3 }$, then $b + c$ = ____ ?", "answer": "4", "steps": "$( { x } ^ { 2 } + bx + 8 ) ( { x } ^ { 2 } - 3 x + c ) = x ^ { 4 } - 3 x ^ { 3 } + cx ^ { 2 } + bx ^ { 3 } - 3 bx ^ { 2 } + bcx + 8 x ^ { 2 } - 24 x + 8 c = x ^ { 4 } + ( b - 3 ) x ^ { 3 } + ( c - 3 b + 8 ) x ^ { 2 } + ( bc - 24 ) x + 8 c$ , because there are no $x ^ { 2 }$ and $x ^ { 3 }$ terms in the product, therefore $b - 3 = 0$, and $c - 3 b + 8 = 0$, which gives $b = 3$, $c = 1$, therefore $b + c = 3 + 1 = 4$.", "expr_cands": ["( { x } ^ { 2 } + bx + 8 ) ( { x } ^ { 2 } - 3 x + c )", "x", "c", "b", "x ^ { 2 }", "x ^ { 3 }", "b + c", "x ^ { 4 } + ( b - 3 ) x ^ { 3 } + ( c - 3 b + 8 ) x ^ { 2 } + ( bc - 24 ) x + 8 c", "b - 3 = 0", "b = 3", "c - 3 b + 8 = 0", "c = 1", "4"], "exprs": ["x ^ { 4 } + ( b - 3 ) x ^ { 3 } + ( c - 3 b + 8 ) x ^ { 2 } + ( bc - 24 ) x + 8 c", "b - 3 = 0", "c - 3 b + 8 = 0", "b = 3", "c = 1", "4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( { x } ^ { 2 } + bx + 8 ) ( { x } ^ { 2 } - 3 x + c )"}, {"id": "x ^ { 4 } + ( b - 3 ) x ^ { 3 } + ( c - 3 b + 8 ) x ^ { 2 } + ( bc - 24 ) x + 8 c"}, {"id": "b - 3 = 0"}, {"id": "$( { x } ^ { 2 } + bx + 8 ) ( { x } ^ { 2 } - 3 x + c )$ 的乘积中不含 $x ^ { 2 }$ 和 $x ^ { 3 }$ 项"}, {"id": "b = 3"}, {"id": "c - 3 b + 8 = 0"}, {"id": "c = 1"}, {"id": "b + c"}, {"id": "4"}], "links": [{"rel": "展开", "source": "( { x } ^ { 2 } + bx + 8 ) ( { x } ^ { 2 } - 3 x + c )", "target": "x ^ { 4 } + ( b - 3 ) x ^ { 3 } + ( c - 3 b + 8 ) x ^ { 2 } + ( bc - 24 ) x + 8 c"}, {"rel": "被描述", "source": "x ^ { 4 } + ( b - 3 ) x ^ { 3 } + ( c - 3 b + 8 ) x ^ { 2 } + ( bc - 24 ) x + 8 c", "target": "b - 3 = 0"}, {"rel": "被描述", "source": "x ^ { 4 } + ( b - 3 ) x ^ { 3 } + ( c - 3 b + 8 ) x ^ { 2 } + ( bc - 24 ) x + 8 c", "target": "c - 3 b + 8 = 0"}, {"rel": "等式方程求解", "source": "b - 3 = 0", "target": "b = 3"}, {"rel": "限制性描述", "source": "$( { x } ^ { 2 } + bx + 8 ) ( { x } ^ { 2 } - 3 x + c )$ 的乘积中不含 $x ^ { 2 }$ 和 $x ^ { 3 }$ 项", "target": "b - 3 = 0"}, {"rel": "限制性描述", "source": "$( { x } ^ { 2 } + bx + 8 ) ( { x } ^ { 2 } - 3 x + c )$ 的乘积中不含 $x ^ { 2 }$ 和 $x ^ { 3 }$ 项", "target": "c - 3 b + 8 = 0"}, {"rel": "联立", "source": "b = 3", "target": "c = 1"}, {"rel": "代入", "source": "b = 3", "target": "4"}, {"rel": "联立", "source": "c - 3 b + 8 = 0", "target": "c = 1"}, {"rel": "代入", "source": "c = 1", "target": "4"}, {"rel": "被代入", "source": "b + c", "target": "4"}]}}
{"content": "Given $a$, $b$ are opposite numbers, $c$, $d$ are reciprocal numbers, $| x | = 1$, then $a + b + x ^ 2 - cd$ = ____?", "answer": "0", "steps": "\\because $| x | = 1$ , \\therefore $x ^ { 2 } = 1$ , \\because $a$ , $b$ are opposite numbers, $c$ , $d$ are reciprocal, \\therefore $a + b = 0$ , $cd = 1$ , \\therefore the original expression = $0 + 1 - 1 = 0$.", "expr_cands": ["a", "b", "c", "d", "| x | = 1", "x", "a + b + x ^ { 2 } - cd", "x = - 1", "x = 1", "x ^ { 2 } = 1", "a + b = 0", "cd = 1", "0 + 1 - 1", "0"], "exprs": ["x ^ { 2 } = 1", "a + b = 0", "cd = 1", "0 + 1 - 1", "0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| x | = 1"}, {"id": "x ^ { 2 } = 1"}, {"id": "绝对值恒大于等于0"}, {"id": "a"}, {"id": "a + b = 0"}, {"id": "b"}, {"id": "$a$ , $b$ 互为相反数"}, {"id": "c"}, {"id": "cd = 1"}, {"id": "d"}, {"id": "$c$ , $d$ 互为倒数"}, {"id": "a + b + x ^ { 2 } - cd"}, {"id": "0 + 1 - 1"}, {"id": "0"}], "links": [{"rel": "被描述", "source": "| x | = 1", "target": "x ^ { 2 } = 1"}, {"rel": "代入", "source": "x ^ { 2 } = 1", "target": "0 + 1 - 1"}, {"rel": "限制性描述", "source": "绝对值恒大于等于0", "target": "x ^ { 2 } = 1"}, {"rel": "被描述", "source": "a", "target": "a + b = 0"}, {"rel": "代入", "source": "a + b = 0", "target": "0 + 1 - 1"}, {"rel": "被描述", "source": "b", "target": "a + b = 0"}, {"rel": "限制性描述", "source": "$a$ , $b$ 互为相反数", "target": "a + b = 0"}, {"rel": "被描述", "source": "c", "target": "cd = 1"}, {"rel": "代入", "source": "cd = 1", "target": "0 + 1 - 1"}, {"rel": "被描述", "source": "d", "target": "cd = 1"}, {"rel": "限制性描述", "source": "$c$ , $d$ 互为倒数", "target": "cd = 1"}, {"rel": "被代入", "source": "a + b + x ^ { 2 } - cd", "target": "0 + 1 - 1"}, {"rel": "计算", "source": "0 + 1 - 1", "target": "0"}]}}
{"content": "If $a$ and $b$ are opposite numbers, and $c$ and $d$ are reciprocal numbers, then $2 cd - 2 a - 2 b$ = ____?", "answer": "2", "steps": "According to the problem, we have $a + b = 0$ and $cd = 1$. Therefore, the original expression is equal to $2 cd - 2 ( a + b ) = 2 - 0 = 2$.", "expr_cands": ["a", "b", "c", "d", "2 cd - 2 a - 2 b", "a + b = 0", "cd = 1", "2 cd - 2 ( a + b )", "2"], "exprs": ["a + b = 0", "cd = 1", "2 cd - 2 ( a + b )", "2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a + b = 0"}, {"id": "b"}, {"id": "$a$ 和 $b$ 互为相反数"}, {"id": "c"}, {"id": "cd = 1"}, {"id": "d"}, {"id": "$c$ 和 $d$ 互为倒数"}, {"id": "2 cd - 2 a - 2 b"}, {"id": "2 cd - 2 ( a + b )"}, {"id": "2"}], "links": [{"rel": "被描述", "source": "a", "target": "a + b = 0"}, {"rel": "提取因式参考", "source": "a + b = 0", "target": "2 cd - 2 ( a + b )"}, {"rel": "代入", "source": "a + b = 0", "target": "2"}, {"rel": "被描述", "source": "b", "target": "a + b = 0"}, {"rel": "限制性描述", "source": "$a$ 和 $b$ 互为相反数", "target": "a + b = 0"}, {"rel": "被描述", "source": "c", "target": "cd = 1"}, {"rel": "提取因式参考", "source": "cd = 1", "target": "2 cd - 2 ( a + b )"}, {"rel": "代入", "source": "cd = 1", "target": "2"}, {"rel": "被描述", "source": "d", "target": "cd = 1"}, {"rel": "限制性描述", "source": "$c$ 和 $d$ 互为倒数", "target": "cd = 1"}, {"rel": "提取因式", "source": "2 cd - 2 a - 2 b", "target": "2 cd - 2 ( a + b )"}, {"rel": "被代入", "source": "2 cd - 2 ( a + b )", "target": "2"}]}}
{"content": "The equation $x ^ { 2 } + mx - 1 = 0$ has two roots $x _ { 1 }$ and $x _ { 2 }$, and $\\frac { 1 } { x _ { 1 } } + \\frac { 1 } { x _ { 2 } } = - 3$. What is the value of $m$?", "answer": "- 3", "steps": "From the given information, we have $x _ 1 \\cdot x _ 2 = - 1$ and $x _ 1 + x _ 2 = - m$. Since $\\frac { 1 } { x _ 1 } + \\frac { 1 } { x _ 2 } = - 3$, we can also write $\\frac { x _ 1 + x _ 2 } { x _ 1 x _ 2 } = - 3$. Solving for $m$, we get $m = - 3$.", "expr_cands": ["x ^ { 2 } + mx - 1 = 0", "m", "x", "x _ { 1 }", "x _ { 2 }", "\\frac { 1 } { x _ { 1 } } + \\frac { 1 } { x _ { 2 } } = - 3", "x _ { 1 } \\cdot x _ { 2 } = - 1", "x _ { 1 } + x _ { 2 } = - m", "\\frac { x _ { 1 } + x _ { 2 } } { x _ { 1 } x _ { 2 } } = - 3", "m = - 3", "\\frac { - m } { - 1 } = - 3"], "exprs": ["x _ { 1 } \\cdot x _ { 2 } = - 1", "x _ { 1 } + x _ { 2 } = - m", "\\frac { x _ { 1 } + x _ { 2 } } { x _ { 1 } x _ { 2 } } = - 3", "m = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } + mx - 1 = 0"}, {"id": "x _ { 1 } \\cdot x _ { 2 } = - 1"}, {"id": "x _ { 1 }"}, {"id": "x _ { 2 }"}, {"id": "方程 $x ^ { 2 } + mx - 1 = 0$ 的两根为 $x _ { 1 }$ , $x _ { 2 }$"}, {"id": "一元二次方程根与系数关系，两根之积"}, {"id": "x _ { 1 } + x _ { 2 } = - m"}, {"id": "一元二次方程根与系数关系，两根之和"}, {"id": "\\frac { 1 } { x _ { 1 } } + \\frac { 1 } { x _ { 2 } } = - 3"}, {"id": "\\frac { x _ { 1 } + x _ { 2 } } { x _ { 1 } x _ { 2 } } = - 3"}, {"id": "m = - 3"}], "links": [{"rel": "被描述", "source": "x ^ { 2 } + mx - 1 = 0", "target": "x _ { 1 } \\cdot x _ { 2 } = - 1"}, {"rel": "被描述", "source": "x ^ { 2 } + mx - 1 = 0", "target": "x _ { 1 } + x _ { 2 } = - m"}, {"rel": "代入", "source": "x _ { 1 } \\cdot x _ { 2 } = - 1", "target": "m = - 3"}, {"rel": "被描述", "source": "x _ { 1 }", "target": "x _ { 1 } \\cdot x _ { 2 } = - 1"}, {"rel": "被描述", "source": "x _ { 1 }", "target": "x _ { 1 } + x _ { 2 } = - m"}, {"rel": "被描述", "source": "x _ { 2 }", "target": "x _ { 1 } \\cdot x _ { 2 } = - 1"}, {"rel": "被描述", "source": "x _ { 2 }", "target": "x _ { 1 } + x _ { 2 } = - m"}, {"rel": "限制性描述", "source": "方程 $x ^ { 2 } + mx - 1 = 0$ 的两根为 $x _ { 1 }$ , $x _ { 2 }$", "target": "x _ { 1 } \\cdot x _ { 2 } = - 1"}, {"rel": "限制性描述", "source": "方程 $x ^ { 2 } + mx - 1 = 0$ 的两根为 $x _ { 1 }$ , $x _ { 2 }$", "target": "x _ { 1 } + x _ { 2 } = - m"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之积", "target": "x _ { 1 } \\cdot x _ { 2 } = - 1"}, {"rel": "代入", "source": "x _ { 1 } + x _ { 2 } = - m", "target": "m = - 3"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "x _ { 1 } + x _ { 2 } = - m"}, {"rel": "计算", "source": "\\frac { 1 } { x _ { 1 } } + \\frac { 1 } { x _ { 2 } } = - 3", "target": "\\frac { x _ { 1 } + x _ { 2 } } { x _ { 1 } x _ { 2 } } = - 3"}, {"rel": "被代入", "source": "\\frac { x _ { 1 } + x _ { 2 } } { x _ { 1 } x _ { 2 } } = - 3", "target": "m = - 3"}]}}
{"content": "If $y = \\sqrt { x - 2017 } + \\sqrt { 2017 - x } + 2016$, then $x - y$ = ____ ?", "answer": "1", "steps": "$\\because y = \\sqrt { x - 2017 } + \\sqrt { 2017 - x } + 2016$ , $\\therefore x - 2017 \\ge 0$ , $2017 - x \\ge 0$ , $\\therefore x = 2017$ , $\\therefore y = 2016$ , so $x - y = 2017 - 2016 = 1$.", "expr_cands": ["y = \\sqrt { x - 2017 } + \\sqrt { 2017 - x } + 2016", "y", "x", "x - y", "x - 2017 \\ge 0", "2017 \\le x", "2017 - x \\ge 0", "x \\le 2017", "x = 2017", "y = 2016", "1"], "exprs": ["x - 2017 \\ge 0", "2017 - x \\ge 0", "x = 2017", "y = 2016", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = \\sqrt { x - 2017 } + \\sqrt { 2017 - x } + 2016"}, {"id": "x - 2017 \\ge 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "2017 - x \\ge 0"}, {"id": "x = 2017"}, {"id": "y = 2016"}, {"id": "x - y"}, {"id": "1"}], "links": [{"rel": "被描述", "source": "y = \\sqrt { x - 2017 } + \\sqrt { 2017 - x } + 2016", "target": "x - 2017 \\ge 0"}, {"rel": "被描述", "source": "y = \\sqrt { x - 2017 } + \\sqrt { 2017 - x } + 2016", "target": "2017 - x \\ge 0"}, {"rel": "被代入", "source": "y = \\sqrt { x - 2017 } + \\sqrt { 2017 - x } + 2016", "target": "y = 2016"}, {"rel": "联立", "source": "x - 2017 \\ge 0", "target": "x = 2017"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 2017 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "2017 - x \\ge 0"}, {"rel": "联立", "source": "2017 - x \\ge 0", "target": "x = 2017"}, {"rel": "代入", "source": "x = 2017", "target": "y = 2016"}, {"rel": "代入", "source": "x = 2017", "target": "1"}, {"rel": "代入", "source": "y = 2016", "target": "1"}, {"rel": "被代入", "source": "x - y", "target": "1"}]}}
{"content": "If $x < 3$, then the value of $\\frac { x - 3 } { | x - 3 | }$ is ____?", "answer": "- 1", "steps": "Since $x < 3$, it follows that $x - 3 < 0$. Therefore, $| x - 3 | = 3 - x$. Thus, $\\frac { x - 3 } { | x - 3 | } = \\frac { x - 3 } { 3 - x } = - 1$.", "expr_cands": ["x < 3", "x", "\\frac { x - 3 } { | x - 3 | }", "x - 3 < 0", "| x - 3 | = 3 - x", "- 1"], "exprs": ["x - 3 < 0", "| x - 3 | = 3 - x", "- 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x < 3"}, {"id": "x - 3 < 0"}, {"id": "\\frac { x - 3 } { | x - 3 | }"}, {"id": "| x - 3 | = 3 - x"}, {"id": "绝对值恒大于等于0"}, {"id": "- 1"}], "links": [{"rel": "移项", "source": "x < 3", "target": "x - 3 < 0"}, {"rel": "被描述", "source": "x - 3 < 0", "target": "| x - 3 | = 3 - x"}, {"rel": "被描述", "source": "\\frac { x - 3 } { | x - 3 | }", "target": "| x - 3 | = 3 - x"}, {"rel": "被代入", "source": "\\frac { x - 3 } { | x - 3 | }", "target": "- 1"}, {"rel": "代入", "source": "| x - 3 | = 3 - x", "target": "- 1"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "| x - 3 | = 3 - x"}]}}
{"content": "If the equation $x ^ 2 + 2 x + a = 0$ has two equal real roots, then the value of the real number $a$ is ____?", "answer": "1", "steps": "Since the equation has two equal real roots, therefore $4 - 4 a = 0$, thus $a = 1$.", "expr_cands": ["x ^ { 2 } + 2 x + a = 0", "a", "x", "4 - 4 a = 0", "a = 1"], "exprs": ["4 - 4 a = 0", "a = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } + 2 x + a = 0"}, {"id": "4 - 4 a = 0"}, {"id": "方程 $x ^ { 2 } + 2 x + a = 0$ 有两个相等的实数根"}, {"id": "一元二次方程根与系数关系，方程有解，则根的判别式大于等于0"}, {"id": "a = 1"}], "links": [{"rel": "被描述", "source": "x ^ { 2 } + 2 x + a = 0", "target": "4 - 4 a = 0"}, {"rel": "等式方程求解", "source": "4 - 4 a = 0", "target": "a = 1"}, {"rel": "限制性描述", "source": "方程 $x ^ { 2 } + 2 x + a = 0$ 有两个相等的实数根", "target": "4 - 4 a = 0"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，方程有解，则根的判别式大于等于0", "target": "4 - 4 a = 0"}]}}
{"content": "When $a$ = ____ ?, the value of $\\sqrt { a }$ is 2.", "answer": "4", "steps": "According to the problem, we have $\\sqrt { a } = 2$, which means $a = 4$.", "expr_cands": ["a", "\\sqrt { a }", "2", "\\sqrt { a } = 2", "a = 4"], "exprs": ["\\sqrt { a } = 2", "a = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { a }"}, {"id": "\\sqrt { a } = 2"}, {"id": "式子 $\\sqrt { a }$ 的值为 $2$"}, {"id": "a = 4"}], "links": [{"rel": "被描述", "source": "\\sqrt { a }", "target": "\\sqrt { a } = 2"}, {"rel": "等式方程求解", "source": "\\sqrt { a } = 2", "target": "a = 4"}, {"rel": "限制性描述", "source": "式子 $\\sqrt { a }$ 的值为 $2$", "target": "\\sqrt { a } = 2"}]}}
{"content": "Given: Two inequalities about $x$ and $y$, $x \\le m$ and $\\frac { 1 + y } { 3 } \\le y - 1$. If $x < y$ holds for any $x$ and $y$, then the range of $m$ is ____?", "answer": "m < 2", "steps": "$\\frac { 1 + y } { 3 } \\le y - 1$ , $1 + y \\le 3 y - 3$ , $- 2 y \\le - 4$ , $\\therefore y \\ge 2$ , because for any $x$ and $y$, $x < y$, therefore $m < 2$.", "expr_cands": ["x", "y", "x \\le m", "m", "\\frac { 1 + y } { 3 } \\le y - 1", "x < y", "2 \\le y", "1 + y \\le 3 y - 3", "- 2 y \\le - 4", "y \\ge 2", "m < 2"], "exprs": ["y \\ge 2", "m < 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 1 + y } { 3 } \\le y - 1"}, {"id": "y \\ge 2"}, {"id": "x \\le m"}, {"id": "m < 2"}, {"id": ": 关于 $x$ , $y$ 的两个不等式 $x \\le m$ 和 $\\frac { 1 + y } { 3 } \\le y - 1$"}, {"id": "对任意的 $x$ , $y$ 均满足 $x < y$"}], "links": [{"rel": "不等式方程求解", "source": "\\frac { 1 + y } { 3 } \\le y - 1", "target": "y \\ge 2"}, {"rel": "被描述", "source": "y \\ge 2", "target": "m < 2"}, {"rel": "被描述", "source": "x \\le m", "target": "m < 2"}, {"rel": "限制性描述", "source": ": 关于 $x$ , $y$ 的两个不等式 $x \\le m$ 和 $\\frac { 1 + y } { 3 } \\le y - 1$", "target": "m < 2"}, {"rel": "限制性描述", "source": "对任意的 $x$ , $y$ 均满足 $x < y$", "target": "m < 2"}]}}
{"content": "If the value of the fraction $\\frac { - 5 } { x - 1 }$ is negative, then the possible range of values for $x$ is ____?", "answer": "x > 1", "steps": "Since the value of $\\frac { - 5 } { x - 1 }$ is negative, it follows that $x - 1 > 0$, which means $x > 1$.", "expr_cands": ["\\frac { - 5 } { x - 1 }", "x", "x - 1 > 0", "1 < x", "x > 1"], "exprs": ["x - 1 > 0", "x > 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { - 5 } { x - 1 }"}, {"id": "x - 1 > 0"}, {"id": "分式 $\\frac { - 5 } { x - 1 }$ 的值是负数"}, {"id": "分式为负数，则分子分母异号"}, {"id": "x > 1"}], "links": [{"rel": "被描述", "source": "\\frac { - 5 } { x - 1 }", "target": "x - 1 > 0"}, {"rel": "不等式方程求解", "source": "x - 1 > 0", "target": "x > 1"}, {"rel": "限制性描述", "source": "分式 $\\frac { - 5 } { x - 1 }$ 的值是负数", "target": "x - 1 > 0"}, {"rel": "属性描述", "source": "分式为负数，则分子分母异号", "target": "x - 1 > 0"}]}}
{"content": "The algebraic expression $\\frac { 2 } { x + 1 }$ is meaningful, then the range of values for $x$ is ____?", "answer": "x \\neq - 1", "steps": "The algebraic expression $\\frac { 2 } { x + 1 }$ is meaningful only if $x + 1 \\neq 0$. Solving for $x$, we get $x \\neq - 1$.", "expr_cands": ["\\frac { 2 } { x + 1 }", "x", "x + 1 \\neq 0", "x \\neq - 1"], "exprs": ["x + 1 \\neq 0", "x \\neq - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 2 } { x + 1 }"}, {"id": "x + 1 \\neq 0"}, {"id": "代数式 $\\frac { 2 } { x + 1 }$ 有意义"}, {"id": "分式有意义，则分母不为0"}, {"id": "x \\neq - 1"}], "links": [{"rel": "被描述", "source": "\\frac { 2 } { x + 1 }", "target": "x + 1 \\neq 0"}, {"rel": "不等式方程求解", "source": "x + 1 \\neq 0", "target": "x \\neq - 1"}, {"rel": "限制性描述", "source": "代数式 $\\frac { 2 } { x + 1 }$ 有意义", "target": "x + 1 \\neq 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "x + 1 \\neq 0"}]}}
{"content": "What are all the values of $x$ that make the value of the fraction $\\frac { x ^ 2 - x } { x ^ 2 - 1 }$ equal to zero?", "answer": "0", "steps": "From the given condition, we have $x ^ 2 - x = 0$ and $x ^ 2 - 1 \\neq 0$. Solving the first equation, we get $x = 0$.", "expr_cands": ["\\frac { x ^ { 2 } - x } { x ^ { 2 } - 1 }", "x", "x ^ { 2 } - x = 0", "x = 0", "x = 1", "x ^ { 2 } - 1 \\neq 0", "( - 1 < x \\wedge x < 1 )", "1 < x", "x < - 1"], "exprs": ["x ^ { 2 } - x = 0", "x ^ { 2 } - 1 \\neq 0", "x = 0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x ^ { 2 } - x } { x ^ { 2 } - 1 }"}, {"id": "x ^ { 2 } - x = 0"}, {"id": "能使分式 $\\frac { x ^ { 2 } - x } { x ^ { 2 } - 1 }$ 的值为零的所有 $x$ 的值"}, {"id": "分式有意义，则分母不为0"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "x ^ { 2 } - 1 \\neq 0"}, {"id": "x = 0"}], "links": [{"rel": "被描述", "source": "\\frac { x ^ { 2 } - x } { x ^ { 2 } - 1 }", "target": "x ^ { 2 } - x = 0"}, {"rel": "被描述", "source": "\\frac { x ^ { 2 } - x } { x ^ { 2 } - 1 }", "target": "x ^ { 2 } - 1 \\neq 0"}, {"rel": "联立", "source": "x ^ { 2 } - x = 0", "target": "x = 0"}, {"rel": "限制性描述", "source": "能使分式 $\\frac { x ^ { 2 } - x } { x ^ { 2 } - 1 }$ 的值为零的所有 $x$ 的值", "target": "x ^ { 2 } - x = 0"}, {"rel": "限制性描述", "source": "能使分式 $\\frac { x ^ { 2 } - x } { x ^ { 2 } - 1 }$ 的值为零的所有 $x$ 的值", "target": "x ^ { 2 } - 1 \\neq 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "x ^ { 2 } - x = 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "x ^ { 2 } - 1 \\neq 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x ^ { 2 } - x = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x ^ { 2 } - 1 \\neq 0"}, {"rel": "联立", "source": "x ^ { 2 } - 1 \\neq 0", "target": "x = 0"}]}}
{"content": "In the equation $a ^ { m + n } \\div ____ ? = a ^ { m - 2 }$, what should be the algebraic expression inside the parentheses?", "answer": "a ^ { n + 2 }", "steps": "$a ^ { m + n } \\div ( ) = a ^ { m - 2 }$ becomes a to the power of m plus n divided by ( ) equals a to the power of m minus 2. Then, Therefore, a to the power of m plus n divided by a to the power of m minus 2 equals a to the power of n plus 2.", "expr_cands": ["a ^ { m + n } \\div [blk] = a ^ { m - 2 }", "b", "k", "m", "a", "l", "n", "a ^ { m + n } \\div ( ) = a ^ { m - 2 }", "a ^ { m + n } \\div a ^ { m - 2 }", "a ^ { n + 2 }"], "exprs": ["a ^ { m + n } \\div a ^ { m - 2 }", "a ^ { n + 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a ^ { m + n } \\div [blk] = a ^ { m - 2 }"}, {"id": "a ^ { m + n } \\div a ^ { m - 2 }"}, {"id": "括号内的代数式应"}, {"id": "a ^ { n + 2 }"}], "links": [{"rel": "被描述", "source": "a ^ { m + n } \\div [blk] = a ^ { m - 2 }", "target": "a ^ { m + n } \\div a ^ { m - 2 }"}, {"rel": "计算", "source": "a ^ { m + n } \\div a ^ { m - 2 }", "target": "a ^ { n + 2 }"}, {"rel": "限制性描述", "source": "括号内的代数式应", "target": "a ^ { m + n } \\div a ^ { m - 2 }"}]}}
{"content": "The equation about $x$, $2 a - x = 6$, has a non-negative solution for $x$. What is the condition for $a$ to satisfy?", "answer": "a \\ge 3", "steps": "Since $2 a - x = 6$, therefore $x = 2 a - 6$. Since $x$ is non-negative, therefore $2 a - 6 \\ge 0$, $a \\ge 3$.", "expr_cands": ["x", "2 a - x = 6", "a", "x = 2 a - 6", "2 a - 6 \\ge 0", "3 \\le a", "a \\ge 3"], "exprs": ["x = 2 a - 6", "2 a - 6 \\ge 0", "a \\ge 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 a - x = 6"}, {"id": "x = 2 a - 6"}, {"id": "2 a - 6 \\ge 0"}, {"id": "关于 $x$ 的方程 $2 a - x = 6$ 的解是非负数"}, {"id": "a \\ge 3"}], "links": [{"rel": "等式方程部分求解", "source": "2 a - x = 6", "target": "x = 2 a - 6"}, {"rel": "被描述", "source": "x = 2 a - 6", "target": "2 a - 6 \\ge 0"}, {"rel": "不等式方程求解", "source": "2 a - 6 \\ge 0", "target": "a \\ge 3"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $2 a - x = 6$ 的解是非负数", "target": "2 a - 6 \\ge 0"}]}}
{"content": "Given $a = 2005 x + 2006$, $b = 2005 x + 2007$, $c = 2005 x + 2008$, find $a ^ { 2 } + b ^ { 2 } + c ^ { 2 } - ab - ac - bc$.", "answer": "3", "steps": "Since $a = 2005 x + 2006$, $b = 2005 x + 2007$, and $c = 2005 x + 2008$, we have $a - b = - 1$, $a - c = - 2$, and $b - c = - 1$. Therefore, the original expression is equal to $\\frac { 1 } { 2 } ( 2 a ^ 2 + 2 b ^ 2 + 2 c ^ 2 - 2 ab - 2 ac - 2 bc ) = \\frac { 1 } { 2 } [( a - b ) ^ 2 + ( a - c ) ^ 2 + ( b - c ) ^ 2 ] = 3$.", "expr_cands": ["a = 2005 x + 2006", "a", "x", "b = 2005 x + 2007", "b", "c = 2005 x + 2008", "c", "a ^ { 2 } + b ^ { 2 } + c ^ { 2 } - ab - ac - bc", "a - b", "- 1", "a - c", "- 2", "b - c", "\\frac { 1 } { 2 } ( 2 a ^ { 2 } + 2 b ^ { 2 } + 2 c ^ { 2 } - 2 ab - 2 ac - 2 bc )", "3"], "exprs": ["3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a = 2005 x + 2006"}, {"id": "3"}, {"id": "b = 2005 x + 2007"}, {"id": "c = 2005 x + 2008"}, {"id": "a ^ { 2 } + b ^ { 2 } + c ^ { 2 } - ab - ac - bc"}], "links": [{"rel": "代入", "source": "a = 2005 x + 2006", "target": "3"}, {"rel": "代入", "source": "b = 2005 x + 2007", "target": "3"}, {"rel": "代入", "source": "c = 2005 x + 2008", "target": "3"}, {"rel": "被代入", "source": "a ^ { 2 } + b ^ { 2 } + c ^ { 2 } - ab - ac - bc", "target": "3"}]}}
{"content": "When $a = 2$, $b = - 1$, and $c = - 3$, what is the value of the algebraic expression $b ^ 2 - 4 ac$?", "answer": "25", "steps": "Since $a = 2$, $b = - 1$, and $c = - 3$, therefore $b ^ 2 - 4 ac = ( - 1 ) ^ 2 - 4 * 2 * ( - 3 ) = 1 + 24 = 25$.", "expr_cands": ["a = 2", "a", "b = - 1", "b", "c = - 3", "c", "b ^ { 2 } - 4 ac", "25"], "exprs": ["25"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "b ^ { 2 } - 4 ac"}, {"id": "25"}, {"id": "a = 2"}, {"id": "b = - 1"}, {"id": "c = - 3"}], "links": [{"rel": "被代入", "source": "b ^ { 2 } - 4 ac", "target": "25"}, {"rel": "代入", "source": "a = 2", "target": "25"}, {"rel": "代入", "source": "b = - 1", "target": "25"}, {"rel": "代入", "source": "c = - 3", "target": "25"}]}}
{"content": "The range of the independent variable for the function $y = \\frac { 2020 + x } { 2021 - x }$ is ____ ?", "answer": "x \\neq 2021", "steps": "To make $\\frac { 2020 + x } { 2021 - x }$ meaningful, we must have $2021 - x \\neq 0$, which implies $x \\neq 2021$.", "expr_cands": ["y = \\frac { 2020 + x } { 2021 - x }", "x", "y", "\\frac { 2020 + x } { 2021 - x }", "2021 - x \\neq 0", "x \\neq 2021"], "exprs": ["2021 - x \\neq 0", "x \\neq 2021"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = \\frac { 2020 + x } { 2021 - x }"}, {"id": "2021 - x \\neq 0"}, {"id": "函数 $y = \\frac { 2020 + x } { 2021 - x }$ 自变量的取值范围"}, {"id": "要使 $\\frac { 2020 + x } { 2021 - x }$ 有意义"}, {"id": "分式有意义，则分母不为0"}, {"id": "x \\neq 2021"}], "links": [{"rel": "被描述", "source": "y = \\frac { 2020 + x } { 2021 - x }", "target": "2021 - x \\neq 0"}, {"rel": "不等式方程求解", "source": "2021 - x \\neq 0", "target": "x \\neq 2021"}, {"rel": "限制性描述", "source": "函数 $y = \\frac { 2020 + x } { 2021 - x }$ 自变量的取值范围", "target": "2021 - x \\neq 0"}, {"rel": "限制性描述", "source": "要使 $\\frac { 2020 + x } { 2021 - x }$ 有意义", "target": "2021 - x \\neq 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "2021 - x \\neq 0"}]}}
{"content": "If $\\frac { m + n } { m } = \\frac { 7 } { 13 }$, then $\\frac { m } { n } =$ ____?", "answer": "- \\frac { 13 } { 6 }", "steps": "Because $\\frac { m + n } { m } = \\frac { 7 } { 13 }$, therefore $13 ( m + n ) = 7 m$, therefore $13 m + 13 n = 7 m$. Solving for $m$, we get $m = - \\frac { 13 } { 6 } n$, therefore $\\frac { m } { n } = - \\frac { 13 } { 6 }$.", "expr_cands": ["\\frac { m + n } { m } = \\frac { 7 } { 13 }", "m", "n", "\\frac { m } { n }", "13 ( m + n ) = 7 m", "13 m + 13 n = 7 m", "m = - \\frac { 13 } { 6 } n", "- \\frac { 13 } { 6 }"], "exprs": ["13 ( m + n ) = 7 m", "13 m + 13 n = 7 m", "m = - \\frac { 13 } { 6 } n", "- \\frac { 13 } { 6 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { m + n } { m } = \\frac { 7 } { 13 }"}, {"id": "13 ( m + n ) = 7 m"}, {"id": "13 m + 13 n = 7 m"}, {"id": "m = - \\frac { 13 } { 6 } n"}, {"id": "\\frac { m } { n }"}, {"id": "- \\frac { 13 } { 6 }"}], "links": [{"rel": "同乘除", "source": "\\frac { m + n } { m } = \\frac { 7 } { 13 }", "target": "13 ( m + n ) = 7 m"}, {"rel": "展开", "source": "13 ( m + n ) = 7 m", "target": "13 m + 13 n = 7 m"}, {"rel": "等式方程部分求解", "source": "13 m + 13 n = 7 m", "target": "m = - \\frac { 13 } { 6 } n"}, {"rel": "代入", "source": "m = - \\frac { 13 } { 6 } n", "target": "- \\frac { 13 } { 6 }"}, {"rel": "被代入", "source": "\\frac { m } { n }", "target": "- \\frac { 13 } { 6 }"}]}}
{"content": "If $a$ and $2 a - 9$ are opposite in sign, then the value of $a$ is ____?", "answer": "3", "steps": "$\\because$ If $a$ and $2 a - 9$ are opposite in sign, $\\therefore$ $a + 2 a - 9 = 0$, solving which gives $a = 3$.", "expr_cands": ["a", "2 a - 9", "a + 2 a - 9 = 0", "a = 3"], "exprs": ["a + 2 a - 9 = 0", "a = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 a - 9"}, {"id": "a + 2 a - 9 = 0"}, {"id": "a"}, {"id": "$a$ 与 $2 a - 9$ 互为相反数"}, {"id": "a = 3"}], "links": [{"rel": "被描述", "source": "2 a - 9", "target": "a + 2 a - 9 = 0"}, {"rel": "等式方程求解", "source": "a + 2 a - 9 = 0", "target": "a = 3"}, {"rel": "被描述", "source": "a", "target": "a + 2 a - 9 = 0"}, {"rel": "限制性描述", "source": "$a$ 与 $2 a - 9$ 互为相反数", "target": "a + 2 a - 9 = 0"}]}}
{"content": "The meaningful condition for the function $y = \\sqrt { x - 1 }$ is ____ ?", "answer": "x \\ge 1", "steps": "From the given condition, we have $x - 1 \\ge 0$, which implies $x \\ge 1$ after solving for $x$.", "expr_cands": ["y = \\sqrt { x - 1 }", "y", "x", "x - 1 \\ge 0", "1 \\le x", "x \\ge 1"], "exprs": ["x - 1 \\ge 0", "x \\ge 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = \\sqrt { x - 1 }"}, {"id": "x - 1 \\ge 0"}, {"id": "函数 $y = \\sqrt { x - 1 }$ 有意义的条件"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x \\ge 1"}], "links": [{"rel": "被描述", "source": "y = \\sqrt { x - 1 }", "target": "x - 1 \\ge 0"}, {"rel": "不等式方程求解", "source": "x - 1 \\ge 0", "target": "x \\ge 1"}, {"rel": "限制性描述", "source": "函数 $y = \\sqrt { x - 1 }$ 有意义的条件", "target": "x - 1 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 1 \\ge 0"}]}}
{"content": "The solution to the equation $6 - 2 ( x - 1 ) = 0$ is ____ ?", "answer": "x = 4", "steps": "Moving terms: $2 ( x - 1 ) = 6$, divide both sides by $2$, we get $x - 1 = 3$, move the term, we get $x = 4$.", "expr_cands": ["6 - 2 ( x - 1 ) = 0", "x", "2 ( x - 1 ) = 6", "x = 4", "2", "x - 1 = 3"], "exprs": ["x = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "6 - 2 ( x - 1 ) = 0"}, {"id": "x = 4"}], "links": [{"rel": "等式方程求解", "source": "6 - 2 ( x - 1 ) = 0", "target": "x = 4"}]}}
{"content": "If the value of the fraction $\\frac { 4 } { m - 2 }$ is negative, then the possible range of values for $m$ is ____?", "answer": "m < 2", "steps": "From the given condition, we have $m - 2 < 0$, which implies that $m < 2$.", "expr_cands": ["\\frac { 4 } { m - 2 }", "m", "m - 2 < 0", "m < 2"], "exprs": ["m - 2 < 0", "m < 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 4 } { m - 2 }"}, {"id": "m - 2 < 0"}, {"id": "分式 $\\frac { 4 } { m - 2 }$ 的值为负数"}, {"id": "分式为负数，则分子分母异号"}, {"id": "m < 2"}], "links": [{"rel": "被描述", "source": "\\frac { 4 } { m - 2 }", "target": "m - 2 < 0"}, {"rel": "不等式方程求解", "source": "m - 2 < 0", "target": "m < 2"}, {"rel": "限制性描述", "source": "分式 $\\frac { 4 } { m - 2 }$ 的值为负数", "target": "m - 2 < 0"}, {"rel": "属性描述", "source": "分式为负数，则分子分母异号", "target": "m - 2 < 0"}]}}
{"content": "If $| 4 - x | + | y + 2 | = 0$, then the value of $x \\div y$ is ____?", "answer": "- 2", "steps": "From the given information, we can deduce that $4 - x = 0$ and $y + 2 = 0$. Solving for $x$ and $y$, we get $x = 4$ and $y = - 2$. Therefore, $x \\div y = 4 \\div ( - 2 ) = - 2$.", "expr_cands": ["| 4 - x | + | y + 2 | = 0", "x", "y", "x \\div y", "4 - x = 0", "x = 4", "y + 2 = 0", "y = - 2", "- 2"], "exprs": ["4 - x = 0", "y + 2 = 0", "x = 4", "y = - 2", "- 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| 4 - x | + | y + 2 | = 0"}, {"id": "4 - x = 0"}, {"id": "绝对值恒大于等于0"}, {"id": "y + 2 = 0"}, {"id": "x = 4"}, {"id": "y = - 2"}, {"id": "x \\div y"}, {"id": "- 2"}], "links": [{"rel": "被描述", "source": "| 4 - x | + | y + 2 | = 0", "target": "4 - x = 0"}, {"rel": "被描述", "source": "| 4 - x | + | y + 2 | = 0", "target": "y + 2 = 0"}, {"rel": "等式方程求解", "source": "4 - x = 0", "target": "x = 4"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "4 - x = 0"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "y + 2 = 0"}, {"rel": "等式方程求解", "source": "y + 2 = 0", "target": "y = - 2"}, {"rel": "代入", "source": "x = 4", "target": "- 2"}, {"rel": "代入", "source": "y = - 2", "target": "- 2"}, {"rel": "被代入", "source": "x \\div y", "target": "- 2"}]}}
{"content": "Given that the equation $ax ^ 2 + bx + c = 0 ( a \\neq 0 )$ has a root of $1$, what is $a + b + c$?", "answer": "0", "steps": "$\\because x = 1$ is a root of the quadratic equation $ax ^ 2 + bx + c = 0$, $\\therefore a \\cdot 1 ^ 2 + b \\cdot 1 + c = 0$, $\\therefore a + b + c = 0$.", "expr_cands": ["ax ^ { 2 } + bx + c = 0 ( a \\neq 0 )", "a", "x", "c", "b", "1", "a + b + c", "x = 1", "ax ^ { 2 } + bx + c = 0", "a + b + c = 0", "a * 1 ^ { 2 } + b * 1 + c = 0", "0"], "exprs": ["x = 1", "a + b + c = 0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "1"}, {"id": "x = 1"}, {"id": "ax ^ { 2 } + bx + c = 0 ( a \\neq 0 )"}, {"id": "方程 $ax ^ { 2 } + bx + c = 0 ( a \\neq 0 )$ 有一根是 $1$"}, {"id": "a + b + c = 0"}], "links": [{"rel": "被描述", "source": "1", "target": "x = 1"}, {"rel": "代入", "source": "x = 1", "target": "a + b + c = 0"}, {"rel": "被描述", "source": "ax ^ { 2 } + bx + c = 0 ( a \\neq 0 )", "target": "x = 1"}, {"rel": "被代入", "source": "ax ^ { 2 } + bx + c = 0 ( a \\neq 0 )", "target": "a + b + c = 0"}, {"rel": "限制性描述", "source": "方程 $ax ^ { 2 } + bx + c = 0 ( a \\neq 0 )$ 有一根是 $1$", "target": "x = 1"}]}}
{"content": "Convert the quadratic equation $\\frac { x + y } { 2 } - \\frac { x - y } { 5 } = 1$ to the form of $y = kx + b$, the result is ____ ?", "answer": "y = - \\frac { 1 } { 5 } x + \\frac { 6 } { 5 }", "steps": "Convert the quadratic equation $\\frac { x + y } { 2 } - \\frac { x - y } { 5 } = 1$ to the form $y = kx + b$, we get $y = - \\frac { 1 } { 5 } x + \\frac { 6 } { 5 }$.", "expr_cands": ["\\frac { x + y } { 2 } - \\frac { x - y } { 5 } = 1", "y", "x", "y = kx + b", "k", "b", "y = - \\frac { 1 } { 5 } x + \\frac { 6 } { 5 }", "b + k x = - \\frac { 1 } { 5 } x + \\frac { 6 } { 5 }", "b + k x"], "exprs": ["y = - \\frac { 1 } { 5 } x + \\frac { 6 } { 5 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x + y } { 2 } - \\frac { x - y } { 5 } = 1"}, {"id": "y = - \\frac { 1 } { 5 } x + \\frac { 6 } { 5 }"}, {"id": "y = kx + b"}, {"id": "把二元一次方程 $\\frac { x + y } { 2 } - \\frac { x - y } { 5 } = 1$ 化为 $y = kx + b$ 的形式"}], "links": [{"rel": "被描述", "source": "\\frac { x + y } { 2 } - \\frac { x - y } { 5 } = 1", "target": "y = - \\frac { 1 } { 5 } x + \\frac { 6 } { 5 }"}, {"rel": "被描述", "source": "y = kx + b", "target": "y = - \\frac { 1 } { 5 } x + \\frac { 6 } { 5 }"}, {"rel": "限制性描述", "source": "把二元一次方程 $\\frac { x + y } { 2 } - \\frac { x - y } { 5 } = 1$ 化为 $y = kx + b$ 的形式", "target": "y = - \\frac { 1 } { 5 } x + \\frac { 6 } { 5 }"}]}}
{"content": "If the equation $mx + x = 1$ has no solution with respect to $x$, then the possible values of $m$ are ____?", "answer": "m = - 1", "steps": "Assuming that the equation $mx + x = 1$ has a solution, then $x = \\frac { 1 } { m + 1 }$. Since the equation $mx + x = 1$ has no solution, we have $m + 1 = 0$. Therefore, when $m = - 1$, the equation has no solution.", "expr_cands": ["x", "mx + x = 1", "m", "x = \\frac { 1 } { m + 1 }", "m + 1 = 0", "m = - 1"], "exprs": ["x = \\frac { 1 } { m + 1 }", "m + 1 = 0", "m = - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "mx + x = 1"}, {"id": "x = \\frac { 1 } { m + 1 }"}, {"id": "m + 1 = 0"}, {"id": "分式方程无解，则分母为0"}, {"id": "关于 $x$ 的方程 $mx + x = 1$ 无解"}, {"id": "m = - 1"}], "links": [{"rel": "等式方程部分求解", "source": "mx + x = 1", "target": "x = \\frac { 1 } { m + 1 }"}, {"rel": "被描述", "source": "x = \\frac { 1 } { m + 1 }", "target": "m + 1 = 0"}, {"rel": "等式方程求解", "source": "m + 1 = 0", "target": "m = - 1"}, {"rel": "属性描述", "source": "分式方程无解，则分母为0", "target": "m + 1 = 0"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $mx + x = 1$ 无解", "target": "m + 1 = 0"}]}}
{"content": "Given: $x - 2 y = - 3$, what is the value of the algebraic expression ${ ( 2 y - x ) } ^ { 2 } - 2 x + 4 y - 1$?", "answer": "14", "steps": "Because $x - 2 y = - 3$, therefore $( 2 y - x ) ^ 2 - 2 x + 4 y - 1 = ( x - 2 y ) ^ 2 - 2 ( x - 2 y ) - 1 = ( - 3 ) ^ 2 - 2 * ( - 3 ) - 1 = 9 + 6 - 1 = 14$.", "expr_cands": ["x - 2 y = - 3", "x", "y", "{ ( 2 y - x ) } ^ { 2 } - 2 x + 4 y - 1", "( x - 2 y ) ^ { 2 } - 2 ( x - 2 y ) - 1", "14"], "exprs": ["( x - 2 y ) ^ { 2 } - 2 ( x - 2 y ) - 1", "14"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "{ ( 2 y - x ) } ^ { 2 } - 2 x + 4 y - 1"}, {"id": "( x - 2 y ) ^ { 2 } - 2 ( x - 2 y ) - 1"}, {"id": "x - 2 y = - 3"}, {"id": "14"}], "links": [{"rel": "提取因式", "source": "{ ( 2 y - x ) } ^ { 2 } - 2 x + 4 y - 1", "target": "( x - 2 y ) ^ { 2 } - 2 ( x - 2 y ) - 1"}, {"rel": "被代入", "source": "( x - 2 y ) ^ { 2 } - 2 ( x - 2 y ) - 1", "target": "14"}, {"rel": "提取因式参考", "source": "x - 2 y = - 3", "target": "( x - 2 y ) ^ { 2 } - 2 ( x - 2 y ) - 1"}, {"rel": "代入", "source": "x - 2 y = - 3", "target": "14"}]}}
{"content": "Given $x + 5 y = 7$, what is the value of the algebraic expression $3 - x - 5 y$?", "answer": "- 4", "steps": "Because $x + 5 y = 7$, therefore the original expression is equal to $3 - ( x + 5 y ) = 3 - 7 = - 4$.", "expr_cands": ["x + 5 y = 7", "y", "x", "3 - x - 5 y", "3 - ( x + 5 y )", "- 4"], "exprs": ["3 - ( x + 5 y )", "- 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 - x - 5 y"}, {"id": "3 - ( x + 5 y )"}, {"id": "x + 5 y = 7"}, {"id": "- 4"}], "links": [{"rel": "提取因式", "source": "3 - x - 5 y", "target": "3 - ( x + 5 y )"}, {"rel": "被代入", "source": "3 - ( x + 5 y )", "target": "- 4"}, {"rel": "提取因式参考", "source": "x + 5 y = 7", "target": "3 - ( x + 5 y )"}, {"rel": "代入", "source": "x + 5 y = 7", "target": "- 4"}]}}
{"content": "If for any two rational numbers $m$ and $n$, we have $m * n = \\frac { m + 3 n } { 4 }$, then the solution to the equation $3 x * 4 = 2$ is ____?", "answer": "- \\frac { 4 } { 3 }", "steps": "From the given problem, we have the equation: $\\frac { 3 x + 3 * 4 } { 4 } = 2$. We can eliminate the denominator by multiplying both sides by 4, which gives us $3 x + 12 = 8$. Solving for $x$, we can subtract 12 from both sides to get $3 x = - 4$. Dividing both sides by 3, we get $x = - \\frac { 4 } { 3 }$.", "expr_cands": ["m", "n", "m * n = \\frac { m + 3 n } { 4 }", "3 x * 4 = 2", "x", "\\frac { 3 x + 3 * 4 } { 4 } = 2", "x = - \\frac { 4 } { 3 }", "3 x + 12 = 8", "3 x = - 4", "1"], "exprs": ["\\frac { 3 x + 3 * 4 } { 4 } = 2", "x = - \\frac { 4 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "m * n = \\frac { m + 3 n } { 4 }"}, {"id": "\\frac { 3 x + 3 * 4 } { 4 } = 2"}, {"id": "3 x * 4 = 2"}, {"id": "对于任意的两个有理数 $m$ , $n$"}, {"id": "都有 $m * n = \\frac { m + 3 n } { 4 }$"}, {"id": "方程 $3 x * 4 = 2$ 的解"}, {"id": "x = - \\frac { 4 } { 3 }"}], "links": [{"rel": "被描述", "source": "m * n = \\frac { m + 3 n } { 4 }", "target": "\\frac { 3 x + 3 * 4 } { 4 } = 2"}, {"rel": "等式方程求解", "source": "\\frac { 3 x + 3 * 4 } { 4 } = 2", "target": "x = - \\frac { 4 } { 3 }"}, {"rel": "被描述", "source": "3 x * 4 = 2", "target": "\\frac { 3 x + 3 * 4 } { 4 } = 2"}, {"rel": "限制性描述", "source": "对于任意的两个有理数 $m$ , $n$", "target": "\\frac { 3 x + 3 * 4 } { 4 } = 2"}, {"rel": "限制性描述", "source": "都有 $m * n = \\frac { m + 3 n } { 4 }$", "target": "\\frac { 3 x + 3 * 4 } { 4 } = 2"}, {"rel": "限制性描述", "source": "方程 $3 x * 4 = 2$ 的解", "target": "\\frac { 3 x + 3 * 4 } { 4 } = 2"}]}}
{"content": "If $( a - 4 ) x ^ 4 - x ^ b + x - b$ is a cubic trinomial in terms of $x$, then $a + b$ = ____?", "answer": "7", "steps": "$\\because$ The polynomial is a cubic trinomial, $\\therefore$ $a - 4 = 0$, $b = 3$, $\\therefore$ $a = 4$, $b = 3$, then $a + b = 4 + 3 = 7$.", "expr_cands": ["( a - 4 ) x ^ { 4 } - x ^ { b } + x - b", "b", "a", "x", "a + b", "a - 4 = 0", "a = 4", "b = 3", "7"], "exprs": ["a - 4 = 0", "b = 3", "a = 4", "7"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( a - 4 ) x ^ { 4 } - x ^ { b } + x - b"}, {"id": "a - 4 = 0"}, {"id": "$( a - 4 ) x ^ { 4 } - x ^ { b } + x - b$ 是关于 $x$ 的三次三项式"}, {"id": "a = 4"}, {"id": "b = 3"}, {"id": "a + b"}, {"id": "7"}], "links": [{"rel": "被描述", "source": "( a - 4 ) x ^ { 4 } - x ^ { b } + x - b", "target": "a - 4 = 0"}, {"rel": "被描述", "source": "( a - 4 ) x ^ { 4 } - x ^ { b } + x - b", "target": "b = 3"}, {"rel": "等式方程求解", "source": "a - 4 = 0", "target": "a = 4"}, {"rel": "限制性描述", "source": "$( a - 4 ) x ^ { 4 } - x ^ { b } + x - b$ 是关于 $x$ 的三次三项式", "target": "a - 4 = 0"}, {"rel": "限制性描述", "source": "$( a - 4 ) x ^ { 4 } - x ^ { b } + x - b$ 是关于 $x$ 的三次三项式", "target": "b = 3"}, {"rel": "代入", "source": "a = 4", "target": "7"}, {"rel": "代入", "source": "b = 3", "target": "7"}, {"rel": "被代入", "source": "a + b", "target": "7"}]}}
{"content": "If $a - b = 3$, $ab = - 1$, what is the value of the algebraic expression $3 ab - a + b - 2$?", "answer": "- 8", "steps": "Since $a - b = 3$ and $ab = - 1$, therefore $3 ab - a + b - 2 = 3 ab - ( a - b ) - 2 = 3 * ( - 1 ) - 3 - 2 = - 3 - 3 - 2 = - 8$.", "expr_cands": ["a - b = 3", "a", "b", "ab = - 1", "3 ab - a + b - 2", "3 ab - ( a - b ) - 2", "- 8"], "exprs": ["3 ab - ( a - b ) - 2", "- 8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 ab - a + b - 2"}, {"id": "3 ab - ( a - b ) - 2"}, {"id": "a - b = 3"}, {"id": "- 8"}, {"id": "ab = - 1"}], "links": [{"rel": "提取因式", "source": "3 ab - a + b - 2", "target": "3 ab - ( a - b ) - 2"}, {"rel": "被代入", "source": "3 ab - ( a - b ) - 2", "target": "- 8"}, {"rel": "提取因式参考", "source": "a - b = 3", "target": "3 ab - ( a - b ) - 2"}, {"rel": "代入", "source": "a - b = 3", "target": "- 8"}, {"rel": "代入", "source": "ab = - 1", "target": "- 8"}]}}
{"content": "If the square root of a positive number is $2 m - 4$ and $3 m - 1$, then the positive number is ____?", "answer": "4", "steps": "From the given information, we have $2 m - 4 + 3 m - 1 = 0$, which gives us $m = 1$. Therefore, $2 m - 4 = 2 - 4 = - 2$ and $3 m - 1 = 3 - 1 = 2$. This means that the positive number has two square roots, which are $2$ and $- 2$. Hence, the positive number is $4$.", "expr_cands": ["2 m - 4", "m", "3 m - 1", "2 m - 4 + 3 m - 1 = 0", "m = 1", "- 2", "2", "4"], "exprs": ["2 m - 4 + 3 m - 1 = 0", "m = 1", "2", "4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 m - 4"}, {"id": "2 m - 4 + 3 m - 1 = 0"}, {"id": "3 m - 1"}, {"id": "一个正数的平方根是 $2 m - 4$ 与 $3 m - 1$"}, {"id": "平方根互为相反数"}, {"id": "m = 1"}, {"id": "2"}, {"id": "4"}, {"id": "平方"}], "links": [{"rel": "被描述", "source": "2 m - 4", "target": "2 m - 4 + 3 m - 1 = 0"}, {"rel": "等式方程求解", "source": "2 m - 4 + 3 m - 1 = 0", "target": "m = 1"}, {"rel": "被描述", "source": "3 m - 1", "target": "2 m - 4 + 3 m - 1 = 0"}, {"rel": "被代入", "source": "3 m - 1", "target": "2"}, {"rel": "限制性描述", "source": "一个正数的平方根是 $2 m - 4$ 与 $3 m - 1$", "target": "2 m - 4 + 3 m - 1 = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "2 m - 4 + 3 m - 1 = 0"}, {"rel": "代入", "source": "m = 1", "target": "2"}, {"rel": "被描述", "source": "2", "target": "4"}, {"rel": "限制性描述", "source": "平方", "target": "4"}]}}
{"content": "If the equation $3 k - 5 x = - 9$ has a non-negative solution for $x$, then the possible values of $k$ are _____.", "answer": "k \\ge - 3", "steps": "$3 k - 5 x = - 9$, $- 5 x = - 9 - 3 k$, $x = \\frac { 9 + 3 k } { 5 }$. Since the solution to the equation $3 k - 5 x = - 9$ with respect to $x$ is non-negative, we have $\\frac { 9 + 3 k } { 5 } \\geq 0$. Solving the inequality, we get $k \\geq - 3$. Therefore, the range of values for $k$ is $k \\geq - 3$.", "expr_cands": ["x", "3 k - 5 x = - 9", "k", "- 5 x = - 9 - 3 k", "x = \\frac { 9 + 3 k } { 5 }", "\\frac { 9 + 3 k } { 5 } \\ge 0", "- 3 \\le k", "k \\ge - 3"], "exprs": ["x = \\frac { 9 + 3 k } { 5 }", "\\frac { 9 + 3 k } { 5 } \\ge 0", "k \\ge - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 k - 5 x = - 9"}, {"id": "x = \\frac { 9 + 3 k } { 5 }"}, {"id": "\\frac { 9 + 3 k } { 5 } \\ge 0"}, {"id": "关于 $x$ 的方程 $3 k - 5 x = - 9$ 的解是非负数"}, {"id": "k \\ge - 3"}], "links": [{"rel": "等式方程部分求解", "source": "3 k - 5 x = - 9", "target": "x = \\frac { 9 + 3 k } { 5 }"}, {"rel": "被描述", "source": "x = \\frac { 9 + 3 k } { 5 }", "target": "\\frac { 9 + 3 k } { 5 } \\ge 0"}, {"rel": "不等式方程求解", "source": "\\frac { 9 + 3 k } { 5 } \\ge 0", "target": "k \\ge - 3"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $3 k - 5 x = - 9$ 的解是非负数", "target": "\\frac { 9 + 3 k } { 5 } \\ge 0"}]}}
{"content": "When $x$ = ____ ?, $5 - | 2 x - 3 |$ has the maximum value.", "answer": "\\frac { 3 } { 2 }", "steps": "If we want to find the maximum value of $5 - | 2 x - 3 |$, then we need to find the minimum value of $| 2 x - 3 |$, which is 0 when $2 x - 3 = 0$. Solving for $x$, we get $x = \\frac { 3 } { 2 }$.", "expr_cands": ["x", "5 - | 2 x - 3 |", "| 2 x - 3 |", "0", "2 x - 3 = 0", "x = \\frac { 3 } { 2 }"], "exprs": ["2 x - 3 = 0", "x = \\frac { 3 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5 - | 2 x - 3 |"}, {"id": "2 x - 3 = 0"}, {"id": "| 2 x - 3 |"}, {"id": "要 $5 - | 2 x - 3 |$ 取得最大值"}, {"id": "$| 2 x - 3 |$ 需取得最小值"}, {"id": "绝对值恒大于等于0"}, {"id": "x = \\frac { 3 } { 2 }"}], "links": [{"rel": "被描述", "source": "5 - | 2 x - 3 |", "target": "2 x - 3 = 0"}, {"rel": "等式方程求解", "source": "2 x - 3 = 0", "target": "x = \\frac { 3 } { 2 }"}, {"rel": "被描述", "source": "| 2 x - 3 |", "target": "2 x - 3 = 0"}, {"rel": "限制性描述", "source": "要 $5 - | 2 x - 3 |$ 取得最大值", "target": "2 x - 3 = 0"}, {"rel": "限制性描述", "source": "$| 2 x - 3 |$ 需取得最小值", "target": "2 x - 3 = 0"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "2 x - 3 = 0"}]}}
{"content": "$8$, if the equation $m ^ 2 - 5 m = 2$ has roots $m _ 1$ and $m _ 2$, then the value of $m _ 1 + m _ 2 + m _ 1 m _ 2$ is:", "answer": "3", "steps": "The original equation can be transformed into: $- 5 m - 2 = 0$. According to the problem, we have $m _ 1 + m _ 2 = 5$ and $m _ 1 m _ 2 = - 2$. Therefore, $m _ 1 + m _ 2 + m _ 1 m _ 2 = 5 + ( - 2 ) = 3$.", "expr_cands": ["8", "m ^ { 2 } - 5 m = 2", "m", "m _ { 1 }", "m _ { 2 }", "m _ { 1 } + m _ { 2 } + m _ { 1 } m _ { 2 }", "- 5 m - 2 = 0", "m = - \\frac { 2 } { 5 }", "m _ { 1 } + m _ { 2 } = 5", "m _ { 1 } m _ { 2 } = - 2", "3"], "exprs": ["m _ { 1 } + m _ { 2 } = 5", "m _ { 1 } m _ { 2 } = - 2", "3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "m ^ { 2 } - 5 m = 2"}, {"id": "m _ { 1 } + m _ { 2 } = 5"}, {"id": "m _ { 1 }"}, {"id": "m _ { 2 }"}, {"id": "方程 $m ^ { 2 } - 5 m = 2$ 两根分别是 $m _ { 1 }$ , $m _ { 2 }$"}, {"id": "一元二次方程根与系数关系，两根之和"}, {"id": "m _ { 1 } m _ { 2 } = - 2"}, {"id": "一元二次方程根与系数关系，两根之积"}, {"id": "m _ { 1 } + m _ { 2 } + m _ { 1 } m _ { 2 }"}, {"id": "3"}], "links": [{"rel": "被描述", "source": "m ^ { 2 } - 5 m = 2", "target": "m _ { 1 } + m _ { 2 } = 5"}, {"rel": "被描述", "source": "m ^ { 2 } - 5 m = 2", "target": "m _ { 1 } m _ { 2 } = - 2"}, {"rel": "代入", "source": "m _ { 1 } + m _ { 2 } = 5", "target": "3"}, {"rel": "被描述", "source": "m _ { 1 }", "target": "m _ { 1 } + m _ { 2 } = 5"}, {"rel": "被描述", "source": "m _ { 1 }", "target": "m _ { 1 } m _ { 2 } = - 2"}, {"rel": "被描述", "source": "m _ { 2 }", "target": "m _ { 1 } + m _ { 2 } = 5"}, {"rel": "被描述", "source": "m _ { 2 }", "target": "m _ { 1 } m _ { 2 } = - 2"}, {"rel": "限制性描述", "source": "方程 $m ^ { 2 } - 5 m = 2$ 两根分别是 $m _ { 1 }$ , $m _ { 2 }$", "target": "m _ { 1 } + m _ { 2 } = 5"}, {"rel": "限制性描述", "source": "方程 $m ^ { 2 } - 5 m = 2$ 两根分别是 $m _ { 1 }$ , $m _ { 2 }$", "target": "m _ { 1 } m _ { 2 } = - 2"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "m _ { 1 } + m _ { 2 } = 5"}, {"rel": "代入", "source": "m _ { 1 } m _ { 2 } = - 2", "target": "3"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之积", "target": "m _ { 1 } m _ { 2 } = - 2"}, {"rel": "被代入", "source": "m _ { 1 } + m _ { 2 } + m _ { 1 } m _ { 2 }", "target": "3"}]}}
{"content": "Given $a$ and $b$ are the square roots of two positive numbers, what is the value of $\\sqrt { 2019 a + 2019 b }$?", "answer": "0", "steps": "$\\because a$ and $b$ are the square roots of a positive number, $\\therefore a$ and $b$ are opposite to each other, $\\therefore a + b = 0$, $\\therefore$ the original expression $= \\sqrt { 2019 ( a + b )} = \\sqrt { 2019 * 0 } = \\sqrt { 0 } = 0$.", "expr_cands": ["a", "b", "\\sqrt { 2019 a + 2019 b }", "a + b = 0", "\\sqrt { 2019 ( a + b ) }", "0"], "exprs": ["a + b = 0", "\\sqrt { 2019 ( a + b ) }", "0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a + b = 0"}, {"id": "b"}, {"id": "$a$ , $b$ 是一个正数的两个平方根"}, {"id": "平方根互为相反数"}, {"id": "\\sqrt { 2019 a + 2019 b }"}, {"id": "\\sqrt { 2019 ( a + b ) }"}, {"id": "0"}], "links": [{"rel": "被描述", "source": "a", "target": "a + b = 0"}, {"rel": "提取因式参考", "source": "a + b = 0", "target": "\\sqrt { 2019 ( a + b ) }"}, {"rel": "代入", "source": "a + b = 0", "target": "0"}, {"rel": "被描述", "source": "b", "target": "a + b = 0"}, {"rel": "限制性描述", "source": "$a$ , $b$ 是一个正数的两个平方根", "target": "a + b = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "a + b = 0"}, {"rel": "提取因式", "source": "\\sqrt { 2019 a + 2019 b }", "target": "\\sqrt { 2019 ( a + b ) }"}, {"rel": "被代入", "source": "\\sqrt { 2019 ( a + b ) }", "target": "0"}]}}
{"content": "If $\\frac { x } { x - 3 } = 2 + \\frac { 3 } { x - 3 }$ has a positive root, then the root is ____?", "answer": "x = 3", "steps": "$\\because$ The equation has a repeated root, $\\therefore$ the simplest common denominator of the equation is $x - 3 = 0$, which means the repeated root is $x = 3$.", "expr_cands": ["\\frac { x } { x - 3 } = 2 + \\frac { 3 } { x - 3 }", "x", "x - 3 = 0", "x = 3"], "exprs": ["x - 3 = 0", "x = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x } { x - 3 } = 2 + \\frac { 3 } { x - 3 }"}, {"id": "x - 3 = 0"}, {"id": "$\\frac { x } { x - 3 } = 2 + \\frac { 3 } { x - 3 }$ 有增根"}, {"id": "分式有增根，则分母为0"}, {"id": "x = 3"}], "links": [{"rel": "被描述", "source": "\\frac { x } { x - 3 } = 2 + \\frac { 3 } { x - 3 }", "target": "x - 3 = 0"}, {"rel": "等式方程求解", "source": "x - 3 = 0", "target": "x = 3"}, {"rel": "限制性描述", "source": "$\\frac { x } { x - 3 } = 2 + \\frac { 3 } { x - 3 }$ 有增根", "target": "x - 3 = 0"}, {"rel": "属性描述", "source": "分式有增根，则分母为0", "target": "x - 3 = 0"}]}}
{"content": "If $- 3 x ^ { m + 1 } y ^ { 2017 }$ and $2 x ^ { 2015 } y ^ { n }$ are similar terms, then the value of $| m - n |$ is ____?", "answer": "3", "steps": "According to the problem, we have $m + 1 = 2015$ and $n = 2017$. Solving for $m$ and $n$, we get $m = 2014$ and $n = 2017$. Therefore, $| m - n | = | 2014 - 2017 | = 3$.", "expr_cands": ["- 3 x ^ { m + 1 } y ^ { 2017 }", "x", "m", "y", "2 x ^ { 2015 } y ^ { n }", "n", "| m - n |", "m + 1 = 2015", "m = 2014", "n = 2017", "3"], "exprs": ["m + 1 = 2015", "n = 2017", "m = 2014", "3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 3 x ^ { m + 1 } y ^ { 2017 }"}, {"id": "m + 1 = 2015"}, {"id": "2 x ^ { 2015 } y ^ { n }"}, {"id": "$- 3 x ^ { m + 1 } y ^ { 2017 }$ 与 $2 x ^ { 2015 } y ^ { n }$ 是同类项"}, {"id": "m = 2014"}, {"id": "n = 2017"}, {"id": "| m - n |"}, {"id": "3"}], "links": [{"rel": "被描述", "source": "- 3 x ^ { m + 1 } y ^ { 2017 }", "target": "m + 1 = 2015"}, {"rel": "被描述", "source": "- 3 x ^ { m + 1 } y ^ { 2017 }", "target": "n = 2017"}, {"rel": "等式方程求解", "source": "m + 1 = 2015", "target": "m = 2014"}, {"rel": "被描述", "source": "2 x ^ { 2015 } y ^ { n }", "target": "m + 1 = 2015"}, {"rel": "被描述", "source": "2 x ^ { 2015 } y ^ { n }", "target": "n = 2017"}, {"rel": "限制性描述", "source": "$- 3 x ^ { m + 1 } y ^ { 2017 }$ 与 $2 x ^ { 2015 } y ^ { n }$ 是同类项", "target": "m + 1 = 2015"}, {"rel": "限制性描述", "source": "$- 3 x ^ { m + 1 } y ^ { 2017 }$ 与 $2 x ^ { 2015 } y ^ { n }$ 是同类项", "target": "n = 2017"}, {"rel": "代入", "source": "m = 2014", "target": "3"}, {"rel": "代入", "source": "n = 2017", "target": "3"}, {"rel": "被代入", "source": "| m - n |", "target": "3"}]}}
{"content": "Given the equation $x + y = 2$, if $x < 0$, then the range of values for $y$ is ____?", "answer": "y > 2", "steps": "$x + y = 2$, therefore $x = 2 - y$. Because $x < 0$, therefore $2 - y < 0$, $- y < - 2$, $y > 2$.", "expr_cands": ["x + y = 2", "y", "x", "x < 0", "x = 2 - y", "2 - y < 0", "2 < y", "- y < - 2", "y > 2"], "exprs": ["x = 2 - y", "2 - y < 0", "y > 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x + y = 2"}, {"id": "x = 2 - y"}, {"id": "x < 0"}, {"id": "2 - y < 0"}, {"id": "y > 2"}], "links": [{"rel": "移项", "source": "x + y = 2", "target": "x = 2 - y"}, {"rel": "联立", "source": "x = 2 - y", "target": "2 - y < 0"}, {"rel": "联立", "source": "x < 0", "target": "2 - y < 0"}, {"rel": "不等式方程求解", "source": "2 - y < 0", "target": "y > 2"}]}}
{"content": "If $m + n = 2$, $mn = 4$, then the value of $\\frac { 1 } { m } + \\frac { 1 } { n }$ is ____?", "answer": "\\frac { 1 } { 2 }", "steps": "Because $m + n = 2$ and $mn = 4$, therefore $\\frac { 1 } { m } + \\frac { 1 } { n } = \\frac { m + n } { mn } = \\frac { 2 } { 4 } = \\frac { 1 } { 2 }$.", "expr_cands": ["m + n = 2", "n", "m", "mn = 4", "\\frac { 1 } { m } + \\frac { 1 } { n }", "\\frac { m + n } { mn }", "\\frac { 1 } { 2 }"], "exprs": ["\\frac { m + n } { mn }", "\\frac { 1 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 1 } { m } + \\frac { 1 } { n }"}, {"id": "\\frac { m + n } { mn }"}, {"id": "\\frac { 1 } { 2 }"}, {"id": "m + n = 2"}, {"id": "mn = 4"}], "links": [{"rel": "计算", "source": "\\frac { 1 } { m } + \\frac { 1 } { n }", "target": "\\frac { m + n } { mn }"}, {"rel": "被代入", "source": "\\frac { m + n } { mn }", "target": "\\frac { 1 } { 2 }"}, {"rel": "代入", "source": "m + n = 2", "target": "\\frac { 1 } { 2 }"}, {"rel": "代入", "source": "mn = 4", "target": "\\frac { 1 } { 2 }"}]}}
{"content": "The equation about $x$, $3 x ^ 2 - 2 ( 3 m - 1 ) x + 2 m = 15$, has a root of $- 2$. What is the value of $m$?", "answer": "\\frac { 1 } { 2 }", "steps": "Substituting $x = - 2$ into the equation $3 x ^ 2 - 2 ( 3 m - 1 ) x + 2 m = 15$ yields $3 * 4 - 2 ( 3 m - 1 ) * ( - 2 ) + 2 m = 15$. Solving for $m$, we get $m = \\frac { 1 } { 2 }$.", "expr_cands": ["x", "3 x ^ { 2 } - 2 ( 3 m - 1 ) x + 2 m = 15", "m", "- 2", "x = - 2", "14 m + 8 = 15", "3 * 4 - 2 ( 3 m - 1 ) * ( - 2 ) + 2 m = 15", "m = \\frac { 1 } { 2 }"], "exprs": ["x = - 2", "3 * 4 - 2 ( 3 m - 1 ) * ( - 2 ) + 2 m = 15", "m = \\frac { 1 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 2"}, {"id": "x = - 2"}, {"id": "x"}, {"id": "3 x ^ { 2 } - 2 ( 3 m - 1 ) x + 2 m = 15"}, {"id": "关于 $x$ 的方程 $3 x ^ { 2 } - 2 ( 3 m - 1 ) x + 2 m = 15$ 有一个根为 $- 2$"}, {"id": "3 * 4 - 2 ( 3 m - 1 ) * ( - 2 ) + 2 m = 15"}, {"id": "m = \\frac { 1 } { 2 }"}], "links": [{"rel": "被描述", "source": "- 2", "target": "x = - 2"}, {"rel": "代入", "source": "x = - 2", "target": "3 * 4 - 2 ( 3 m - 1 ) * ( - 2 ) + 2 m = 15"}, {"rel": "被描述", "source": "x", "target": "x = - 2"}, {"rel": "被描述", "source": "3 x ^ { 2 } - 2 ( 3 m - 1 ) x + 2 m = 15", "target": "x = - 2"}, {"rel": "被代入", "source": "3 x ^ { 2 } - 2 ( 3 m - 1 ) x + 2 m = 15", "target": "3 * 4 - 2 ( 3 m - 1 ) * ( - 2 ) + 2 m = 15"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $3 x ^ { 2 } - 2 ( 3 m - 1 ) x + 2 m = 15$ 有一个根为 $- 2$", "target": "x = - 2"}, {"rel": "等式方程求解", "source": "3 * 4 - 2 ( 3 m - 1 ) * ( - 2 ) + 2 m = 15", "target": "m = \\frac { 1 } { 2 }"}]}}
{"content": "If $a$ and $b$ are rational numbers and $\\sqrt { 3 } + \\sqrt { 12 } + \\sqrt { \\frac { 1 } { 9 }} = a + b \\sqrt { 3 }$, then $ab$ = ____?", "answer": "1", "steps": "Since $a$ and $b$ are rational numbers and $\\sqrt { 3 } + \\sqrt { 12 } + \\sqrt { \\frac { 1 } { 9 }} = a + b \\sqrt { 3 }$, we have $\\sqrt { 3 } + 2 \\sqrt { 3 } + \\frac { 1 } { 3 } = a + b \\sqrt { 3 }$. Therefore, $3 \\sqrt { 3 } + \\frac { 1 } { 3 } = a + b \\sqrt { 3 }$. Hence, $b = 3$ and $a = \\frac { 1 } { 3 }$. Thus, $ab = 1$.", "expr_cands": ["a", "b", "\\sqrt { 3 } + \\sqrt { 12 } + \\sqrt { \\frac { 1 } { 9 } } = a + b \\sqrt { 3 }", "ab", "\\sqrt { 3 } + 2 \\sqrt { 3 } + \\frac { 1 } { 3 } = a + b \\sqrt { 3 }", "3 \\sqrt { 3 } + \\frac { 1 } { 3 } = a + b \\sqrt { 3 }", "b = 3", "a = \\frac { 1 } { 3 }", "1"], "exprs": ["3 \\sqrt { 3 } + \\frac { 1 } { 3 } = a + b \\sqrt { 3 }", "b = 3", "a = \\frac { 1 } { 3 }", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { 3 } + \\sqrt { 12 } + \\sqrt { \\frac { 1 } { 9 } } = a + b \\sqrt { 3 }"}, {"id": "3 \\sqrt { 3 } + \\frac { 1 } { 3 } = a + b \\sqrt { 3 }"}, {"id": "b = 3"}, {"id": "$a$ , $b$ 为有理数"}, {"id": "且 $\\sqrt { 3 } + \\sqrt { 12 } + \\sqrt { \\frac { 1 } { 9 } } = a + b \\sqrt { 3 }$"}, {"id": "a = \\frac { 1 } { 3 }"}, {"id": "ab"}, {"id": "1"}], "links": [{"rel": "计算", "source": "\\sqrt { 3 } + \\sqrt { 12 } + \\sqrt { \\frac { 1 } { 9 } } = a + b \\sqrt { 3 }", "target": "3 \\sqrt { 3 } + \\frac { 1 } { 3 } = a + b \\sqrt { 3 }"}, {"rel": "被描述", "source": "3 \\sqrt { 3 } + \\frac { 1 } { 3 } = a + b \\sqrt { 3 }", "target": "b = 3"}, {"rel": "被描述", "source": "3 \\sqrt { 3 } + \\frac { 1 } { 3 } = a + b \\sqrt { 3 }", "target": "a = \\frac { 1 } { 3 }"}, {"rel": "代入", "source": "b = 3", "target": "1"}, {"rel": "限制性描述", "source": "$a$ , $b$ 为有理数", "target": "b = 3"}, {"rel": "限制性描述", "source": "$a$ , $b$ 为有理数", "target": "a = \\frac { 1 } { 3 }"}, {"rel": "限制性描述", "source": "且 $\\sqrt { 3 } + \\sqrt { 12 } + \\sqrt { \\frac { 1 } { 9 } } = a + b \\sqrt { 3 }$", "target": "b = 3"}, {"rel": "限制性描述", "source": "且 $\\sqrt { 3 } + \\sqrt { 12 } + \\sqrt { \\frac { 1 } { 9 } } = a + b \\sqrt { 3 }$", "target": "a = \\frac { 1 } { 3 }"}, {"rel": "代入", "source": "a = \\frac { 1 } { 3 }", "target": "1"}, {"rel": "被代入", "source": "ab", "target": "1"}]}}
{"content": "When $| 2 x + y | + 1$ takes its minimum value, the value of the algebraic expression $4 x + 2 y + 3$ is ____?", "answer": "3", "steps": "$\\because$ When $| 2 x + y | + 1$ takes the minimum value, $\\therefore$ $2 x + y = 0$, then $4 x + 2 y + 3 = 2 ( 2 x + y ) + 3 = 2 * 0 + 3 = 3$.", "expr_cands": ["| 2 x + y | + 1", "y", "x", "4 x + 2 y + 3", "2 x + y = 0", "2 ( 2 x + y ) + 3", "3"], "exprs": ["2 x + y = 0", "2 ( 2 x + y ) + 3", "3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| 2 x + y | + 1"}, {"id": "2 x + y = 0"}, {"id": "当 $| 2 x + y | + 1$ 取最小值时"}, {"id": "绝对值恒大于等于0"}, {"id": "4 x + 2 y + 3"}, {"id": "2 ( 2 x + y ) + 3"}, {"id": "3"}], "links": [{"rel": "被描述", "source": "| 2 x + y | + 1", "target": "2 x + y = 0"}, {"rel": "提取因式参考", "source": "2 x + y = 0", "target": "2 ( 2 x + y ) + 3"}, {"rel": "代入", "source": "2 x + y = 0", "target": "3"}, {"rel": "限制性描述", "source": "当 $| 2 x + y | + 1$ 取最小值时", "target": "2 x + y = 0"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "2 x + y = 0"}, {"rel": "提取因式", "source": "4 x + 2 y + 3", "target": "2 ( 2 x + y ) + 3"}, {"rel": "被代入", "source": "2 ( 2 x + y ) + 3", "target": "3"}]}}
{"content": "If $\\sqrt { 3 x - 1 }$ is meaningful in the real number range, then the range of values ​​for $x$ is ____?", "answer": "x \\ge \\frac { 1 } { 3 }", "steps": "From the given condition, we have $3 x - 1 \\geq 0$, which implies $x \\geq \\frac { 1 } { 3 }$.", "expr_cands": ["\\sqrt { 3 x - 1 }", "x", "3 x - 1 \\ge 0", "\\frac { 1 } { 3 } \\le x", "x \\ge \\frac { 1 } { 3 }"], "exprs": ["3 x - 1 \\ge 0", "x \\ge \\frac { 1 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { 3 x - 1 }"}, {"id": "3 x - 1 \\ge 0"}, {"id": "$\\sqrt { 3 x - 1 }$ 在实数范围内有意义"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x \\ge \\frac { 1 } { 3 }"}], "links": [{"rel": "被描述", "source": "\\sqrt { 3 x - 1 }", "target": "3 x - 1 \\ge 0"}, {"rel": "不等式方程求解", "source": "3 x - 1 \\ge 0", "target": "x \\ge \\frac { 1 } { 3 }"}, {"rel": "限制性描述", "source": "$\\sqrt { 3 x - 1 }$ 在实数范围内有意义", "target": "3 x - 1 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "3 x - 1 \\ge 0"}]}}
{"content": "If $a \\div b = \\frac { 2 } { 3 }$, what is the ratio of $a$ to $b$?", "answer": "2 : 3", "steps": "Since $a \\div b = \\frac { 2 } { 3 }$, it follows that the ratio of $a$ to $b$ is $2 : 3$.", "expr_cands": ["a \\div b = \\frac { 2 } { 3 }", "b", "a", "a : b", "\\frac { 2 } { 3 }"], "exprs": ["\\frac { 2 } { 3 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a \\div b = \\frac { 2 } { 3 }"}, {"id": "\\frac { 2 } { 3 }"}, {"id": "$a$ 与 $b$ 的比"}], "links": [{"rel": "被描述", "source": "a \\div b = \\frac { 2 } { 3 }", "target": "\\frac { 2 } { 3 }"}, {"rel": "限制性描述", "source": "$a$ 与 $b$ 的比", "target": "\\frac { 2 } { 3 }"}]}}
{"content": "The common factor of the polynomial $4 ( x - y ) ^ { 3 } - 6 ( y - x ) ^ { 2 }$ is ____ ?", "answer": "2 ( x - y ) ^ { 2 }", "steps": "$4 ( x - y ) ^ { 3 } - 6 ( y - x ) ^ { 2 } = 2 ( x - y ) ^ { 2 } ( 2 x - 2 y - 3 )$ Therefore, the common factor of the polynomial $4 ( x - y ) ^ { 3 } - 6 ( y - x ) ^ { 2 }$ is $2 ( x - y ) ^ { 2 }$.", "expr_cands": ["4 ( x - y ) ^ { 3 } - 6 ( y - x ) ^ { 2 }", "y", "x", "2 ( x - y ) ^ { 2 } ( 2 x - 2 y - 3 )", "2 ( x - y ) ^ { 2 }"], "exprs": ["2 ( x - y ) ^ { 2 } ( 2 x - 2 y - 3 )", "2 ( x - y ) ^ { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 ( x - y ) ^ { 3 } - 6 ( y - x ) ^ { 2 }"}, {"id": "2 ( x - y ) ^ { 2 } ( 2 x - 2 y - 3 )"}, {"id": "2 ( x - y ) ^ { 2 }"}, {"id": "多项式 $4 ( x - y ) ^ { 3 } - 6 ( y - x ) ^ { 2 }$ 的公因式是 $2 ( x - y ) ^ { 2 }$"}], "links": [{"rel": "提取因式", "source": "4 ( x - y ) ^ { 3 } - 6 ( y - x ) ^ { 2 }", "target": "2 ( x - y ) ^ { 2 } ( 2 x - 2 y - 3 )"}, {"rel": "被描述", "source": "2 ( x - y ) ^ { 2 } ( 2 x - 2 y - 3 )", "target": "2 ( x - y ) ^ { 2 }"}, {"rel": "限制性描述", "source": "多项式 $4 ( x - y ) ^ { 3 } - 6 ( y - x ) ^ { 2 }$ 的公因式是 $2 ( x - y ) ^ { 2 }$", "target": "2 ( x - y ) ^ { 2 }"}]}}
{"content": "If the value of the fraction $\\frac { 4 } { 3 x + 4 }$ is $1$, then the value of $x$ should be ____?", "answer": "0", "steps": "According to the problem, we can write the fractional equation $\\frac { 4 } { 3 x + 4 } = 1$. By eliminating the denominator, we get $4 = 3 x + 4$. Solving the equation, we get $x = 0$. After checking, we find that $x = 0$ is a solution to the equation.", "expr_cands": ["\\frac { 4 } { 3 x + 4 }", "x", "1", "\\frac { 4 } { 3 x + 4 } = 1", "x = 0", "4 = 3 x + 4"], "exprs": ["\\frac { 4 } { 3 x + 4 } = 1", "x = 0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 4 } { 3 x + 4 }"}, {"id": "\\frac { 4 } { 3 x + 4 } = 1"}, {"id": "1"}, {"id": "分式 $\\frac { 4 } { 3 x + 4 }$ 的值为 $1$"}, {"id": "分式有意义，则分母不为0"}, {"id": "x = 0"}], "links": [{"rel": "被描述", "source": "\\frac { 4 } { 3 x + 4 }", "target": "\\frac { 4 } { 3 x + 4 } = 1"}, {"rel": "等式方程求解", "source": "\\frac { 4 } { 3 x + 4 } = 1", "target": "x = 0"}, {"rel": "被描述", "source": "1", "target": "\\frac { 4 } { 3 x + 4 } = 1"}, {"rel": "限制性描述", "source": "分式 $\\frac { 4 } { 3 x + 4 }$ 的值为 $1$", "target": "\\frac { 4 } { 3 x + 4 } = 1"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "\\frac { 4 } { 3 x + 4 } = 1"}]}}
{"content": "The negative integer $x$ that satisfies $4 x - 2 < 5 x$ is _____.", "answer": "- 1", "steps": "$\\because$ $4 x - 2 < 5 x$, $\\therefore$ $x > - 2$, thus the negative integer that satisfies this inequality is $- 1$.", "expr_cands": ["4 x - 2 < 5 x", "x", "- 2 < x", "x > - 2", "- 1"], "exprs": ["x > - 2", "- 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 x - 2 < 5 x"}, {"id": "x > - 2"}, {"id": "- 1"}, {"id": "满足 $4 x - 2 < 5 x$ 的负整数 $x$"}, {"id": "满足该不等式的负整数为 $- 1$"}], "links": [{"rel": "不等式方程求解", "source": "4 x - 2 < 5 x", "target": "x > - 2"}, {"rel": "被描述", "source": "x > - 2", "target": "- 1"}, {"rel": "限制性描述", "source": "满足 $4 x - 2 < 5 x$ 的负整数 $x$", "target": "- 1"}, {"rel": "限制性描述", "source": "满足该不等式的负整数为 $- 1$", "target": "- 1"}]}}
{"content": "The solution to the inequality $2 x - 3 > 1$ is ____ ?", "answer": "x > 2", "steps": "Moving terms, we get: $2 x > 1 + 3$. Combining like terms, we get: $2 x > 4$. Dividing by the coefficient, we get: $x > 2$.", "expr_cands": ["2 x - 3 > 1", "x", "2 x > 1 + 3", "2 < x", "2 x > 4", "1", "x > 2"], "exprs": ["x > 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x - 3 > 1"}, {"id": "x > 2"}], "links": [{"rel": "不等式方程求解", "source": "2 x - 3 > 1", "target": "x > 2"}]}}
{"content": "Given that the equation $x ^ 2 + kx + 2 = 0$ has two roots $x _ 1$ and $x _ 2$, and $\\frac { 1 } { x _ 1 } + \\frac { 1 } { x _ 2 } + x _ 1 x _ 2 = 0$, what is the value of $k$?", "answer": "4", "steps": "From the given information, we know that $x _ 1 + x _ 2 = - k$ and $x _ 1 \\times x _ 2 = 2$. Then, using $\\frac { 1 } { x _ 1 } + \\frac { 1 } { x _ 2 } + x _ 1 x _ 2 = 0$, we get $\\frac { x _ 2 + x _ 1 } { x _ 1 \\cdot x _ 2 } + x _ 1 x _ 2 = \\frac { - k } { 2 } + 2 = 0$, which implies $\\frac { - k } { 2 } + 2 = 0$. Solving for $k$, we get $k = 4$.", "expr_cands": ["x", "x ^ { 2 } + kx + 2 = 0", "k", "x _ { 1 }", "x _ { 2 }", "\\frac { 1 } { x _ { 1 } } + \\frac { 1 } { x _ { 2 } } + x _ { 1 } x _ { 2 } = 0", "x _ { 1 } + x _ { 2 } = - k", "x _ { 1 } \\times x _ { 2 } = 2", "2 + \\frac { 1 } { x _ { 2 }} + \\frac { 1 } { x _ { 1 }} = 0", "\\frac { x _ { 2 } + x _ { 1 } } { x _ { 1 } \\cdot x _ { 2 } } + x _ { 1 } x _ { 2 } = 0", "\\frac { - k } { 2 } + 2 = 0", "k = 4"], "exprs": ["x _ { 1 } + x _ { 2 } = - k", "x _ { 1 } \\times x _ { 2 } = 2", "\\frac { x _ { 2 } + x _ { 1 } } { x _ { 1 } \\cdot x _ { 2 } } + x _ { 1 } x _ { 2 } = 0", "\\frac { - k } { 2 } + 2 = 0", "k = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } + kx + 2 = 0"}, {"id": "x _ { 1 } + x _ { 2 } = - k"}, {"id": "x _ { 1 }"}, {"id": "x _ { 2 }"}, {"id": "关于 $x$ 的方程 $x ^ { 2 } + kx + 2 = 0$ 的两个根为 $x _ { 1 }$ , $x _ { 2 }$"}, {"id": "一元二次方程根与系数关系，两根之和"}, {"id": "x _ { 1 } \\times x _ { 2 } = 2"}, {"id": "一元二次方程根与系数关系，两根之积"}, {"id": "\\frac { 1 } { x _ { 1 } } + \\frac { 1 } { x _ { 2 } } + x _ { 1 } x _ { 2 } = 0"}, {"id": "\\frac { x _ { 2 } + x _ { 1 } } { x _ { 1 } \\cdot x _ { 2 } } + x _ { 1 } x _ { 2 } = 0"}, {"id": "\\frac { - k } { 2 } + 2 = 0"}, {"id": "k = 4"}], "links": [{"rel": "被描述", "source": "x ^ { 2 } + kx + 2 = 0", "target": "x _ { 1 } + x _ { 2 } = - k"}, {"rel": "被描述", "source": "x ^ { 2 } + kx + 2 = 0", "target": "x _ { 1 } \\times x _ { 2 } = 2"}, {"rel": "联立", "source": "x _ { 1 } + x _ { 2 } = - k", "target": "\\frac { - k } { 2 } + 2 = 0"}, {"rel": "被描述", "source": "x _ { 1 }", "target": "x _ { 1 } + x _ { 2 } = - k"}, {"rel": "被描述", "source": "x _ { 1 }", "target": "x _ { 1 } \\times x _ { 2 } = 2"}, {"rel": "被描述", "source": "x _ { 2 }", "target": "x _ { 1 } + x _ { 2 } = - k"}, {"rel": "被描述", "source": "x _ { 2 }", "target": "x _ { 1 } \\times x _ { 2 } = 2"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $x ^ { 2 } + kx + 2 = 0$ 的两个根为 $x _ { 1 }$ , $x _ { 2 }$", "target": "x _ { 1 } + x _ { 2 } = - k"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $x ^ { 2 } + kx + 2 = 0$ 的两个根为 $x _ { 1 }$ , $x _ { 2 }$", "target": "x _ { 1 } \\times x _ { 2 } = 2"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "x _ { 1 } + x _ { 2 } = - k"}, {"rel": "联立", "source": "x _ { 1 } \\times x _ { 2 } = 2", "target": "\\frac { - k } { 2 } + 2 = 0"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之积", "target": "x _ { 1 } \\times x _ { 2 } = 2"}, {"rel": "计算", "source": "\\frac { 1 } { x _ { 1 } } + \\frac { 1 } { x _ { 2 } } + x _ { 1 } x _ { 2 } = 0", "target": "\\frac { x _ { 2 } + x _ { 1 } } { x _ { 1 } \\cdot x _ { 2 } } + x _ { 1 } x _ { 2 } = 0"}, {"rel": "联立", "source": "\\frac { x _ { 2 } + x _ { 1 } } { x _ { 1 } \\cdot x _ { 2 } } + x _ { 1 } x _ { 2 } = 0", "target": "\\frac { - k } { 2 } + 2 = 0"}, {"rel": "等式方程求解", "source": "\\frac { - k } { 2 } + 2 = 0", "target": "k = 4"}]}}
{"content": "Given that $x = 3$ is a solution to the fractional equation $\\frac { 3 } { 2 x } = \\frac { m } { x - 1 }$, then ____?", "answer": "m = 1", "steps": "Substituting $x = 3$ into the fractional equation, we get $\\frac { 3 } { 2 \\times 3 } = \\frac { m } { 3 - 1 }$, so $m = 1$.", "expr_cands": ["x = 3", "x", "\\frac { 3 } { 2 x } = \\frac { m } { x - 1 }", "m", "\\frac { 3 } { 2 * 3 } = \\frac { m } { 3 - 1 }", "m = 1"], "exprs": ["\\frac { 3 } { 2 * 3 } = \\frac { m } { 3 - 1 }", "m = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x = 3"}, {"id": "\\frac { 3 } { 2 * 3 } = \\frac { m } { 3 - 1 }"}, {"id": "\\frac { 3 } { 2 x } = \\frac { m } { x - 1 }"}, {"id": "m = 1"}], "links": [{"rel": "代入", "source": "x = 3", "target": "\\frac { 3 } { 2 * 3 } = \\frac { m } { 3 - 1 }"}, {"rel": "等式方程求解", "source": "\\frac { 3 } { 2 * 3 } = \\frac { m } { 3 - 1 }", "target": "m = 1"}, {"rel": "被代入", "source": "\\frac { 3 } { 2 x } = \\frac { m } { x - 1 }", "target": "\\frac { 3 } { 2 * 3 } = \\frac { m } { 3 - 1 }"}]}}
{"content": "If the sum of $2 x$ and $y$ is $0$, and $( 2 x + y + 3 ) ( 5 x - y - 7 ) = 6$, then $x$ = ____ ?", "answer": "\\frac { 9 } { 7 }", "steps": "$\\because$ the sum of $2 x$ and $y$ is $0$, $\\therefore$ $2 x + y = 0$, $\\therefore$ $y = - 2 x$. Then the equation can be transformed into $3 ( 7 x - 7 ) = 6$. Expanding the brackets, we get $21 x - 21 = 6$. Moving the constant term to the right-hand side and combining like terms, we get $21 x = 27$. Solving for $x$, we get $x = \\frac { 9 } { 7 }$.", "expr_cands": ["2 x", "x", "y", "0", "( 2 x + y + 3 ) ( 5 x - y - 7 ) = 6", "2 x + y = 0", "y = - 2 x", "3 ( 7 x - 7 ) = 6", "x = \\frac { 9 } { 7 }", "21 x - 21 = 6", "21 x = 27"], "exprs": ["2 x + y = 0", "y = - 2 x", "3 ( 7 x - 7 ) = 6", "x = \\frac { 9 } { 7 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x"}, {"id": "2 x + y = 0"}, {"id": "y"}, {"id": "0"}, {"id": "$2 x$ 与 $y$ 的和为 $0$"}, {"id": "y = - 2 x"}, {"id": "( 2 x + y + 3 ) ( 5 x - y - 7 ) = 6"}, {"id": "3 ( 7 x - 7 ) = 6"}, {"id": "x = \\frac { 9 } { 7 }"}], "links": [{"rel": "被描述", "source": "2 x", "target": "2 x + y = 0"}, {"rel": "等式方程部分求解", "source": "2 x + y = 0", "target": "y = - 2 x"}, {"rel": "被描述", "source": "y", "target": "2 x + y = 0"}, {"rel": "被描述", "source": "0", "target": "2 x + y = 0"}, {"rel": "限制性描述", "source": "$2 x$ 与 $y$ 的和为 $0$", "target": "2 x + y = 0"}, {"rel": "代入", "source": "y = - 2 x", "target": "3 ( 7 x - 7 ) = 6"}, {"rel": "被代入", "source": "( 2 x + y + 3 ) ( 5 x - y - 7 ) = 6", "target": "3 ( 7 x - 7 ) = 6"}, {"rel": "等式方程求解", "source": "3 ( 7 x - 7 ) = 6", "target": "x = \\frac { 9 } { 7 }"}]}}
{"content": "The polynomial $x ^ 2 - 4 x + m$ can be factored as $( x + 3 ) ( x - n )$. What is the value of $\\frac { m } { n }$?", "answer": "- 3", "steps": "According to the problem, we have $x ^ 2 - 4 x + m = ( x + 3 ) ( x - n ) = x ^ 2 + ( 3 - n ) x - 3 n$. Therefore, $3 - n = - 4$ and $m = - 3 n$. Solving for $m$ and $n$, we get $m = - 21$ and $n = 7$. Thus, the original expression is equal to $- 3$.", "expr_cands": ["x ^ { 2 } - 4 x + m", "x", "m", "( x + 3 ) ( x - n )", "n", "\\frac { m } { n }", "x ^ { 2 } - 4 x + m = x ^ { 2 } + ( 3 - n ) x - 3 n", "3 - n = - 4", "n = 7", "m = - 3 n", "m = - 21", "- 3"], "exprs": ["x ^ { 2 } - 4 x + m = x ^ { 2 } + ( 3 - n ) x - 3 n", "3 - n = - 4", "m = - 3 n", "n = 7", "m = - 21", "- 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } - 4 x + m"}, {"id": "x ^ { 2 } - 4 x + m = x ^ { 2 } + ( 3 - n ) x - 3 n"}, {"id": "( x + 3 ) ( x - n )"}, {"id": "多项式 $x ^ { 2 } - 4 x + m$ 分解因式的结果是 $( x + 3 ) ( x - n )$"}, {"id": "3 - n = - 4"}, {"id": "m = - 3 n"}, {"id": "n = 7"}, {"id": "m = - 21"}, {"id": "\\frac { m } { n }"}, {"id": "- 3"}], "links": [{"rel": "被描述", "source": "x ^ { 2 } - 4 x + m", "target": "x ^ { 2 } - 4 x + m = x ^ { 2 } + ( 3 - n ) x - 3 n"}, {"rel": "被描述", "source": "x ^ { 2 } - 4 x + m = x ^ { 2 } + ( 3 - n ) x - 3 n", "target": "3 - n = - 4"}, {"rel": "被描述", "source": "x ^ { 2 } - 4 x + m = x ^ { 2 } + ( 3 - n ) x - 3 n", "target": "m = - 3 n"}, {"rel": "被描述", "source": "( x + 3 ) ( x - n )", "target": "x ^ { 2 } - 4 x + m = x ^ { 2 } + ( 3 - n ) x - 3 n"}, {"rel": "限制性描述", "source": "多项式 $x ^ { 2 } - 4 x + m$ 分解因式的结果是 $( x + 3 ) ( x - n )$", "target": "x ^ { 2 } - 4 x + m = x ^ { 2 } + ( 3 - n ) x - 3 n"}, {"rel": "限制性描述", "source": "多项式 $x ^ { 2 } - 4 x + m$ 分解因式的结果是 $( x + 3 ) ( x - n )$", "target": "3 - n = - 4"}, {"rel": "限制性描述", "source": "多项式 $x ^ { 2 } - 4 x + m$ 分解因式的结果是 $( x + 3 ) ( x - n )$", "target": "m = - 3 n"}, {"rel": "等式方程求解", "source": "3 - n = - 4", "target": "n = 7"}, {"rel": "被代入", "source": "m = - 3 n", "target": "m = - 21"}, {"rel": "代入", "source": "n = 7", "target": "m = - 21"}, {"rel": "代入", "source": "n = 7", "target": "- 3"}, {"rel": "代入", "source": "m = - 21", "target": "- 3"}, {"rel": "被代入", "source": "\\frac { m } { n }", "target": "- 3"}]}}
{"content": "The solution set of the inequality $( { m } ^ { 2 } + 2 ) x > { m } ^ { 2 } + 2$ is ____ ?", "answer": "x > 1", "steps": "$\\because$ $( m ^ 2 + 2 ) x > m ^ 2 + 2$, and $m ^ 2 + 2 \\ge 2$, $\\therefore$ $x > \\frac { m ^ 2 + 2 } { m ^ 2 + 2 }$, which means $x > 1$.", "expr_cands": ["( { m } ^ { 2 } + 2 ) x > { m } ^ { 2 } + 2", "x", "m", "{ m } ^ { 2 } + 2 \\ge 2", "x > \\frac { { m } ^ { 2 } + 2 } { { m } ^ { 2 } + 2 }", "1 < x", "x > 1"], "exprs": ["{ m } ^ { 2 } + 2 \\ge 2", "x > \\frac { { m } ^ { 2 } + 2 } { { m } ^ { 2 } + 2 }", "x > 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( { m } ^ { 2 } + 2 ) x > { m } ^ { 2 } + 2"}, {"id": "{ m } ^ { 2 } + 2 \\ge 2"}, {"id": "多项式偶次方项恒大于等于0"}, {"id": "x > \\frac { { m } ^ { 2 } + 2 } { { m } ^ { 2 } + 2 }"}, {"id": "x > 1"}], "links": [{"rel": "被描述", "source": "( { m } ^ { 2 } + 2 ) x > { m } ^ { 2 } + 2", "target": "{ m } ^ { 2 } + 2 \\ge 2"}, {"rel": "联立", "source": "( { m } ^ { 2 } + 2 ) x > { m } ^ { 2 } + 2", "target": "x > \\frac { { m } ^ { 2 } + 2 } { { m } ^ { 2 } + 2 }"}, {"rel": "联立", "source": "{ m } ^ { 2 } + 2 \\ge 2", "target": "x > \\frac { { m } ^ { 2 } + 2 } { { m } ^ { 2 } + 2 }"}, {"rel": "属性描述", "source": "多项式偶次方项恒大于等于0", "target": "{ m } ^ { 2 } + 2 \\ge 2"}, {"rel": "计算", "source": "x > \\frac { { m } ^ { 2 } + 2 } { { m } ^ { 2 } + 2 }", "target": "x > 1"}]}}
{"content": "Given $3 a - 2 b = 2$, then $9 a - 6 b$ = ____ ?", "answer": "6", "steps": "Because $3 a - 2 b = 2$, therefore $9 a - 6 b = 3 ( 3 a - 2 b ) = 3 * 2 = 6$.", "expr_cands": ["3 a - 2 b = 2", "a", "b", "9 a - 6 b", "3 ( 3 a - 2 b )", "6"], "exprs": ["3 ( 3 a - 2 b )", "6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "9 a - 6 b"}, {"id": "3 ( 3 a - 2 b )"}, {"id": "3 a - 2 b = 2"}, {"id": "6"}], "links": [{"rel": "提取因式", "source": "9 a - 6 b", "target": "3 ( 3 a - 2 b )"}, {"rel": "被代入", "source": "3 ( 3 a - 2 b )", "target": "6"}, {"rel": "提取因式参考", "source": "3 a - 2 b = 2", "target": "3 ( 3 a - 2 b )"}, {"rel": "代入", "source": "3 a - 2 b = 2", "target": "6"}]}}
{"content": "The expression of the new parabola after translating the parabola $y = { x } ^ { 2 } - 2$ one unit upward is ____?", "answer": "y = x ^ { 2 } - 1", "steps": "According to the principle of adding up and subtracting down, it can be known that if the parabola $y = x ^ 2 - 2$ is shifted up by one unit, then the expression of the new parabola is $y = x ^ 2 - 2 + 1$, that is, $y = x ^ 2 - 1$.", "expr_cands": ["y = { x } ^ { 2 } - 2", "y", "x", "y = x ^ { 2 } - 2", "y = x ^ { 2 } - 2 + 1", "x ^ { 2 } - 2 = x ^ { 2 } - 2 + 1", "x ^ { 2 } - 1"], "exprs": ["y = x ^ { 2 } - 2 + 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = { x } ^ { 2 } - 2"}, {"id": "y = x ^ { 2 } - 2 + 1"}, {"id": "将抛物线 $y = { x } ^ { 2 } - 2$ 向上平移一个单位后"}, {"id": "新的抛物线的表达式"}, {"id": "\" 上加下减 \" 的原可"}, {"id": "将抛物线 $y = x ^ { 2 } - 2$ 向上平移一个单位后"}], "links": [{"rel": "被描述", "source": "y = { x } ^ { 2 } - 2", "target": "y = x ^ { 2 } - 2 + 1"}, {"rel": "限制性描述", "source": "将抛物线 $y = { x } ^ { 2 } - 2$ 向上平移一个单位后", "target": "y = x ^ { 2 } - 2 + 1"}, {"rel": "限制性描述", "source": "新的抛物线的表达式", "target": "y = x ^ { 2 } - 2 + 1"}, {"rel": "属性描述", "source": "\" 上加下减 \" 的原可", "target": "y = x ^ { 2 } - 2 + 1"}, {"rel": "限制性描述", "source": "将抛物线 $y = x ^ { 2 } - 2$ 向上平移一个单位后", "target": "y = x ^ { 2 } - 2 + 1"}]}}
{"content": "If the solution set of the inequality $3 x - a \\leq 0$ is $x \\leq 5$, then the value of $a$ is ____?", "answer": "15", "steps": "Solve the inequality $3 x - a \\leq 0$ to get $x \\leq \\frac { a } { 3 }$. Since the solution set of the inequality is $x \\leq 5$, we have $\\frac { a } { 3 } = 5$. Solving for $a$, we get $a = 15$.", "expr_cands": ["3 x - a \\le 0", "x", "a", "x \\le 5", "x \\le \\frac { a } { 3 }", "\\frac { a } { 3 } = 5", "a = 15"], "exprs": ["x \\le \\frac { a } { 3 }", "\\frac { a } { 3 } = 5", "a = 15"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 x - a \\le 0"}, {"id": "x \\le \\frac { a } { 3 }"}, {"id": "\\frac { a } { 3 } = 5"}, {"id": "x \\le 5"}, {"id": "不等式 $3 x - a \\le 0$ 的解集为 $x \\le 5$"}, {"id": "a = 15"}], "links": [{"rel": "不等式方程部分求解", "source": "3 x - a \\le 0", "target": "x \\le \\frac { a } { 3 }"}, {"rel": "被描述", "source": "x \\le \\frac { a } { 3 }", "target": "\\frac { a } { 3 } = 5"}, {"rel": "等式方程求解", "source": "\\frac { a } { 3 } = 5", "target": "a = 15"}, {"rel": "被描述", "source": "x \\le 5", "target": "\\frac { a } { 3 } = 5"}, {"rel": "限制性描述", "source": "不等式 $3 x - a \\le 0$ 的解集为 $x \\le 5$", "target": "\\frac { a } { 3 } = 5"}]}}
{"content": "The analytical expression of the parabola obtained by translating the parabola $y = x ^ 2 + 1$ down by $3$ units is ____ ?", "answer": "y = x ^ { 2 } - 2", "steps": "The parabola $y = x ^ { 2 } + 1$ is translated down $3$ units to become $y = x ^ { 2 } + 1 - 3$, which is equivalent to $y = x ^ { 2 } - 2$.", "expr_cands": ["y = x ^ { 2 } + 1", "y", "x", "3", "y = x ^ { 2 } + 1 - 3", "x ^ { 2 } + 1 = x ^ { 2 } + 1 - 3", "x ^ { 2 } - 2"], "exprs": ["y = x ^ { 2 } + 1 - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = x ^ { 2 } + 1"}, {"id": "y = x ^ { 2 } + 1 - 3"}, {"id": "3"}, {"id": "将抛物线 $y = x ^ { 2 } + 1$ 向下平移 $3$ 个单位长度得到的抛物线的解析式"}], "links": [{"rel": "被描述", "source": "y = x ^ { 2 } + 1", "target": "y = x ^ { 2 } + 1 - 3"}, {"rel": "被描述", "source": "3", "target": "y = x ^ { 2 } + 1 - 3"}, {"rel": "限制性描述", "source": "将抛物线 $y = x ^ { 2 } + 1$ 向下平移 $3$ 个单位长度得到的抛物线的解析式", "target": "y = x ^ { 2 } + 1 - 3"}]}}
{"content": "Line segments $a$, $b$, $c$, and $d$ are in proportion, where $a = 4$, $b = 2$, $c = 2$. What is the length of $d$?", "answer": "1", "steps": "$\\because a$, $b$, $c$, $d$ are proportional line segments, $\\therefore a : b = c : d$, which means $4 : 2 = 2 : d$, $\\therefore d = 1$.", "expr_cands": ["a", "b", "c", "d", "a = 4", "b = 2", "c = 2", "a : b = c : d", "4 : 2 = 2 : d", "d = 1"], "exprs": ["a : b = c : d", "4 : 2 = 2 : d", "d = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a : b = c : d"}, {"id": "b"}, {"id": "c"}, {"id": "d"}, {"id": "线段 $a$ , $b$ , $c$ , $d$ 是成比例线段"}, {"id": "a = 4"}, {"id": "4 : 2 = 2 : d"}, {"id": "b = 2"}, {"id": "c = 2"}, {"id": "d = 1"}], "links": [{"rel": "被描述", "source": "a", "target": "a : b = c : d"}, {"rel": "被代入", "source": "a : b = c : d", "target": "4 : 2 = 2 : d"}, {"rel": "被描述", "source": "b", "target": "a : b = c : d"}, {"rel": "被描述", "source": "c", "target": "a : b = c : d"}, {"rel": "被描述", "source": "d", "target": "a : b = c : d"}, {"rel": "限制性描述", "source": "线段 $a$ , $b$ , $c$ , $d$ 是成比例线段", "target": "a : b = c : d"}, {"rel": "代入", "source": "a = 4", "target": "4 : 2 = 2 : d"}, {"rel": "等式方程求解", "source": "4 : 2 = 2 : d", "target": "d = 1"}, {"rel": "代入", "source": "b = 2", "target": "4 : 2 = 2 : d"}, {"rel": "代入", "source": "c = 2", "target": "4 : 2 = 2 : d"}]}}
{"content": "If $x = - 1$ is a solution of the equation $2 x + a = 0$, then $a$ = ____ ?", "answer": "2", "steps": "Substituting $x = - 1$ into the equation, we get $- 2 + a = 0$, which gives us $a = 2$.", "expr_cands": ["x = - 1", "x", "2 x + a = 0", "a", "- 2 + a = 0", "a = 2"], "exprs": ["- 2 + a = 0", "a = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x = - 1"}, {"id": "- 2 + a = 0"}, {"id": "2 x + a = 0"}, {"id": "a = 2"}], "links": [{"rel": "代入", "source": "x = - 1", "target": "- 2 + a = 0"}, {"rel": "等式方程求解", "source": "- 2 + a = 0", "target": "a = 2"}, {"rel": "被代入", "source": "2 x + a = 0", "target": "- 2 + a = 0"}]}}
{"content": "If the equation $x ^ { 2 } - c = 0$ has one root as $1$, then the other root is ____?", "answer": "- 1", "steps": "Substituting $x = 1$ into the equation gives $1 - c = 0$, which yields $c = 1$. The equation becomes $x ^ 2 - 1 = 0$, which can be rewritten as $x ^ 2 = 1$. Taking the square root gives $x = 1$ or $x = - 1$, so the other root is $- 1$.", "expr_cands": ["x ^ { 2 } - c = 0", "x", "c", "1", "x = 1", "1 - c = 0", "c = 1", "x ^ { 2 } - 1", "0", "x ^ { 2 }", "x = - 1", "- 1"], "exprs": ["x = 1", "1 - c = 0", "c = 1", "- 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "1"}, {"id": "x = 1"}, {"id": "x ^ { 2 } - c = 0"}, {"id": "方程 $x ^ { 2 } - c = 0$ 有一个根是 $1$"}, {"id": "1 - c = 0"}, {"id": "c = 1"}, {"id": "- 1"}, {"id": "另一根"}], "links": [{"rel": "被描述", "source": "1", "target": "x = 1"}, {"rel": "代入", "source": "x = 1", "target": "1 - c = 0"}, {"rel": "被描述", "source": "x = 1", "target": "- 1"}, {"rel": "被描述", "source": "x ^ { 2 } - c = 0", "target": "x = 1"}, {"rel": "被代入", "source": "x ^ { 2 } - c = 0", "target": "1 - c = 0"}, {"rel": "被描述", "source": "x ^ { 2 } - c = 0", "target": "- 1"}, {"rel": "限制性描述", "source": "方程 $x ^ { 2 } - c = 0$ 有一个根是 $1$", "target": "x = 1"}, {"rel": "等式方程求解", "source": "1 - c = 0", "target": "c = 1"}, {"rel": "被描述", "source": "c = 1", "target": "- 1"}, {"rel": "限制性描述", "source": "另一根", "target": "- 1"}]}}
{"content": "Given that $x = 1$ is a solution to the quadratic equation $x ^ 2 - mx - 2 = 0$ in terms of $x$, what is the value of $m$?", "answer": "- 1", "steps": "Substituting $x = 1$ into the equation, we get $1 - m - 2 = 0$, which yields $m = - 1$.", "expr_cands": ["x = 1", "x", "x ^ { 2 } - mx - 2 = 0", "m", "1 - m - 2 = 0", "m = - 1"], "exprs": ["1 - m - 2 = 0", "m = - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x = 1"}, {"id": "1 - m - 2 = 0"}, {"id": "x ^ { 2 } - mx - 2 = 0"}, {"id": "m = - 1"}], "links": [{"rel": "代入", "source": "x = 1", "target": "1 - m - 2 = 0"}, {"rel": "等式方程求解", "source": "1 - m - 2 = 0", "target": "m = - 1"}, {"rel": "被代入", "source": "x ^ { 2 } - mx - 2 = 0", "target": "1 - m - 2 = 0"}]}}
{"content": "The proportional function $y = ( m + 3 ) x$ increases as $x$ increases. What is the range of values for $m$?", "answer": "m > - 3", "steps": "$\\because$ In the proportional function $y = ( m + 3 ) x$, $y$ increases as $x$ increases. $\\therefore$ $m + 3 > 0$, which means $m > - 3$ after solving for $m$.", "expr_cands": ["y = ( m + 3 ) x", "y", "m", "x", "m + 3 > 0", "- 3 < m", "m > - 3"], "exprs": ["m + 3 > 0", "m > - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = ( m + 3 ) x"}, {"id": "m + 3 > 0"}, {"id": "正比例函数 $y = ( m + 3 ) x$ , $y$ 随 $x$ 的增大而增大"}, {"id": "m > - 3"}], "links": [{"rel": "被描述", "source": "y = ( m + 3 ) x", "target": "m + 3 > 0"}, {"rel": "不等式方程求解", "source": "m + 3 > 0", "target": "m > - 3"}, {"rel": "限制性描述", "source": "正比例函数 $y = ( m + 3 ) x$ , $y$ 随 $x$ 的增大而增大", "target": "m + 3 > 0"}]}}
{"content": "Given the algebraic expression $3 x ^ 2 + mx - 2 y + 1 - 6 nx ^ 2 + 3 x$ has a value independent of the value of the variable $x$, then the value of the algebraic expression $\\frac { 1 } { 2 } m ^ 3 - n ^ 2 - \\frac { 1 } { 3 } m ^ 3 + 3 n ^ 2 + 1$ is ____?", "answer": "- 3", "steps": "$\\because$ The value of the algebraic expression $3 x ^ { 2 } + mx - 2 y + 1 - 6 nx ^ { 2 } + 3 x$ is independent of the value of the variable $x$, $\\therefore$ $3 - 6 n = 0$, $m + 3 = 0$, which gives $m = - 3$, $n = \\frac { 1 } { 2 }$. Therefore, the original expression $= \\frac { 1 } { 6 } m ^ { 3 } + 2 n ^ { 2 } + 1 = - \\frac { 9 } { 2 } + \\frac { 1 } { 2 } + 1 = - 3$.", "expr_cands": ["3 x ^ { 2 } + mx - 2 y + 1 - 6 nx ^ { 2 } + 3 x", "y", "x", "m", "n", "\\frac { 1 } { 2 } m ^ { 3 } - n ^ { 2 } - \\frac { 1 } { 3 } m ^ { 3 } + 3 n ^ { 2 } + 1", "3 - 6 n = 0", "n = \\frac { 1 } { 2 }", "m + 3 = 0", "m = - 3", "\\frac { 1 } { 6 } m ^ { 3 } + 2 n ^ { 2 } + 1", "- 3"], "exprs": ["3 - 6 n = 0", "m + 3 = 0", "\\frac { 1 } { 6 } m ^ { 3 } + 2 n ^ { 2 } + 1", "n = \\frac { 1 } { 2 }", "m = - 3", "- 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 x ^ { 2 } + mx - 2 y + 1 - 6 nx ^ { 2 } + 3 x"}, {"id": "3 - 6 n = 0"}, {"id": "代数式 $3 x ^ { 2 } + mx - 2 y + 1 - 6 nx ^ { 2 } + 3 x$ 的值与字母 $x$ 的取值无关"}, {"id": "m + 3 = 0"}, {"id": "m = - 3"}, {"id": "n = \\frac { 1 } { 2 }"}, {"id": "\\frac { 1 } { 2 } m ^ { 3 } - n ^ { 2 } - \\frac { 1 } { 3 } m ^ { 3 } + 3 n ^ { 2 } + 1"}, {"id": "\\frac { 1 } { 6 } m ^ { 3 } + 2 n ^ { 2 } + 1"}, {"id": "- 3"}], "links": [{"rel": "被描述", "source": "3 x ^ { 2 } + mx - 2 y + 1 - 6 nx ^ { 2 } + 3 x", "target": "3 - 6 n = 0"}, {"rel": "被描述", "source": "3 x ^ { 2 } + mx - 2 y + 1 - 6 nx ^ { 2 } + 3 x", "target": "m + 3 = 0"}, {"rel": "等式方程求解", "source": "3 - 6 n = 0", "target": "n = \\frac { 1 } { 2 }"}, {"rel": "限制性描述", "source": "代数式 $3 x ^ { 2 } + mx - 2 y + 1 - 6 nx ^ { 2 } + 3 x$ 的值与字母 $x$ 的取值无关", "target": "3 - 6 n = 0"}, {"rel": "限制性描述", "source": "代数式 $3 x ^ { 2 } + mx - 2 y + 1 - 6 nx ^ { 2 } + 3 x$ 的值与字母 $x$ 的取值无关", "target": "m + 3 = 0"}, {"rel": "等式方程求解", "source": "m + 3 = 0", "target": "m = - 3"}, {"rel": "代入", "source": "m = - 3", "target": "- 3"}, {"rel": "代入", "source": "n = \\frac { 1 } { 2 }", "target": "- 3"}, {"rel": "计算", "source": "\\frac { 1 } { 2 } m ^ { 3 } - n ^ { 2 } - \\frac { 1 } { 3 } m ^ { 3 } + 3 n ^ { 2 } + 1", "target": "\\frac { 1 } { 6 } m ^ { 3 } + 2 n ^ { 2 } + 1"}, {"rel": "被代入", "source": "\\frac { 1 } { 6 } m ^ { 3 } + 2 n ^ { 2 } + 1", "target": "- 3"}]}}
{"content": "If $a ^ { m - 2 } b ^ { n + 9 }$ and $- 3 ab ^ { 7 }$ are like terms, then $n ^ { m }$ = ____ ?", "answer": "- 8", "steps": "$\\because a ^ { m - 2 } b ^ { n + 9 }$ and $- 3 ab ^ 7$ are like terms, $\\therefore m - 2 = 1$, $n + 9 = 7$, solving for $m$ and $n$ gives $m = 3$, $n = - 2$. $\\therefore {( - 2 )} ^ 3 = - 8$.", "expr_cands": ["a ^ { m - 2 } b ^ { n + 9 }", "n", "m", "a", "b", "- 3 ab ^ { 7 }", "n ^ { m }", "m - 2 = 1", "m = 3", "n + 9 = 7", "n = - 2", "{ ( - 2 ) } ^ { 3 }", "- 8"], "exprs": ["m - 2 = 1", "n + 9 = 7", "m = 3", "n = - 2", "- 8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a ^ { m - 2 } b ^ { n + 9 }"}, {"id": "m - 2 = 1"}, {"id": "- 3 ab ^ { 7 }"}, {"id": "$a ^ { m - 2 } b ^ { n + 9 }$ 与 $- 3 ab ^ { 7 }$ 是同类项"}, {"id": "n + 9 = 7"}, {"id": "m = 3"}, {"id": "n = - 2"}, {"id": "n ^ { m }"}, {"id": "- 8"}], "links": [{"rel": "被描述", "source": "a ^ { m - 2 } b ^ { n + 9 }", "target": "m - 2 = 1"}, {"rel": "被描述", "source": "a ^ { m - 2 } b ^ { n + 9 }", "target": "n + 9 = 7"}, {"rel": "等式方程求解", "source": "m - 2 = 1", "target": "m = 3"}, {"rel": "被描述", "source": "- 3 ab ^ { 7 }", "target": "m - 2 = 1"}, {"rel": "被描述", "source": "- 3 ab ^ { 7 }", "target": "n + 9 = 7"}, {"rel": "限制性描述", "source": "$a ^ { m - 2 } b ^ { n + 9 }$ 与 $- 3 ab ^ { 7 }$ 是同类项", "target": "m - 2 = 1"}, {"rel": "限制性描述", "source": "$a ^ { m - 2 } b ^ { n + 9 }$ 与 $- 3 ab ^ { 7 }$ 是同类项", "target": "n + 9 = 7"}, {"rel": "等式方程求解", "source": "n + 9 = 7", "target": "n = - 2"}, {"rel": "代入", "source": "m = 3", "target": "- 8"}, {"rel": "代入", "source": "n = - 2", "target": "- 8"}, {"rel": "被代入", "source": "n ^ { m }", "target": "- 8"}]}}
{"content": "If $x - y = 4$, $xy = 2$, then $( x + y ) ^ 2$ = ____?", "answer": "24", "steps": "Since $x - y = 4$ and $xy = 2$, therefore $( x + y ) ^ 2 = ( x - y ) ^ 2 + 4 xy = 4 ^ 2 + 4 * 2 = 16 + 8 = 24$.", "expr_cands": ["x - y = 4", "x", "y", "xy = 2", "( x + y ) ^ { 2 }", "( x - y ) ^ { 2 } + 4 xy", "24"], "exprs": ["( x - y ) ^ { 2 } + 4 xy", "24"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( x + y ) ^ { 2 }"}, {"id": "( x - y ) ^ { 2 } + 4 xy"}, {"id": "x - y = 4"}, {"id": "xy = 2"}, {"id": "24"}], "links": [{"rel": "提取因式", "source": "( x + y ) ^ { 2 }", "target": "( x - y ) ^ { 2 } + 4 xy"}, {"rel": "被代入", "source": "( x - y ) ^ { 2 } + 4 xy", "target": "24"}, {"rel": "提取因式参考", "source": "x - y = 4", "target": "( x - y ) ^ { 2 } + 4 xy"}, {"rel": "代入", "source": "x - y = 4", "target": "24"}, {"rel": "提取因式参考", "source": "xy = 2", "target": "( x - y ) ^ { 2 } + 4 xy"}, {"rel": "代入", "source": "xy = 2", "target": "24"}]}}
{"content": "Given $x = 2015$, what is the value of $\\frac { x ^ 2 + 1 } { x + 1 } + \\frac { 2 x } { x + 1 }$?", "answer": "2016", "steps": "$\\frac { x ^ { 2 } + 1 } { x + 1 } + \\frac { 2 x } { x + 1 } = \\frac { x ^ { 2 } + 2 x + 1 } { x + 1 } = \\frac { ( x + 1 ) ^ { 2 } } { x + 1 } = x + 1$ , when $x = 2015$, $x + 1 = 2015 + 1 = 2016$.", "expr_cands": ["x = 2015", "x", "\\frac { x ^ { 2 } + 1 } { x + 1 } + \\frac { 2 x } { x + 1 }", "x + 1", "2016"], "exprs": ["x + 1", "2016"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x ^ { 2 } + 1 } { x + 1 } + \\frac { 2 x } { x + 1 }"}, {"id": "x + 1"}, {"id": "2016"}, {"id": "x = 2015"}], "links": [{"rel": "计算", "source": "\\frac { x ^ { 2 } + 1 } { x + 1 } + \\frac { 2 x } { x + 1 }", "target": "x + 1"}, {"rel": "被代入", "source": "x + 1", "target": "2016"}, {"rel": "代入", "source": "x = 2015", "target": "2016"}]}}
{"content": "If $a ^ { 3 } \\cdot a ^ { 3 n } \\cdot a ^ { n + 1 } = a ^ { 32 }$, then $n$ = ____ ?", "answer": "7", "steps": "\\because $a ^ { 3 } \\cdot a ^ { 3 n } \\cdot a ^ { n + 1 } = a ^ { 4 n + 4 } = a ^ { 32 }$ , \\therefore $4 n + 4 = 32$ , so we get: $n = 7$ .", "expr_cands": ["a ^ { 3 } \\cdot a ^ { 3 n } \\cdot a ^ { n + 1 } = a ^ { 32 }", "n", "a", "4 n + 4 = 32", "n = 7"], "exprs": ["4 n + 4 = 32", "n = 7"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a ^ { 3 } \\cdot a ^ { 3 n } \\cdot a ^ { n + 1 } = a ^ { 32 }"}, {"id": "4 n + 4 = 32"}, {"id": "n = 7"}], "links": [{"rel": "同取对数", "source": "a ^ { 3 } \\cdot a ^ { 3 n } \\cdot a ^ { n + 1 } = a ^ { 32 }", "target": "4 n + 4 = 32"}, {"rel": "等式方程求解", "source": "4 n + 4 = 32", "target": "n = 7"}]}}
{"content": "If the solution to $ax - 5 \\ge 0$ is $x \\le - 2.5$, then the value of $a$ is", "answer": "a = - 2", "steps": "Since $ax - 5 \\ge 0$, it follows that $ax \\ge 5$. Since the solution to $ax - 5 \\ge 0$ is $x \\le - 2.5$, it follows that $a < 0$. Solving $\\frac { 5 } { a } = - 2.5$, we get $a = - 2$.", "expr_cands": ["ax - 5 \\ge 0", "x", "a", "x \\le - 2.5", "ax \\ge 5", "a < 0", "\\frac { 5 } { a } = - 2.5", "a = - 2.0", "a = - 2"], "exprs": ["a < 0", "\\frac { 5 } { a } = - 2.5", "a = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "ax - 5 \\ge 0"}, {"id": "a < 0"}, {"id": "x \\le - 2.5"}, {"id": "$ax - 5 \\ge 0$ 的解是 $x \\le - 2.5$"}, {"id": "不等式两边都乘或除同一个负数，不等号的方向改变"}, {"id": "\\frac { 5 } { a } = - 2.5"}, {"id": "a = - 2"}], "links": [{"rel": "被描述", "source": "ax - 5 \\ge 0", "target": "a < 0"}, {"rel": "被描述", "source": "ax - 5 \\ge 0", "target": "\\frac { 5 } { a } = - 2.5"}, {"rel": "被描述", "source": "a < 0", "target": "\\frac { 5 } { a } = - 2.5"}, {"rel": "被描述", "source": "x \\le - 2.5", "target": "a < 0"}, {"rel": "被描述", "source": "x \\le - 2.5", "target": "\\frac { 5 } { a } = - 2.5"}, {"rel": "限制性描述", "source": "$ax - 5 \\ge 0$ 的解是 $x \\le - 2.5$", "target": "a < 0"}, {"rel": "限制性描述", "source": "$ax - 5 \\ge 0$ 的解是 $x \\le - 2.5$", "target": "\\frac { 5 } { a } = - 2.5"}, {"rel": "属性描述", "source": "不等式两边都乘或除同一个负数，不等号的方向改变", "target": "a < 0"}, {"rel": "等式方程求解", "source": "\\frac { 5 } { a } = - 2.5", "target": "a = - 2"}]}}
{"content": "The maximum integer solution that satisfies the inequality $x + 3 < 0$ is ____ ?", "answer": "- 4", "steps": "From the inequality $x + 3 < 0$, we can solve for $x$ to get $x < - 3$. Therefore, the largest integer solution to the inequality is $- 4$.", "expr_cands": ["x + 3 < 0", "x", "x < - 3", "- 4"], "exprs": ["x < - 3", "- 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x + 3 < 0"}, {"id": "x < - 3"}, {"id": "- 4"}, {"id": "满足不等式 $x + 3 < 0$ 的最大整数解"}], "links": [{"rel": "不等式方程求解", "source": "x + 3 < 0", "target": "x < - 3"}, {"rel": "被描述", "source": "x < - 3", "target": "- 4"}, {"rel": "限制性描述", "source": "满足不等式 $x + 3 < 0$ 的最大整数解", "target": "- 4"}]}}
{"content": "Given the function $y = ( m - 1 ) x + m ^ { 2 } - 1$ is a proportional function, then $m$ = ____ ?", "answer": "- 1", "steps": "From the definition of a proportional function, we have: $m ^ 2 - 1 = 0$, and $m - 1 \\neq 0$. Solving for $m$, we get: $m = - 1$.", "expr_cands": ["y = ( m - 1 ) x + m ^ { 2 } - 1", "m", "y", "x", "m ^ { 2 } - 1 = 0", "m = - 1", "m = 1", "m - 1 \\neq 0", "m \\neq 1"], "exprs": ["m ^ { 2 } - 1 = 0", "m - 1 \\neq 0", "m = - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = ( m - 1 ) x + m ^ { 2 } - 1"}, {"id": "m ^ { 2 } - 1 = 0"}, {"id": "函数 $y = ( m - 1 ) x + m ^ { 2 } - 1$ 是正比例函数"}, {"id": "正比例函数的定义可得 : $m ^ { 2 } - 1 = 0$"}, {"id": "m - 1 \\neq 0"}, {"id": "m = - 1"}], "links": [{"rel": "被描述", "source": "y = ( m - 1 ) x + m ^ { 2 } - 1", "target": "m ^ { 2 } - 1 = 0"}, {"rel": "被描述", "source": "y = ( m - 1 ) x + m ^ { 2 } - 1", "target": "m - 1 \\neq 0"}, {"rel": "联立", "source": "m ^ { 2 } - 1 = 0", "target": "m = - 1"}, {"rel": "限制性描述", "source": "函数 $y = ( m - 1 ) x + m ^ { 2 } - 1$ 是正比例函数", "target": "m ^ { 2 } - 1 = 0"}, {"rel": "限制性描述", "source": "函数 $y = ( m - 1 ) x + m ^ { 2 } - 1$ 是正比例函数", "target": "m - 1 \\neq 0"}, {"rel": "限制性描述", "source": "正比例函数的定义可得 : $m ^ { 2 } - 1 = 0$", "target": "m ^ { 2 } - 1 = 0"}, {"rel": "联立", "source": "m - 1 \\neq 0", "target": "m = - 1"}]}}
{"content": "If $2 x - 4$ and $1 - 3 x$ are two different square roots of the same number, then the value of $x$ is ____?", "answer": "- 3", "steps": "Because $2 x - 4$ and $1 - 3 x$ are two different square roots of the same number, therefore $( 2 x - 4 ) + ( 1 - 3 x ) = 0$, solving which gives $x = - 3$.", "expr_cands": ["2 x - 4", "x", "1 - 3 x", "( 2 x - 4 ) + ( 1 - 3 x ) = 0", "x = - 3"], "exprs": ["( 2 x - 4 ) + ( 1 - 3 x ) = 0", "x = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x - 4"}, {"id": "( 2 x - 4 ) + ( 1 - 3 x ) = 0"}, {"id": "1 - 3 x"}, {"id": "$2 x - 4$ 与 $1 - 3 x$ 是同一个数的两个不同的平方根"}, {"id": "平方根互为相反数"}, {"id": "x = - 3"}], "links": [{"rel": "被描述", "source": "2 x - 4", "target": "( 2 x - 4 ) + ( 1 - 3 x ) = 0"}, {"rel": "等式方程求解", "source": "( 2 x - 4 ) + ( 1 - 3 x ) = 0", "target": "x = - 3"}, {"rel": "被描述", "source": "1 - 3 x", "target": "( 2 x - 4 ) + ( 1 - 3 x ) = 0"}, {"rel": "限制性描述", "source": "$2 x - 4$ 与 $1 - 3 x$ 是同一个数的两个不同的平方根", "target": "( 2 x - 4 ) + ( 1 - 3 x ) = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "( 2 x - 4 ) + ( 1 - 3 x ) = 0"}]}}
{"content": "The equation $3 ax - a = 2$ has the same solution as the equation $2 x - 1 = 4$. What is the value of $a$?", "answer": "\\frac { 4 } { 13 }", "steps": "From $2 x - 1 = 4$, we get $x = \\frac { 5 } { 2 }$. Since the solution to $3 ax - a = 2$ is the same as the solution to $2 x - 1 = 4$, we have $\\frac { 15 } { 2 } a - a = 2$, which gives us $a = \\frac { 4 } { 13 }$.", "expr_cands": ["3 ax - a = 2", "x", "a", "2 x - 1 = 4", "x = \\frac { 5 } { 2 }", "\\frac { 13 a } { 2 } = 2", "\\frac { 15 } { 2 } a - a = 2", "a = \\frac { 4 } { 13 }"], "exprs": ["x = \\frac { 5 } { 2 }", "\\frac { 15 } { 2 } a - a = 2", "a = \\frac { 4 } { 13 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x - 1 = 4"}, {"id": "x = \\frac { 5 } { 2 }"}, {"id": "3 ax - a = 2"}, {"id": "\\frac { 15 } { 2 } a - a = 2"}, {"id": "a = \\frac { 4 } { 13 }"}], "links": [{"rel": "等式方程求解", "source": "2 x - 1 = 4", "target": "x = \\frac { 5 } { 2 }"}, {"rel": "代入", "source": "x = \\frac { 5 } { 2 }", "target": "\\frac { 15 } { 2 } a - a = 2"}, {"rel": "被代入", "source": "3 ax - a = 2", "target": "\\frac { 15 } { 2 } a - a = 2"}, {"rel": "等式方程求解", "source": "\\frac { 15 } { 2 } a - a = 2", "target": "a = \\frac { 4 } { 13 }"}]}}
{"content": "Given a quadratic equation $x ^ 2 + k - 3 = 0$ with one root being $- 2$, the value of $k$ is ____?", "answer": "- 1", "steps": "$\\because$ The quadratic equation $x ^ 2 + k - 3 = 0$ has a root of $- 2$, $\\therefore$ substituting $x = - 2$ gives $4 + k - 3 = 0$, solving for $k$ gives: $k = - 1$.", "expr_cands": ["x ^ { 2 } + k - 3 = 0", "x", "k", "- 2", "x = - 2", "4 + k - 3 = 0", "k = - 1"], "exprs": ["x = - 2", "4 + k - 3 = 0", "k = - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 2"}, {"id": "x = - 2"}, {"id": "x ^ { 2 } + k - 3 = 0"}, {"id": "一元二次方程 $x ^ { 2 } + k - 3 = 0$ 有一个根为 $- 2$"}, {"id": "4 + k - 3 = 0"}, {"id": "k = - 1"}], "links": [{"rel": "被描述", "source": "- 2", "target": "x = - 2"}, {"rel": "代入", "source": "x = - 2", "target": "4 + k - 3 = 0"}, {"rel": "被描述", "source": "x ^ { 2 } + k - 3 = 0", "target": "x = - 2"}, {"rel": "被代入", "source": "x ^ { 2 } + k - 3 = 0", "target": "4 + k - 3 = 0"}, {"rel": "限制性描述", "source": "一元二次方程 $x ^ { 2 } + k - 3 = 0$ 有一个根为 $- 2$", "target": "x = - 2"}, {"rel": "等式方程求解", "source": "4 + k - 3 = 0", "target": "k = - 1"}]}}
{"content": "Given $a ^ { x } \\cdot a ^ { y } = a ^ { 5 }$ and $a ^ { x } \\div a ^ { y } = a$, what is the value of $x ^ { 2 } - y ^ { 2 }$?", "answer": "5", "steps": "From $a ^ { x } \\cdot a ^ { y } = a ^ { 5 }$ and $a ^ { x } \\div a ^ { y } = a$, we get: $x + y = 5$, $x - y = 1$. Since the original expression is equal to $( x + y ) ( x - y )$, substituting $x + y = 5$ and $x - y = 1$ gives the original expression as $5 * 1 = 5$.", "expr_cands": ["a ^ { x } \\cdot a ^ { y } = a ^ { 5 }", "a", "x", "y", "a ^ { x } \\div a ^ { y } = a", "x ^ { 2 } - y ^ { 2 }", "x + y = 5", "x - y = 1", "( x + y ) ( x - y )", "5 * 1", "5"], "exprs": ["x + y = 5", "x - y = 1", "( x + y ) ( x - y )", "5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a ^ { x } \\cdot a ^ { y } = a ^ { 5 }"}, {"id": "x + y = 5"}, {"id": "a ^ { x } \\div a ^ { y } = a"}, {"id": "x - y = 1"}, {"id": "x ^ { 2 } - y ^ { 2 }"}, {"id": "( x + y ) ( x - y )"}, {"id": "5"}], "links": [{"rel": "同取对数", "source": "a ^ { x } \\cdot a ^ { y } = a ^ { 5 }", "target": "x + y = 5"}, {"rel": "提取因式参考", "source": "x + y = 5", "target": "( x + y ) ( x - y )"}, {"rel": "代入", "source": "x + y = 5", "target": "5"}, {"rel": "同取对数", "source": "a ^ { x } \\div a ^ { y } = a", "target": "x - y = 1"}, {"rel": "提取因式参考", "source": "x - y = 1", "target": "( x + y ) ( x - y )"}, {"rel": "代入", "source": "x - y = 1", "target": "5"}, {"rel": "提取因式", "source": "x ^ { 2 } - y ^ { 2 }", "target": "( x + y ) ( x - y )"}, {"rel": "被代入", "source": "( x + y ) ( x - y )", "target": "5"}]}}
{"content": "Given a linear function $y = kx + 3$, when $x = 2$, $y = 5$. What is the value of $k$?", "answer": "1", "steps": "$\\because$ When $x = 2$, $y = 5$, $\\therefore$ $5 = 2 k + 3$, so $k = 1$, $\\therefore$ the value of $k$ is $1$.", "expr_cands": ["y = kx + 3", "k", "y", "x", "x = 2", "y = 5", "5 = 2 k + 3", "k = 1", "1"], "exprs": ["5 = 2 k + 3", "k = 1", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = kx + 3"}, {"id": "5 = 2 k + 3"}, {"id": "x = 2"}, {"id": "y = 5"}, {"id": "k = 1"}, {"id": "1"}, {"id": "$k$ 的值"}], "links": [{"rel": "被代入", "source": "y = kx + 3", "target": "5 = 2 k + 3"}, {"rel": "等式方程求解", "source": "5 = 2 k + 3", "target": "k = 1"}, {"rel": "代入", "source": "x = 2", "target": "5 = 2 k + 3"}, {"rel": "代入", "source": "y = 5", "target": "5 = 2 k + 3"}, {"rel": "被描述", "source": "k = 1", "target": "1"}, {"rel": "属性描述", "source": "$k$ 的值", "target": "1"}]}}
{"content": "The monomial $\\frac { 1 } { 5 } { a } ^ { 2 x + 1 }$ and the sum $- 8 { a } ^ { 3 x - 2 }$ are like terms. What is the value of $x$?", "answer": "3", "steps": "From the given information, we have $2 x + 1 = 3 x - 2$. Solving for $x$, we get $x = 3$.", "expr_cands": ["\\frac { 1 } { 5 } { a } ^ { 2 x + 1 }", "x", "a", "- 8 { a } ^ { 3 x - 2 }", "2 x + 1 = 3 x - 2", "x = 3"], "exprs": ["2 x + 1 = 3 x - 2", "x = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 1 } { 5 } { a } ^ { 2 x + 1 }"}, {"id": "2 x + 1 = 3 x - 2"}, {"id": "- 8 { a } ^ { 3 x - 2 }"}, {"id": "单项式 $\\frac { 1 } { 5 } { a } ^ { 2 x + 1 }$ 和是 $- 8 { a } ^ { 3 x - 2 }$ 同类项"}, {"id": "x = 3"}], "links": [{"rel": "被描述", "source": "\\frac { 1 } { 5 } { a } ^ { 2 x + 1 }", "target": "2 x + 1 = 3 x - 2"}, {"rel": "等式方程求解", "source": "2 x + 1 = 3 x - 2", "target": "x = 3"}, {"rel": "被描述", "source": "- 8 { a } ^ { 3 x - 2 }", "target": "2 x + 1 = 3 x - 2"}, {"rel": "限制性描述", "source": "单项式 $\\frac { 1 } { 5 } { a } ^ { 2 x + 1 }$ 和是 $- 8 { a } ^ { 3 x - 2 }$ 同类项", "target": "2 x + 1 = 3 x - 2"}]}}
{"content": "Given: $2 a - 3 b - 6 = 0$, then ${ 9 } ^ { a } \\div { 27 } ^ { b }$ = ____ ?", "answer": "729", "steps": "${ 9 } ^ { a } \\div { 27 } ^ { b } = { ( { 3 } ^ { 2 } ) } ^ { a } \\div { ( { 3 } ^ { 3 } ) } ^ { b } = { 3 } ^ { 2 a } \\div { 3 } ^ { 3 b } = { 3 } ^ { 2 a - 3 b }$ , because $2 a - 3 b - 6 = 0$ , therefore $2 a - 3 b = 6$ , therefore ${ 9 } ^ { a } \\div { 27 } ^ { b } = 3 ^ { 6 } = 729$ .", "expr_cands": ["2 a - 3 b - 6 = 0", "b", "a", "{ 9 } ^ { a } \\div { 27 } ^ { b }", "{ 3 } ^ { 2 a - 3 b }", "2 a - 3 b = 6", "{ 9 } ^ { a } \\div { 27 } ^ { b } = 729", "729"], "exprs": ["{ 3 } ^ { 2 a - 3 b }", "2 a - 3 b = 6", "729"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "{ 9 } ^ { a } \\div { 27 } ^ { b }"}, {"id": "{ 3 } ^ { 2 a - 3 b }"}, {"id": "2 a - 3 b - 6 = 0"}, {"id": "2 a - 3 b = 6"}, {"id": "729"}], "links": [{"rel": "计算", "source": "{ 9 } ^ { a } \\div { 27 } ^ { b }", "target": "{ 3 } ^ { 2 a - 3 b }"}, {"rel": "被代入", "source": "{ 3 } ^ { 2 a - 3 b }", "target": "729"}, {"rel": "移项", "source": "2 a - 3 b - 6 = 0", "target": "2 a - 3 b = 6"}, {"rel": "代入", "source": "2 a - 3 b = 6", "target": "729"}]}}
{"content": "If $x _ 1$ and $x _ 2$ are two solutions of the equation $x ^ 2 - 5 x + 6 = 0$, then the value of the algebraic expression $( x _ 1 + 1 ) ( x _ 2 + 1 )$ is ____?", "answer": "12", "steps": "According to the problem, we have $x _ 1 + x _ 2 = 5$ and $x _ 1 x _ 2 = 6$. Therefore, $( x _ 1 + 1 ) ( x _ 2 + 1 ) = x _ 1 x _ 2 + x _ 1 + x _ 2 + 1 = 6 + 5 + 1 = 12$.", "expr_cands": ["x _ { 1 }", "x _ { 2 }", "x ^ { 2 } - 5 x + 6 = 0", "x", "( x _ { 1 } + 1 ) ( x _ { 2 } + 1 )", "x _ { 1 } + x _ { 2 } = 5", "x _ { 1 } x _ { 2 } = 6", "x _ { 1 } x _ { 2 } + x _ { 1 } + x _ { 2 } + 1", "12"], "exprs": ["x _ { 1 } + x _ { 2 } = 5", "x _ { 1 } x _ { 2 } = 6", "x _ { 1 } x _ { 2 } + x _ { 1 } + x _ { 2 } + 1", "12"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x _ { 1 }"}, {"id": "x _ { 1 } + x _ { 2 } = 5"}, {"id": "x _ { 2 }"}, {"id": "x ^ { 2 } - 5 x + 6 = 0"}, {"id": "$x _ { 1 }$ , $x _ { 2 }$ 是方程 $x ^ { 2 } - 5 x + 6 = 0$ 的两个解"}, {"id": "一元二次方程根与系数关系，两根之和"}, {"id": "x _ { 1 } x _ { 2 } = 6"}, {"id": "一元二次方程根与系数关系，两根之积"}, {"id": "( x _ { 1 } + 1 ) ( x _ { 2 } + 1 )"}, {"id": "x _ { 1 } x _ { 2 } + x _ { 1 } + x _ { 2 } + 1"}, {"id": "12"}], "links": [{"rel": "被描述", "source": "x _ { 1 }", "target": "x _ { 1 } + x _ { 2 } = 5"}, {"rel": "被描述", "source": "x _ { 1 }", "target": "x _ { 1 } x _ { 2 } = 6"}, {"rel": "代入", "source": "x _ { 1 } + x _ { 2 } = 5", "target": "12"}, {"rel": "被描述", "source": "x _ { 2 }", "target": "x _ { 1 } + x _ { 2 } = 5"}, {"rel": "被描述", "source": "x _ { 2 }", "target": "x _ { 1 } x _ { 2 } = 6"}, {"rel": "被描述", "source": "x ^ { 2 } - 5 x + 6 = 0", "target": "x _ { 1 } + x _ { 2 } = 5"}, {"rel": "被描述", "source": "x ^ { 2 } - 5 x + 6 = 0", "target": "x _ { 1 } x _ { 2 } = 6"}, {"rel": "限制性描述", "source": "$x _ { 1 }$ , $x _ { 2 }$ 是方程 $x ^ { 2 } - 5 x + 6 = 0$ 的两个解", "target": "x _ { 1 } + x _ { 2 } = 5"}, {"rel": "限制性描述", "source": "$x _ { 1 }$ , $x _ { 2 }$ 是方程 $x ^ { 2 } - 5 x + 6 = 0$ 的两个解", "target": "x _ { 1 } x _ { 2 } = 6"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "x _ { 1 } + x _ { 2 } = 5"}, {"rel": "代入", "source": "x _ { 1 } x _ { 2 } = 6", "target": "12"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之积", "target": "x _ { 1 } x _ { 2 } = 6"}, {"rel": "展开", "source": "( x _ { 1 } + 1 ) ( x _ { 2 } + 1 )", "target": "x _ { 1 } x _ { 2 } + x _ { 1 } + x _ { 2 } + 1"}, {"rel": "被代入", "source": "x _ { 1 } x _ { 2 } + x _ { 1 } + x _ { 2 } + 1", "target": "12"}]}}
{"content": "If $x = 3$, $y = 1$ is a solution of the equation $3 x - ay = 2$, then $a$ = ____?", "answer": "7", "steps": "Substituting $x = 3$ and $y = 1$ into the equation $3 x - ay = 2$, we get $3 \\cdot 3 - a \\cdot 1 = 2$. Solving for $a$, we get $a = 7$.", "expr_cands": ["x = 3", "x", "y = 1", "y", "3 x - ay = 2", "a", "9 - a = 2", "3 * 3 - a = 2", "a = 7"], "exprs": ["3 * 3 - a = 2", "a = 7"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 x - ay = 2"}, {"id": "3 * 3 - a = 2"}, {"id": "x = 3"}, {"id": "y = 1"}, {"id": "a = 7"}], "links": [{"rel": "被代入", "source": "3 x - ay = 2", "target": "3 * 3 - a = 2"}, {"rel": "等式方程求解", "source": "3 * 3 - a = 2", "target": "a = 7"}, {"rel": "代入", "source": "x = 3", "target": "3 * 3 - a = 2"}, {"rel": "代入", "source": "y = 1", "target": "3 * 3 - a = 2"}]}}
{"content": "If $a - b = 2010$, $c + d = 2011$, then the value of $( b + c ) - ( a - d )$ is ____?", "answer": "1", "steps": "Original expression = $b + c - a + d = - ( a - b ) + ( c + d )$, when $a - b = 2010$, $c + d = 2011$, the original expression $= - 2010 + 2011 = 1$.", "expr_cands": ["a - b = 2010", "b", "a", "c + d = 2011", "c", "d", "( b + c ) - ( a - d )", "b + c - a + d", "- ( a - b ) + ( c + d )", "- 2010 + 2011", "1"], "exprs": ["- ( a - b ) + ( c + d )", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( b + c ) - ( a - d )"}, {"id": "- ( a - b ) + ( c + d )"}, {"id": "a - b = 2010"}, {"id": "c + d = 2011"}, {"id": "1"}], "links": [{"rel": "提取因式", "source": "( b + c ) - ( a - d )", "target": "- ( a - b ) + ( c + d )"}, {"rel": "被代入", "source": "- ( a - b ) + ( c + d )", "target": "1"}, {"rel": "提取因式参考", "source": "a - b = 2010", "target": "- ( a - b ) + ( c + d )"}, {"rel": "代入", "source": "a - b = 2010", "target": "1"}, {"rel": "提取因式参考", "source": "c + d = 2011", "target": "- ( a - b ) + ( c + d )"}, {"rel": "代入", "source": "c + d = 2011", "target": "1"}]}}
{"content": "If the solution set of the inequality $2 x + a < 2$ and $2 x < 4$ with respect to $x$ is the same, then the value of $a$ is ____?", "answer": "- 2", "steps": "From $2 x < 4$, we can obtain that $x < 2$. Also, from $2 x + a < 2$, we can obtain that $x < \\frac { 2 - a } { 2 }$. Therefore, $\\frac { 2 - a } { 2 } = 2$, which means $2 - a = 4$, and $a = - 2$.", "expr_cands": ["x", "2 x + a < 2", "a", "2 x < 4", "x < 2", "x < \\frac { 2 - a } { 2 }", "\\frac { 2 - a } { 2 } = 2", "a = - 2"], "exprs": ["x < 2", "x < \\frac { 2 - a } { 2 }", "\\frac { 2 - a } { 2 } = 2", "a = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x < 4"}, {"id": "x < 2"}, {"id": "2 x + a < 2"}, {"id": "x < \\frac { 2 - a } { 2 }"}, {"id": "\\frac { 2 - a } { 2 } = 2"}, {"id": "关于 $x$ 的不等式 $2 x + a < 2$ 和 $2 x < 4$ 的解集相同"}, {"id": "a = - 2"}], "links": [{"rel": "不等式方程求解", "source": "2 x < 4", "target": "x < 2"}, {"rel": "被描述", "source": "x < 2", "target": "\\frac { 2 - a } { 2 } = 2"}, {"rel": "不等式方程部分求解", "source": "2 x + a < 2", "target": "x < \\frac { 2 - a } { 2 }"}, {"rel": "被描述", "source": "x < \\frac { 2 - a } { 2 }", "target": "\\frac { 2 - a } { 2 } = 2"}, {"rel": "等式方程求解", "source": "\\frac { 2 - a } { 2 } = 2", "target": "a = - 2"}, {"rel": "限制性描述", "source": "关于 $x$ 的不等式 $2 x + a < 2$ 和 $2 x < 4$ 的解集相同", "target": "\\frac { 2 - a } { 2 } = 2"}]}}
{"content": "Given $\\frac { x } { y } = \\frac { 2 } { 5 }$, what is the value of $\\frac { x + y } { y }$?", "answer": "\\frac { 7 } { 5 }", "steps": "Because $\\frac { x } { y } = \\frac { 2 } { 5 }$, therefore let $x = 2 a$, then $y = 5 a$, so $\\frac { x + y } { y } = \\frac { 2 a + 5 a } { 5 a } = \\frac { 7 } { 5 }$.", "expr_cands": ["\\frac { x } { y } = \\frac { 2 } { 5 }", "x", "y", "\\frac { x + y } { y }", "x = 2 a", "a", "y = 5 a", "\\frac { 7 } { 5 }"], "exprs": ["x = 2 a", "y = 5 a", "\\frac { 7 } { 5 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "设 $x = 2 a$"}, {"id": "x = 2 a"}, {"id": "\\frac { x } { y } = \\frac { 2 } { 5 }"}, {"id": "y = 5 a"}, {"id": "\\frac { x + y } { y }"}, {"id": "\\frac { 7 } { 5 }"}], "links": [{"rel": "假设描述", "source": "设 $x = 2 a$", "target": "x = 2 a"}, {"rel": "联立", "source": "x = 2 a", "target": "y = 5 a"}, {"rel": "代入", "source": "x = 2 a", "target": "\\frac { 7 } { 5 }"}, {"rel": "联立", "source": "\\frac { x } { y } = \\frac { 2 } { 5 }", "target": "y = 5 a"}, {"rel": "代入", "source": "y = 5 a", "target": "\\frac { 7 } { 5 }"}, {"rel": "被代入", "source": "\\frac { x + y } { y }", "target": "\\frac { 7 } { 5 }"}]}}
{"content": "Given the equation $( k - 2 ) x ^ { | k - 1 | } - 2017 = 2021$ is a linear equation in one variable $x$, find the value of $k$.", "answer": "0", "steps": "From the given condition, we have $| k - 1 | = 1$ and $k - 2 \\neq 0$. Solving for $k$, we get $k = 0$.", "expr_cands": ["( k - 2 ) x ^ { | k - 1 | } - 2017 = 2021", "k", "x", "| k - 1 | = 1", "k = 0", "k = 2", "k - 2 \\neq 0", "k \\neq 2"], "exprs": ["| k - 1 | = 1", "k - 2 \\neq 0", "k = 0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( k - 2 ) x ^ { | k - 1 | } - 2017 = 2021"}, {"id": "| k - 1 | = 1"}, {"id": "方程 $( k - 2 ) x ^ { | k - 1 | } - 2017 = 2021$ 是关于 $x$ 的一元一次方程"}, {"id": "k - 2 \\neq 0"}, {"id": "k = 0"}], "links": [{"rel": "被描述", "source": "( k - 2 ) x ^ { | k - 1 | } - 2017 = 2021", "target": "| k - 1 | = 1"}, {"rel": "被描述", "source": "( k - 2 ) x ^ { | k - 1 | } - 2017 = 2021", "target": "k - 2 \\neq 0"}, {"rel": "联立", "source": "| k - 1 | = 1", "target": "k = 0"}, {"rel": "限制性描述", "source": "方程 $( k - 2 ) x ^ { | k - 1 | } - 2017 = 2021$ 是关于 $x$ 的一元一次方程", "target": "| k - 1 | = 1"}, {"rel": "限制性描述", "source": "方程 $( k - 2 ) x ^ { | k - 1 | } - 2017 = 2021$ 是关于 $x$ 的一元一次方程", "target": "k - 2 \\neq 0"}, {"rel": "联立", "source": "k - 2 \\neq 0", "target": "k = 0"}]}}
{"content": "If the value of the fraction $\\frac { x - 1 } { x - 4 }$ is $0$, then the value of $x$ is ____?", "answer": "1", "steps": "According to the problem, we have $x - 1 = 0$ and $x - 4 \\neq 0$. Solving for $x$, we get $x = 1$.", "expr_cands": ["\\frac { x - 1 } { x - 4 }", "x", "0", "x - 1 = 0", "x = 1", "x - 4 \\neq 0", "x \\neq 4"], "exprs": ["x - 1 = 0", "x - 4 \\neq 0", "x = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x - 1 } { x - 4 }"}, {"id": "x - 1 = 0"}, {"id": "0"}, {"id": "分式 $\\frac { x - 1 } { x - 4 }$ 的值为 $0$"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "x - 4 \\neq 0"}, {"id": "x = 1"}], "links": [{"rel": "被描述", "source": "\\frac { x - 1 } { x - 4 }", "target": "x - 1 = 0"}, {"rel": "被描述", "source": "\\frac { x - 1 } { x - 4 }", "target": "x - 4 \\neq 0"}, {"rel": "联立", "source": "x - 1 = 0", "target": "x = 1"}, {"rel": "被描述", "source": "0", "target": "x - 1 = 0"}, {"rel": "被描述", "source": "0", "target": "x - 4 \\neq 0"}, {"rel": "限制性描述", "source": "分式 $\\frac { x - 1 } { x - 4 }$ 的值为 $0$", "target": "x - 1 = 0"}, {"rel": "限制性描述", "source": "分式 $\\frac { x - 1 } { x - 4 }$ 的值为 $0$", "target": "x - 4 \\neq 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x - 1 = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x - 4 \\neq 0"}, {"rel": "联立", "source": "x - 4 \\neq 0", "target": "x = 1"}]}}
{"content": "If the inequality $( a + 1 ) x < a + 1$ can be transformed into $x > 1$, then $a$ must satisfy ____?", "answer": "a < - 1", "steps": "From $x > 1$, we know that $a + 1 < 0$, which implies that $a < - 1$.", "expr_cands": ["( a + 1 ) x < a + 1", "a", "x", "x > 1", "a + 1 < 0", "a < - 1"], "exprs": ["a + 1 < 0", "a < - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( a + 1 ) x < a + 1"}, {"id": "a + 1 < 0"}, {"id": "不等式两边都乘或除同一个负数，不等号的方向改变"}, {"id": "不等式 $( a + 1 ) x < a + 1$ 可变形为 $x > 1$"}, {"id": "a < - 1"}], "links": [{"rel": "被描述", "source": "( a + 1 ) x < a + 1", "target": "a + 1 < 0"}, {"rel": "不等式方程求解", "source": "a + 1 < 0", "target": "a < - 1"}, {"rel": "限制性描述", "source": "不等式两边都乘或除同一个负数，不等号的方向改变", "target": "a + 1 < 0"}, {"rel": "限制性描述", "source": "不等式 $( a + 1 ) x < a + 1$ 可变形为 $x > 1$", "target": "a + 1 < 0"}]}}
{"content": "Given that $x$ and $y$ are real numbers and $\\sqrt { x - 1 }$ is opposite in sign to $( y - 2 ) ^ 2$, what is the value of $x - y$?", "answer": "- 1", "steps": "From $\\sqrt { x - 1 }$ and $( y - 2 ) ^ { 2 }$ being opposite, we can obtain: $\\sqrt { x - 1 } + ( y - 2 ) ^ { 2 } = 0$. Therefore, $x - 1 = 0$, $y - 2 = 0$, and we get: $x = 1$, $y = 2$. Hence, $x - y = 1 - 2 = - 1$.", "expr_cands": ["x", "y", "\\sqrt { x - 1 }", "( y - 2 ) ^ { 2 }", "x - y", "\\sqrt { x - 1 } + ( y - 2 ) ^ { 2 } = 0", "x - 1 = 0", "x = 1", "y - 2 = 0", "y = 2", "- 1"], "exprs": ["\\sqrt { x - 1 } + ( y - 2 ) ^ { 2 } = 0", "x - 1 = 0", "y - 2 = 0", "x = 1", "y = 2", "- 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x - 1 }"}, {"id": "\\sqrt { x - 1 } + ( y - 2 ) ^ { 2 } = 0"}, {"id": "( y - 2 ) ^ { 2 }"}, {"id": "$x$ , $y$ 为实数"}, {"id": "且 $\\sqrt { x - 1 }$ { 与 } $( y - 2 ) ^ { 2 }$ 互为相反数"}, {"id": "x - 1 = 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "y - 2 = 0"}, {"id": "多项式偶次方项恒大于等于0"}, {"id": "x = 1"}, {"id": "y = 2"}, {"id": "x - y"}, {"id": "- 1"}], "links": [{"rel": "被描述", "source": "\\sqrt { x - 1 }", "target": "\\sqrt { x - 1 } + ( y - 2 ) ^ { 2 } = 0"}, {"rel": "被描述", "source": "\\sqrt { x - 1 } + ( y - 2 ) ^ { 2 } = 0", "target": "x - 1 = 0"}, {"rel": "被描述", "source": "\\sqrt { x - 1 } + ( y - 2 ) ^ { 2 } = 0", "target": "y - 2 = 0"}, {"rel": "被描述", "source": "( y - 2 ) ^ { 2 }", "target": "\\sqrt { x - 1 } + ( y - 2 ) ^ { 2 } = 0"}, {"rel": "限制性描述", "source": "$x$ , $y$ 为实数", "target": "\\sqrt { x - 1 } + ( y - 2 ) ^ { 2 } = 0"}, {"rel": "限制性描述", "source": "且 $\\sqrt { x - 1 }$ { 与 } $( y - 2 ) ^ { 2 }$ 互为相反数", "target": "\\sqrt { x - 1 } + ( y - 2 ) ^ { 2 } = 0"}, {"rel": "等式方程求解", "source": "x - 1 = 0", "target": "x = 1"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 1 = 0"}, {"rel": "等式方程求解", "source": "y - 2 = 0", "target": "y = 2"}, {"rel": "属性描述", "source": "多项式偶次方项恒大于等于0", "target": "y - 2 = 0"}, {"rel": "代入", "source": "x = 1", "target": "- 1"}, {"rel": "代入", "source": "y = 2", "target": "- 1"}, {"rel": "被代入", "source": "x - y", "target": "- 1"}]}}
{"content": "If $( m + 1 ) x ^ { | m | } > 2$ is a linear inequality in one variable, then $m$ = ____ ?", "answer": "1", "steps": "$\\because$ $( m + 1 ) x ^ { | m | } > 2$ is a one-variable linear inequality in $x$, $\\therefore$ $m + 1 \\neq 0$, $| m | = 1$, and solving yields: $m = 1$.", "expr_cands": ["( m + 1 ) x ^ { | m | } > 2", "m", "x", "m + 1 \\neq 0", "m \\neq - 1", "| m | = 1", "m = - 1", "m = 1"], "exprs": ["m + 1 \\neq 0", "| m | = 1", "m = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( m + 1 ) x ^ { | m | } > 2"}, {"id": "m + 1 \\neq 0"}, {"id": ", $( m + 1 ) x ^ { | m | } > 2$ 是关于 $x$ 的一元一次不等式"}, {"id": "| m | = 1"}, {"id": "m = 1"}], "links": [{"rel": "被描述", "source": "( m + 1 ) x ^ { | m | } > 2", "target": "m + 1 \\neq 0"}, {"rel": "被描述", "source": "( m + 1 ) x ^ { | m | } > 2", "target": "| m | = 1"}, {"rel": "联立", "source": "m + 1 \\neq 0", "target": "m = 1"}, {"rel": "限制性描述", "source": ", $( m + 1 ) x ^ { | m | } > 2$ 是关于 $x$ 的一元一次不等式", "target": "m + 1 \\neq 0"}, {"rel": "限制性描述", "source": ", $( m + 1 ) x ^ { | m | } > 2$ 是关于 $x$ 的一元一次不等式", "target": "| m | = 1"}, {"rel": "联立", "source": "| m | = 1", "target": "m = 1"}]}}
{"content": "If $\\frac { | x | - 2 } {{ x } ^ { 2 } - x - 6 } = 0$, then $x$ is ____?", "answer": "2", "steps": "From the given information, we can deduce that $| x | - 2 = 0$ and $x ^ 2 - x - 6 \\neq 0$. Solving for $x$, we get $x = 2$.", "expr_cands": ["\\frac { | x | - 2 } { { x } ^ { 2 } - x - 6 } = 0", "x", "| x | - 2 = 0", "x = - 2", "x = 2", "x ^ { 2 } - x - 6 \\neq 0", "( - 2 < x \\wedge x < 3 )", "3 < x", "x < - 2"], "exprs": ["| x | - 2 = 0", "x ^ { 2 } - x - 6 \\neq 0", "x = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { | x | - 2 } { { x } ^ { 2 } - x - 6 } = 0"}, {"id": "| x | - 2 = 0"}, {"id": "分式有意义，则分母不为0"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "x = 2"}, {"id": "x ^ { 2 } - x - 6 \\neq 0"}], "links": [{"rel": "被描述", "source": "\\frac { | x | - 2 } { { x } ^ { 2 } - x - 6 } = 0", "target": "| x | - 2 = 0"}, {"rel": "被描述", "source": "\\frac { | x | - 2 } { { x } ^ { 2 } - x - 6 } = 0", "target": "x ^ { 2 } - x - 6 \\neq 0"}, {"rel": "联立", "source": "| x | - 2 = 0", "target": "x = 2"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "| x | - 2 = 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "x ^ { 2 } - x - 6 \\neq 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "| x | - 2 = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x ^ { 2 } - x - 6 \\neq 0"}, {"rel": "联立", "source": "x ^ { 2 } - x - 6 \\neq 0", "target": "x = 2"}]}}
{"content": "The equation $x + y = 7$ is a linear equation in two variables. When $x = 3$, the value of $y$ is ____?", "answer": "4", "steps": "When $x = 3$, $3 + y = 7$, so $y = 4$.", "expr_cands": ["x + y = 7", "y", "x", "x = 3", "3 + y = 7", "y = 4"], "exprs": ["y = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 + y = 7"}, {"id": "y = 4"}, {"id": "x = 3"}], "links": [{"rel": "被代入", "source": "3 + y = 7", "target": "y = 4"}, {"rel": "代入", "source": "x = 3", "target": "y = 4"}]}}
{"content": "The polynomial $5 x ^ { 3 } y ^ { m } - 3 x ^ { 2 } y + 6$ has the same degree as the monomial $7 ax ^ { 4 } y ^ { 3 }$. What is the value of $m$?", "answer": "5", "steps": "According to the problem, we have $m + 3 = 1 + 4 + 3$, which implies $m = 5$.", "expr_cands": ["5 x ^ { 3 } y ^ { m } - 3 x ^ { 2 } y + 6", "x", "m", "y", "7 ax ^ { 4 } y ^ { 3 }", "a", "m + 3 = 1 + 4 + 3", "m = 5"], "exprs": ["m + 3 = 1 + 4 + 3", "m = 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5 x ^ { 3 } y ^ { m } - 3 x ^ { 2 } y + 6"}, {"id": "m + 3 = 1 + 4 + 3"}, {"id": "7 ax ^ { 4 } y ^ { 3 }"}, {"id": "多项式 $5 x ^ { 3 } y ^ { m } - 3 x ^ { 2 } y + 6$ 与单项式 $7 ax ^ { 4 } y ^ { 3 }$ 的次数相同"}, {"id": "m = 5"}], "links": [{"rel": "被描述", "source": "5 x ^ { 3 } y ^ { m } - 3 x ^ { 2 } y + 6", "target": "m + 3 = 1 + 4 + 3"}, {"rel": "等式方程求解", "source": "m + 3 = 1 + 4 + 3", "target": "m = 5"}, {"rel": "被描述", "source": "7 ax ^ { 4 } y ^ { 3 }", "target": "m + 3 = 1 + 4 + 3"}, {"rel": "限制性描述", "source": "多项式 $5 x ^ { 3 } y ^ { m } - 3 x ^ { 2 } y + 6$ 与单项式 $7 ax ^ { 4 } y ^ { 3 }$ 的次数相同", "target": "m + 3 = 1 + 4 + 3"}]}}
{"content": "If the value of the polynomial $3 a - 2$ is $2$, then the value of the polynomial $6 a + 2$ is ____?", "answer": "10", "steps": "From the given information, we can obtain $3 a - 2 = 2$, which yields $a = \\frac { 4 } { 3 }$. When $a = \\frac { 4 } { 3 }$, we have $6 a + 2 = 6 * \\frac { 4 } { 3 } + 2 = 10$.", "expr_cands": ["3 a - 2", "a", "2", "6 a + 2", "3 a - 2 = 2", "a = \\frac { 4 } { 3 }", "10"], "exprs": ["3 a - 2 = 2", "a = \\frac { 4 } { 3 }", "10"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 a - 2"}, {"id": "3 a - 2 = 2"}, {"id": "2"}, {"id": "多项式 $3 a - 2$ 的值为 $2$"}, {"id": "a = \\frac { 4 } { 3 }"}, {"id": "6 a + 2"}, {"id": "10"}], "links": [{"rel": "被描述", "source": "3 a - 2", "target": "3 a - 2 = 2"}, {"rel": "等式方程求解", "source": "3 a - 2 = 2", "target": "a = \\frac { 4 } { 3 }"}, {"rel": "被描述", "source": "2", "target": "3 a - 2 = 2"}, {"rel": "限制性描述", "source": "多项式 $3 a - 2$ 的值为 $2$", "target": "3 a - 2 = 2"}, {"rel": "代入", "source": "a = \\frac { 4 } { 3 }", "target": "10"}, {"rel": "被代入", "source": "6 a + 2", "target": "10"}]}}
{"content": "Given $x - y = 1$, what is the value of the algebraic expression $5 - x + y$?", "answer": "4", "steps": "Because $x - y = 1$, then $5 - x + y = 5 - ( x - y ) = 5 - 1 = 4$.", "expr_cands": ["x - y = 1", "y", "x", "5 - x + y", "5 - ( x - y )", "4"], "exprs": ["5 - ( x - y )", "4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5 - x + y"}, {"id": "5 - ( x - y )"}, {"id": "x - y = 1"}, {"id": "4"}], "links": [{"rel": "提取因式", "source": "5 - x + y", "target": "5 - ( x - y )"}, {"rel": "被代入", "source": "5 - ( x - y )", "target": "4"}, {"rel": "提取因式参考", "source": "x - y = 1", "target": "5 - ( x - y )"}, {"rel": "代入", "source": "x - y = 1", "target": "4"}]}}
{"content": "The solution to the one-variable linear equation $9 x - 7 a = 2$ for $x = 1$ is $a =$ ____ ?", "answer": "1", "steps": "Substituting $x = 1$ into the equation $9 x - 7 a = 2$, we get $9 - 7 a = 2$. Solving for $a$, we get $a = 1$.", "expr_cands": ["x", "9 x - 7 a = 2", "a", "x = 1", "9 - 7 a = 2", "a = 1"], "exprs": ["9 - 7 a = 2", "a = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "9 x - 7 a = 2"}, {"id": "9 - 7 a = 2"}, {"id": "x = 1"}, {"id": "a = 1"}], "links": [{"rel": "被代入", "source": "9 x - 7 a = 2", "target": "9 - 7 a = 2"}, {"rel": "等式方程求解", "source": "9 - 7 a = 2", "target": "a = 1"}, {"rel": "代入", "source": "x = 1", "target": "9 - 7 a = 2"}]}}
{"content": "Given that $- 1$ is a root of the equation $x ^ 2 - 3 x + C = 0$ with respect to $x$, what is the value of $C$?", "answer": "- 4", "steps": "Substituting $x = - 1$ into $x ^ 2 - 3 x + C = 0$ yields $1 + 3 + C = 0$, so $C = - 4$.", "expr_cands": ["- 1", "x", "x ^ { 2 } - 3 x + C = 0", "C", "x = - 1", "C + 4 = 0", "1 + 3 + C = 0", "C = - 4"], "exprs": ["x = - 1", "1 + 3 + C = 0", "C = - 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 1"}, {"id": "x = - 1"}, {"id": "x ^ { 2 } - 3 x + C = 0"}, {"id": "x"}, {"id": "$- 1$ 是关于 $x$ 的方程 $x ^ { 2 } - 3 x + C = 0$ 的一个根"}, {"id": "1 + 3 + C = 0"}, {"id": "C = - 4"}], "links": [{"rel": "被描述", "source": "- 1", "target": "x = - 1"}, {"rel": "代入", "source": "x = - 1", "target": "1 + 3 + C = 0"}, {"rel": "被描述", "source": "x ^ { 2 } - 3 x + C = 0", "target": "x = - 1"}, {"rel": "被代入", "source": "x ^ { 2 } - 3 x + C = 0", "target": "1 + 3 + C = 0"}, {"rel": "被描述", "source": "x", "target": "x = - 1"}, {"rel": "限制性描述", "source": "$- 1$ 是关于 $x$ 的方程 $x ^ { 2 } - 3 x + C = 0$ 的一个根", "target": "x = - 1"}, {"rel": "等式方程求解", "source": "1 + 3 + C = 0", "target": "C = - 4"}]}}
{"content": "$7$, given the function $y = ( m + 2 ) x - 3$, to make the function value $y$ increase as the independent variable $x$ increases, the range of values for $m$ is ____?", "answer": "m > - 2", "steps": "$\\because$ In a linear function $y = ( m + 2 ) x - 3$, the value of $y$ increases as the value of $x$ increases. $\\therefore$ $m + 2 > 0$, which implies $m > - 2$.", "expr_cands": ["7", "y = ( m + 2 ) x - 3", "m", "y", "x", "m + 2 > 0", "- 2 < m", "m > - 2"], "exprs": ["m + 2 > 0", "m > - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = ( m + 2 ) x - 3"}, {"id": "m + 2 > 0"}, {"id": "函数 $y = ( m + 2 ) x - 3$"}, {"id": "要使函数值 $y$ 随自变量 $x$ 的增大而增大"}, {"id": "$m$ 的取值范围"}, {"id": "m > - 2"}], "links": [{"rel": "被描述", "source": "y = ( m + 2 ) x - 3", "target": "m + 2 > 0"}, {"rel": "不等式方程求解", "source": "m + 2 > 0", "target": "m > - 2"}, {"rel": "限制性描述", "source": "函数 $y = ( m + 2 ) x - 3$", "target": "m + 2 > 0"}, {"rel": "限制性描述", "source": "要使函数值 $y$ 随自变量 $x$ 的增大而增大", "target": "m + 2 > 0"}, {"rel": "限制性描述", "source": "$m$ 的取值范围", "target": "m + 2 > 0"}]}}
{"content": "Given the function $y = 2 x - 1$, if the function value is $1$ when $x = a$, then the value of $a$ is ____?", "answer": "1", "steps": "Substituting $x = a$, $y = 1$, we get $2 a - 1 = 1$. Solving for $a$, we get $a = 1$.", "expr_cands": ["y = 2 x - 1", "y", "x", "x = a", "a", "1", "y = 1", "2 a - 1 = 1", "a = 1"], "exprs": ["2 a - 1 = 1", "a = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = 2 x - 1"}, {"id": "2 a - 1 = 1"}, {"id": "x = a"}, {"id": "y = 1"}, {"id": "函数 $y = 2 x - 1$"}, {"id": "$x = a$ 时的函数值为 $1$"}, {"id": "a = 1"}], "links": [{"rel": "被描述", "source": "y = 2 x - 1", "target": "2 a - 1 = 1"}, {"rel": "等式方程求解", "source": "2 a - 1 = 1", "target": "a = 1"}, {"rel": "被描述", "source": "x = a", "target": "2 a - 1 = 1"}, {"rel": "被描述", "source": "y = 1", "target": "2 a - 1 = 1"}, {"rel": "限制性描述", "source": "函数 $y = 2 x - 1$", "target": "2 a - 1 = 1"}, {"rel": "限制性描述", "source": "$x = a$ 时的函数值为 $1$", "target": "2 a - 1 = 1"}]}}
{"content": "The equation about $x$, $- 5 x ^ { 3 m - 2 } + 2 m = 0$, is a one-variable linear equation about $x$. What is the solution to this equation? ____", "answer": "x = \\frac { 2 } { 5 }", "steps": "According to the problem, we have $3 m - 2 = 1$. Solving for $m$, we get $m = 1$. Substituting $m = 1$ back into the original equation, we have $- 5 x + 2 = 0$. Solving for $x$, we get $x = \\frac { 2 } { 5 }$.", "expr_cands": ["x", "- 5 x ^ { 3 m - 2 } + 2 m = 0", "m", "3 m - 2 = 1", "m = 1", "- 5 x + 2 = 0", "x = \\frac { 2 } { 5 }"], "exprs": ["3 m - 2 = 1", "m = 1", "- 5 x + 2 = 0", "x = \\frac { 2 } { 5 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 5 x ^ { 3 m - 2 } + 2 m = 0"}, {"id": "3 m - 2 = 1"}, {"id": "关于 $x$ 的方程 $- 5 x ^ { 3 m - 2 } + 2 m = 0$ 是关于 $x$ 的一元一次方程"}, {"id": "m = 1"}, {"id": "- 5 x + 2 = 0"}, {"id": "x = \\frac { 2 } { 5 }"}], "links": [{"rel": "被描述", "source": "- 5 x ^ { 3 m - 2 } + 2 m = 0", "target": "3 m - 2 = 1"}, {"rel": "被代入", "source": "- 5 x ^ { 3 m - 2 } + 2 m = 0", "target": "- 5 x + 2 = 0"}, {"rel": "等式方程求解", "source": "3 m - 2 = 1", "target": "m = 1"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $- 5 x ^ { 3 m - 2 } + 2 m = 0$ 是关于 $x$ 的一元一次方程", "target": "3 m - 2 = 1"}, {"rel": "代入", "source": "m = 1", "target": "- 5 x + 2 = 0"}, {"rel": "等式方程求解", "source": "- 5 x + 2 = 0", "target": "x = \\frac { 2 } { 5 }"}]}}
{"content": "Given $mn - n = 15$ and $m - mn = 6$, what is $- 2 mn + m + n$?", "answer": "- 9", "steps": "$\\because mn - n = 15$, $m - mn = 6$, $\\therefore$ the original expression is equal to $- ( mn - n ) + ( m - mn ) = - 15 + 6 = - 9$.", "expr_cands": ["mn - n = 15", "m", "n", "m - mn = 6", "- 2 mn + m + n", "- ( mn - n ) + ( m - mn )", "- 9"], "exprs": ["- ( mn - n ) + ( m - mn )", "- 9"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 2 mn + m + n"}, {"id": "- ( mn - n ) + ( m - mn )"}, {"id": "mn - n = 15"}, {"id": "m - mn = 6"}, {"id": "- 9"}], "links": [{"rel": "提取因式", "source": "- 2 mn + m + n", "target": "- ( mn - n ) + ( m - mn )"}, {"rel": "被代入", "source": "- ( mn - n ) + ( m - mn )", "target": "- 9"}, {"rel": "提取因式参考", "source": "mn - n = 15", "target": "- ( mn - n ) + ( m - mn )"}, {"rel": "代入", "source": "mn - n = 15", "target": "- 9"}, {"rel": "提取因式参考", "source": "m - mn = 6", "target": "- ( mn - n ) + ( m - mn )"}, {"rel": "代入", "source": "m - mn = 6", "target": "- 9"}]}}
{"content": "Given that the value of the algebraic expression $3 x - 12$ is the reciprocal of $- \\frac { 1 } { 3 }$, what is the value of $x$?", "answer": "3", "steps": "$\\because$ The value of the algebraic expression $3 x - 12$ is the reciprocal of $- \\frac { 1 } { 3 }$, $\\therefore$ $3 x - 12 = - 3$, solving for $x$ gives $x = 3$.", "expr_cands": ["3 x - 12", "x", "- \\frac { 1 } { 3 }", "3 x - 12 = - 3", "x = 3"], "exprs": ["3 x - 12 = - 3", "x = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 x - 12"}, {"id": "3 x - 12 = - 3"}, {"id": "- \\frac { 1 } { 3 }"}, {"id": "代数式 $3 x - 12$ 的值与 $- \\frac { 1 } { 3 }$ 互为倒数"}, {"id": "x = 3"}], "links": [{"rel": "被描述", "source": "3 x - 12", "target": "3 x - 12 = - 3"}, {"rel": "等式方程求解", "source": "3 x - 12 = - 3", "target": "x = 3"}, {"rel": "被描述", "source": "- \\frac { 1 } { 3 }", "target": "3 x - 12 = - 3"}, {"rel": "限制性描述", "source": "代数式 $3 x - 12$ 的值与 $- \\frac { 1 } { 3 }$ 互为倒数", "target": "3 x - 12 = - 3"}]}}
{"content": "If the fraction $\\frac { 3 x - 9 } { x + 2 }$ is undefined, then the value of $x$ is ____?", "answer": "- 2", "steps": "$\\because$ The fraction $\\frac { 3 x - 9 } { x + 2 }$ is undefined, $\\therefore$ $x + 2 = 0$, solving for $x$ gives $x = - 2$.", "expr_cands": ["\\frac { 3 x - 9 } { x + 2 }", "x", "x + 2 = 0", "x = - 2"], "exprs": ["x + 2 = 0", "x = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 3 x - 9 } { x + 2 }"}, {"id": "x + 2 = 0"}, {"id": "分式 $\\frac { 3 x - 9 } { x + 2 }$ 无意义"}, {"id": "分式有意义，则分母不为0"}, {"id": "x = - 2"}], "links": [{"rel": "被描述", "source": "\\frac { 3 x - 9 } { x + 2 }", "target": "x + 2 = 0"}, {"rel": "等式方程求解", "source": "x + 2 = 0", "target": "x = - 2"}, {"rel": "限制性描述", "source": "分式 $\\frac { 3 x - 9 } { x + 2 }$ 无意义", "target": "x + 2 = 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "x + 2 = 0"}]}}
{"content": "Given $| 1 - a | = a - 1$, the possible values of $a$ are ____?", "answer": "a \\ge 1", "steps": "Since $| 1 - a | = a - 1$, it follows that $1 - a \\leq 0$, which implies that $a \\geq 1$.", "expr_cands": ["| 1 - a | = a - 1", "a", "1 - a \\le 0", "1 \\le a", "a \\ge 1"], "exprs": ["1 - a \\le 0", "a \\ge 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| 1 - a | = a - 1"}, {"id": "1 - a \\le 0"}, {"id": "绝对值恒大于等于0"}, {"id": "a \\ge 1"}], "links": [{"rel": "被描述", "source": "| 1 - a | = a - 1", "target": "1 - a \\le 0"}, {"rel": "不等式方程求解", "source": "1 - a \\le 0", "target": "a \\ge 1"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "1 - a \\le 0"}]}}
{"content": "The two square roots of a positive number are $2 a + 1$ and $4 - 3 a$. What is this positive number?", "answer": "121", "steps": "According to the problem, we have $2 a + 1 + 4 - 3 a = 0$. Solving for $a$, we get $a = 5$. Therefore, the two square roots of this positive number are $11$ and $- 11$. Thus, the positive number is $121$.", "expr_cands": ["2 a + 1", "a", "4 - 3 a", "2 a + 1 + 4 - 3 a = 0", "a = 5", "11", "- 11", "121"], "exprs": ["2 a + 1 + 4 - 3 a = 0", "a = 5", "11", "121"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 a + 1"}, {"id": "2 a + 1 + 4 - 3 a = 0"}, {"id": "4 - 3 a"}, {"id": "正数的两个平方根分别是 $2 a + 1$ 和 $4 - 3 a$"}, {"id": "平方根互为相反数"}, {"id": "a = 5"}, {"id": "11"}, {"id": "121"}, {"id": "平方"}], "links": [{"rel": "被描述", "source": "2 a + 1", "target": "2 a + 1 + 4 - 3 a = 0"}, {"rel": "被代入", "source": "2 a + 1", "target": "11"}, {"rel": "等式方程求解", "source": "2 a + 1 + 4 - 3 a = 0", "target": "a = 5"}, {"rel": "被描述", "source": "4 - 3 a", "target": "2 a + 1 + 4 - 3 a = 0"}, {"rel": "限制性描述", "source": "正数的两个平方根分别是 $2 a + 1$ 和 $4 - 3 a$", "target": "2 a + 1 + 4 - 3 a = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "2 a + 1 + 4 - 3 a = 0"}, {"rel": "代入", "source": "a = 5", "target": "11"}, {"rel": "被描述", "source": "11", "target": "121"}, {"rel": "限制性描述", "source": "平方", "target": "121"}]}}
{"content": "If $\\sqrt { 20 m }$ is a positive integer, what is the smallest positive integer value of $m$?", "answer": "5", "steps": "$\\because \\sqrt { 20 m }$ is a positive integer, $\\sqrt { 20 m } = 2 \\sqrt { 5 m }$, $\\therefore \\sqrt { 5 m }$ is also a positive integer. According to the problem, we can conclude that $5 m$ is the smallest perfect square. $\\therefore m = 5$.", "expr_cands": ["\\sqrt { 20 m }", "m", "2 \\sqrt { 5 m }", "\\sqrt { 5 m }", "5 m", "m = 5"], "exprs": ["2 \\sqrt { 5 m }", "m = 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { 20 m }"}, {"id": "2 \\sqrt { 5 m }"}, {"id": "$\\sqrt { 20 m }$ 是一个正整数"}, {"id": ", $\\sqrt { 5 m }$ 也是一个正整数"}, {"id": "m"}, {"id": "m = 5"}, {"id": "\\sqrt { 5 m }"}, {"id": "根据题意可得 $5 m$ 是一个最小的完全平方数"}, {"id": "正整数 $m$ 的最小值"}], "links": [{"rel": "被描述", "source": "\\sqrt { 20 m }", "target": "2 \\sqrt { 5 m }"}, {"rel": "限制性描述", "source": "$\\sqrt { 20 m }$ 是一个正整数", "target": "2 \\sqrt { 5 m }"}, {"rel": "限制性描述", "source": ", $\\sqrt { 5 m }$ 也是一个正整数", "target": "2 \\sqrt { 5 m }"}, {"rel": "被描述", "source": "m", "target": "m = 5"}, {"rel": "被描述", "source": "\\sqrt { 5 m }", "target": "m = 5"}, {"rel": "限制性描述", "source": "根据题意可得 $5 m$ 是一个最小的完全平方数", "target": "m = 5"}, {"rel": "限制性描述", "source": "正整数 $m$ 的最小值", "target": "m = 5"}]}}
{"content": "Given $x$ satisfies $( x + 3 ) ^ 3 = 27$, then $x$ equals ____?", "answer": "0", "steps": "Because the cube root of 27 is 3, therefore x + 3 = 3, therefore x = 0.", "expr_cands": ["x", "( x + 3 ) ^ { 3 } = 27", "27", "3", "x + 3 = 3", "x = 0"], "exprs": ["x = 0"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( x + 3 ) ^ { 3 } = 27"}, {"id": "x = 0"}], "links": [{"rel": "等式方程求解", "source": "( x + 3 ) ^ { 3 } = 27", "target": "x = 0"}]}}
{"content": "If real numbers $x$ and $y$ satisfy $y = \\sqrt { x - 9 } + \\sqrt { 9 - x } - 3$, then the cube root of $xy$ is ____?", "answer": "- 3", "steps": "From the given information, we can obtain that $x - 9 = 0$, which leads to $x = 9$. Therefore, $y = - 3$, and the cube root of $xy = - 27$ is $- 3$.", "expr_cands": ["x", "y", "y = \\sqrt { x - 9 } + \\sqrt { 9 - x } - 3", "xy", "x - 9 = 0", "x = 9", "y = - 3", "- 27", "- 3"], "exprs": ["x - 9 = 0", "x = 9", "y = - 3", "- 27", "- 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = \\sqrt { x - 9 } + \\sqrt { 9 - x } - 3"}, {"id": "x - 9 = 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x = 9"}, {"id": "y = - 3"}, {"id": "xy"}, {"id": "- 27"}, {"id": "- 3"}, {"id": "$xy$ 的立方根"}], "links": [{"rel": "被描述", "source": "y = \\sqrt { x - 9 } + \\sqrt { 9 - x } - 3", "target": "x - 9 = 0"}, {"rel": "被代入", "source": "y = \\sqrt { x - 9 } + \\sqrt { 9 - x } - 3", "target": "y = - 3"}, {"rel": "等式方程求解", "source": "x - 9 = 0", "target": "x = 9"}, {"rel": "限制性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 9 = 0"}, {"rel": "代入", "source": "x = 9", "target": "y = - 3"}, {"rel": "代入", "source": "x = 9", "target": "- 27"}, {"rel": "代入", "source": "y = - 3", "target": "- 27"}, {"rel": "被代入", "source": "xy", "target": "- 27"}, {"rel": "被描述", "source": "- 27", "target": "- 3"}, {"rel": "限制性描述", "source": "$xy$ 的立方根", "target": "- 3"}]}}
{"content": "Given that the equation $2 x - a + 3 = 0$ has a solution of $x = - 3$, what is the value of $a$?", "answer": "- 3", "steps": "Substituting $x = - 3$ into the equation $2 x - a + 3 = 0$ yields $- 6 - a + 3 = 0$, which can be solved to obtain $a = - 3$.", "expr_cands": ["x", "2 x - a + 3 = 0", "a", "x = - 3", "- a - 3 = 0", "- 6 - a + 3 = 0", "a = - 3"], "exprs": ["- 6 - a + 3 = 0", "a = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x - a + 3 = 0"}, {"id": "- 6 - a + 3 = 0"}, {"id": "x = - 3"}, {"id": "a = - 3"}], "links": [{"rel": "被代入", "source": "2 x - a + 3 = 0", "target": "- 6 - a + 3 = 0"}, {"rel": "等式方程求解", "source": "- 6 - a + 3 = 0", "target": "a = - 3"}, {"rel": "代入", "source": "x = - 3", "target": "- 6 - a + 3 = 0"}]}}
{"content": "The equation $( a - 3 ) x + 1 = 5$ is a linear equation in one variable $x$, then ____ ?", "answer": "a \\neq 3", "steps": "The equation $( a - 3 ) x + 1 = 5$ about $x$ is a linear equation with one variable. It follows that $a - 3 \\neq 0$, and we can solve for $a$ to get $a \\neq 3$.", "expr_cands": ["x", "( a - 3 ) x + 1 = 5", "a", "a - 3 \\neq 0", "a \\neq 3"], "exprs": ["a - 3 \\neq 0", "a \\neq 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( a - 3 ) x + 1 = 5"}, {"id": "a - 3 \\neq 0"}, {"id": "关于 $x$ 的方程 $( a - 3 ) x + 1 = 5$ 是一元一次方程"}, {"id": "a \\neq 3"}], "links": [{"rel": "被描述", "source": "( a - 3 ) x + 1 = 5", "target": "a - 3 \\neq 0"}, {"rel": "不等式方程求解", "source": "a - 3 \\neq 0", "target": "a \\neq 3"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 $( a - 3 ) x + 1 = 5$ 是一元一次方程", "target": "a - 3 \\neq 0"}]}}
{"content": "The degree of the monomial $4 xyz ^ 3$ is ____?", "answer": "5", "steps": "The degree of the monomial $4 xyz ^ 3$ is $1 + 1 + 3 = 5$.", "expr_cands": ["4 xyz ^ { 3 }", "x", "z", "y", "1 + 1 + 3", "5"], "exprs": ["1 + 1 + 3", "5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 xyz ^ { 3 }"}, {"id": "1 + 1 + 3"}, {"id": "单项式 $4 xyz ^ { 3 }$ 的次数"}, {"id": "5"}], "links": [{"rel": "被描述", "source": "4 xyz ^ { 3 }", "target": "1 + 1 + 3"}, {"rel": "计算", "source": "1 + 1 + 3", "target": "5"}, {"rel": "限制性描述", "source": "单项式 $4 xyz ^ { 3 }$ 的次数", "target": "1 + 1 + 3"}]}}
{"content": "When $a = 3$, $b = - 4$, and $c = - 5$, what is the value of $a + ( - b ) - ( - c )$?", "answer": "2", "steps": "When $a = 3$, $b = - 4$, and $c = - 5$, the original expression $a - b + c = 3 - ( - 4 ) - 5 = 3 + 4 - 5 = 2$.", "expr_cands": ["a = 3", "a", "b = - 4", "b", "c = - 5", "c", "a + ( - b ) - ( - c )", "a - b + c", "2"], "exprs": ["2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a = 3"}, {"id": "2"}, {"id": "b = - 4"}, {"id": "c = - 5"}, {"id": "a + ( - b ) - ( - c )"}], "links": [{"rel": "代入", "source": "a = 3", "target": "2"}, {"rel": "代入", "source": "b = - 4", "target": "2"}, {"rel": "代入", "source": "c = - 5", "target": "2"}, {"rel": "被代入", "source": "a + ( - b ) - ( - c )", "target": "2"}]}}
{"content": "If the value of the fraction $\\frac { x ^ 2 - 9 } { x + 3 }$ is $0$, then the possible values of $x$ are ____?", "answer": "3", "steps": "According to the problem, we have ${ x } ^ { 2 } - 9 = 0$ and $x + 3 \\neq 0$. Solving for $x$, we get $x = 3$.", "expr_cands": ["\\frac { x ^ { 2 } - 9 } { x + 3 }", "x", "0", "{ x } ^ { 2 } - 9 = 0", "x = - 3", "x = 3", "x + 3 \\neq 0", "x \\neq - 3"], "exprs": ["{ x } ^ { 2 } - 9 = 0", "x + 3 \\neq 0", "x = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x ^ { 2 } - 9 } { x + 3 }"}, {"id": "{ x } ^ { 2 } - 9 = 0"}, {"id": "0"}, {"id": "分式 $\\frac { x ^ { 2 } - 9 } { x + 3 }$ 的值为 $0$"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "x + 3 \\neq 0"}, {"id": "x = 3"}], "links": [{"rel": "被描述", "source": "\\frac { x ^ { 2 } - 9 } { x + 3 }", "target": "{ x } ^ { 2 } - 9 = 0"}, {"rel": "被描述", "source": "\\frac { x ^ { 2 } - 9 } { x + 3 }", "target": "x + 3 \\neq 0"}, {"rel": "联立", "source": "{ x } ^ { 2 } - 9 = 0", "target": "x = 3"}, {"rel": "被描述", "source": "0", "target": "{ x } ^ { 2 } - 9 = 0"}, {"rel": "被描述", "source": "0", "target": "x + 3 \\neq 0"}, {"rel": "限制性描述", "source": "分式 $\\frac { x ^ { 2 } - 9 } { x + 3 }$ 的值为 $0$", "target": "{ x } ^ { 2 } - 9 = 0"}, {"rel": "限制性描述", "source": "分式 $\\frac { x ^ { 2 } - 9 } { x + 3 }$ 的值为 $0$", "target": "x + 3 \\neq 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "{ x } ^ { 2 } - 9 = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x + 3 \\neq 0"}, {"rel": "联立", "source": "x + 3 \\neq 0", "target": "x = 3"}]}}
{"content": "After factoring out the common factor $x ^ n$, the other factor of the polynomial $x ^ { 2 n } - x ^ n$ is ____?", "answer": "x ^ { n } - 1", "steps": "$\\because x ^ { 2 n } - x ^ n = x ^ n ( x ^ n - 1 )$, $\\therefore$ the polynomial $x ^ { 2 n } - x ^ n$ can be factored as $x ^ n ( x ^ n - 1 )$, where the other factor is $x ^ n - 1$.", "expr_cands": ["x ^ { 2 n } - x ^ { n }", "n", "x", "x ^ { n }", "x ^ { n } ( x ^ { n } - 1 )", "x ^ { n } - 1"], "exprs": ["x ^ { n } - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { n }"}, {"id": "x ^ { n } - 1"}, {"id": "x ^ { 2 n } - x ^ { n }"}, {"id": "多项式 $x ^ { 2 n } - x ^ { n }$ 提取公因式 $x ^ { n }$ 后"}], "links": [{"rel": "被描述", "source": "x ^ { n }", "target": "x ^ { n } - 1"}, {"rel": "被描述", "source": "x ^ { 2 n } - x ^ { n }", "target": "x ^ { n } - 1"}, {"rel": "限制性描述", "source": "多项式 $x ^ { 2 n } - x ^ { n }$ 提取公因式 $x ^ { n }$ 后", "target": "x ^ { n } - 1"}]}}
{"content": "If $x = - 1$ is a solution of the equation $2 x - 3 a = 7$, then $a$ = ____ ?", "answer": "- 3", "steps": "Substituting $x = - 1$ into the equation gives $- 2 - 3 a = 7$, solving for $a$ gives $a = - 3$.", "expr_cands": ["x = - 1", "x", "2 x - 3 a = 7", "a", "- 2 - 3 a = 7", "a = - 3"], "exprs": ["- 2 - 3 a = 7", "a = - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x = - 1"}, {"id": "- 2 - 3 a = 7"}, {"id": "2 x - 3 a = 7"}, {"id": "$x = - 1$ 是方程 $2 x - 3 a = 7$ 的解"}, {"id": "a = - 3"}], "links": [{"rel": "被描述", "source": "x = - 1", "target": "- 2 - 3 a = 7"}, {"rel": "等式方程求解", "source": "- 2 - 3 a = 7", "target": "a = - 3"}, {"rel": "被描述", "source": "2 x - 3 a = 7", "target": "- 2 - 3 a = 7"}, {"rel": "限制性描述", "source": "$x = - 1$ 是方程 $2 x - 3 a = 7$ 的解", "target": "- 2 - 3 a = 7"}]}}
{"content": "Given that the reciprocal of $\\frac { 2 } { a + 1 }$ is the opposite of $\\frac { 5 - 2 a } { 3 } - 1$, find the value of $a$.", "answer": "7", "steps": "The reciprocal of $\\frac { 2 } { a + 1 }$ is $\\frac { a + 1 } { 2 }$. From the given information, we have $\\frac { a + 1 } { 2 } + \\frac { 5 - 2 a } { 3 } - 1 = 0$. Solving for $a$, we get $a = 7$.", "expr_cands": ["\\frac { 2 } { a + 1 }", "a", "\\frac { 5 - 2 a } { 3 } - 1", "\\frac { a + 1 } { 2 }", "\\frac { a + 1 } { 2 } + \\frac { 5 - 2 a } { 3 } - 1 = 0", "a = 7"], "exprs": ["\\frac { a + 1 } { 2 } + \\frac { 5 - 2 a } { 3 } - 1 = 0", "a = 7"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 5 - 2 a } { 3 } - 1"}, {"id": "\\frac { a + 1 } { 2 } + \\frac { 5 - 2 a } { 3 } - 1 = 0"}, {"id": "\\frac { 2 } { a + 1 }"}, {"id": "$\\frac { 2 } { a + 1 }$ 的倒数与 $\\frac { 5 - 2 a } { 3 } - 1$ 互为相反数"}, {"id": "a = 7"}], "links": [{"rel": "被描述", "source": "\\frac { 5 - 2 a } { 3 } - 1", "target": "\\frac { a + 1 } { 2 } + \\frac { 5 - 2 a } { 3 } - 1 = 0"}, {"rel": "等式方程求解", "source": "\\frac { a + 1 } { 2 } + \\frac { 5 - 2 a } { 3 } - 1 = 0", "target": "a = 7"}, {"rel": "被描述", "source": "\\frac { 2 } { a + 1 }", "target": "\\frac { a + 1 } { 2 } + \\frac { 5 - 2 a } { 3 } - 1 = 0"}, {"rel": "限制性描述", "source": "$\\frac { 2 } { a + 1 }$ 的倒数与 $\\frac { 5 - 2 a } { 3 } - 1$ 互为相反数", "target": "\\frac { a + 1 } { 2 } + \\frac { 5 - 2 a } { 3 } - 1 = 0"}]}}
{"content": "Given that the equation $x ^ 2 - 6 x - m = 0$ has a root of $- 2$ for $x$, the other root is:", "answer": "8", "steps": "Assuming the other root of the equation is $x _ 1$, we have $x _ 1 + ( - 2 ) = 6$. Solving for $x _ 1$, we get $x _ 1 = 8$.", "expr_cands": ["x", "{ x } ^ { 2 } - 6 x - m = 0", "m", "- 2", "x _ { 1 }", "x _ { 1 } + ( - 2 ) = 6", "x_{1} = 8", "x _ { 1 } = 8"], "exprs": ["x _ { 1 }", "x _ { 1 } + ( - 2 ) = 6", "x _ { 1 } = 8"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "设方程另一个根为 $x _ { 1 }$"}, {"id": "x _ { 1 }"}, {"id": "- 2"}, {"id": "x _ { 1 } + ( - 2 ) = 6"}, {"id": "{ x } ^ { 2 } - 6 x - m = 0"}, {"id": "x"}, {"id": "关于 $x$ 的方程 ${ x } ^ { 2 } - 6 x - m = 0$ 有一个根为 $- 2$"}, {"id": "一元二次方程根与系数关系，两根之和"}, {"id": "x _ { 1 } = 8"}], "links": [{"rel": "假设描述", "source": "设方程另一个根为 $x _ { 1 }$", "target": "x _ { 1 }"}, {"rel": "限制性描述", "source": "设方程另一个根为 $x _ { 1 }$", "target": "x _ { 1 } + ( - 2 ) = 6"}, {"rel": "被描述", "source": "x _ { 1 }", "target": "x _ { 1 } + ( - 2 ) = 6"}, {"rel": "被描述", "source": "- 2", "target": "x _ { 1 } + ( - 2 ) = 6"}, {"rel": "等式方程求解", "source": "x _ { 1 } + ( - 2 ) = 6", "target": "x _ { 1 } = 8"}, {"rel": "被描述", "source": "{ x } ^ { 2 } - 6 x - m = 0", "target": "x _ { 1 } + ( - 2 ) = 6"}, {"rel": "被描述", "source": "x", "target": "x _ { 1 } + ( - 2 ) = 6"}, {"rel": "限制性描述", "source": "关于 $x$ 的方程 ${ x } ^ { 2 } - 6 x - m = 0$ 有一个根为 $- 2$", "target": "x _ { 1 } + ( - 2 ) = 6"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "x _ { 1 } + ( - 2 ) = 6"}]}}
{"content": "$11$, If the value of $x ^ 2 - 2 x - 1$ is $2$, then the value of $3 x ^ 2 - 6 x$ is ____?", "answer": "9", "steps": "$\\because$ The value of the algebraic expression $x ^ 2 - 2 x - 1$ is equal to $2$, $\\therefore$ $x ^ 2 - 2 x = 3$, $\\therefore$ $3 x ^ 2 - 6 x = 3 ( x ^ 2 - 2 x ) = 3 * 3 = 9$.", "expr_cands": ["11", "x ^ { 2 } - 2 x - 1", "x", "2", "3 x ^ { 2 } - 6 x", "x ^ { 2 } - 2 x = 3", "x = - 1", "x = 3", "3 ( x ^ { 2 } - 2 x )", "9"], "exprs": ["x ^ { 2 } - 2 x = 3", "3 ( x ^ { 2 } - 2 x )", "9"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } - 2 x - 1"}, {"id": "x ^ { 2 } - 2 x = 3"}, {"id": "2"}, {"id": "$x ^ { 2 } - 2 x - 1$ 的值为 $2$"}, {"id": "3 x ^ { 2 } - 6 x"}, {"id": "3 ( x ^ { 2 } - 2 x )"}, {"id": "9"}], "links": [{"rel": "被描述", "source": "x ^ { 2 } - 2 x - 1", "target": "x ^ { 2 } - 2 x = 3"}, {"rel": "提取因式参考", "source": "x ^ { 2 } - 2 x = 3", "target": "3 ( x ^ { 2 } - 2 x )"}, {"rel": "代入", "source": "x ^ { 2 } - 2 x = 3", "target": "9"}, {"rel": "被描述", "source": "2", "target": "x ^ { 2 } - 2 x = 3"}, {"rel": "限制性描述", "source": "$x ^ { 2 } - 2 x - 1$ 的值为 $2$", "target": "x ^ { 2 } - 2 x = 3"}, {"rel": "提取因式", "source": "3 x ^ { 2 } - 6 x", "target": "3 ( x ^ { 2 } - 2 x )"}, {"rel": "被代入", "source": "3 ( x ^ { 2 } - 2 x )", "target": "9"}]}}
{"content": "Given that the algebraic expressions $- 6 a ^ mb ^ 3$ and $\\frac { 1 } { 3 } ab ^ n$ are like terms, what is the value of $m - n$?", "answer": "- 2", "steps": "From the given information, we have $m = 1$ and $n = 3$. Therefore, $m - n = 1 - 3 = - 2$.", "expr_cands": ["- 6 a ^ { m } b ^ { 3 }", "b", "a", "m", "\\frac { 1 } { 3 } ab ^ { n }", "n", "m - n", "m = 1", "n = 3", "- 2"], "exprs": ["m = 1", "n = 3", "- 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 6 a ^ { m } b ^ { 3 }"}, {"id": "m = 1"}, {"id": "\\frac { 1 } { 3 } ab ^ { n }"}, {"id": "代数式 $- 6 a ^ { m } b ^ { 3 }$ 和 $\\frac { 1 } { 3 } ab ^ { n }$ 是同类项"}, {"id": "n = 3"}, {"id": "m - n"}, {"id": "- 2"}], "links": [{"rel": "被描述", "source": "- 6 a ^ { m } b ^ { 3 }", "target": "m = 1"}, {"rel": "被描述", "source": "- 6 a ^ { m } b ^ { 3 }", "target": "n = 3"}, {"rel": "代入", "source": "m = 1", "target": "- 2"}, {"rel": "被描述", "source": "\\frac { 1 } { 3 } ab ^ { n }", "target": "m = 1"}, {"rel": "被描述", "source": "\\frac { 1 } { 3 } ab ^ { n }", "target": "n = 3"}, {"rel": "限制性描述", "source": "代数式 $- 6 a ^ { m } b ^ { 3 }$ 和 $\\frac { 1 } { 3 } ab ^ { n }$ 是同类项", "target": "m = 1"}, {"rel": "限制性描述", "source": "代数式 $- 6 a ^ { m } b ^ { 3 }$ 和 $\\frac { 1 } { 3 } ab ^ { n }$ 是同类项", "target": "n = 3"}, {"rel": "代入", "source": "n = 3", "target": "- 2"}, {"rel": "被代入", "source": "m - n", "target": "- 2"}]}}
{"content": "To make the square root expression $\\sqrt { 2 x - 4 }$ meaningful, the range of values for the variable $x$ must satisfy the condition ____?", "answer": "x \\ge 2", "steps": "From the given condition, we have $2 x - 4 \\geq 0$, which implies $x \\geq 2$.", "expr_cands": ["\\sqrt { 2 x - 4 }", "x", "2 x - 4 \\ge 0", "2 \\le x", "x \\ge 2"], "exprs": ["2 x - 4 \\ge 0", "x \\ge 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { 2 x - 4 }"}, {"id": "2 x - 4 \\ge 0"}, {"id": "要使二次根式 $\\sqrt { 2 x - 4 }$ 有意义"}, {"id": "字母 $x$ 的取值范围必须满足条件"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x \\ge 2"}], "links": [{"rel": "被描述", "source": "\\sqrt { 2 x - 4 }", "target": "2 x - 4 \\ge 0"}, {"rel": "不等式方程求解", "source": "2 x - 4 \\ge 0", "target": "x \\ge 2"}, {"rel": "限制性描述", "source": "要使二次根式 $\\sqrt { 2 x - 4 }$ 有意义", "target": "2 x - 4 \\ge 0"}, {"rel": "限制性描述", "source": "字母 $x$ 的取值范围必须满足条件", "target": "2 x - 4 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "2 x - 4 \\ge 0"}]}}
{"content": "Solve the equation $\\frac { x + 2 } { x - 1 } = \\frac { m + 1 } { x - 1 }$ for $x$ and find the value of $m$ that produces an increase in the number of roots.", "answer": "2", "steps": "Multiply both sides of the equation by $( x - 1 )$, we get $x + 2 = m + 1$. Since the equation has an extraneous root, the simplest common denominator is $x - 1 = 0$, which means the extraneous root is $x = 1$. Substituting $x = 1$ into the polynomial equation, we get $m = 2$.", "expr_cands": ["x", "\\frac { x + 2 } { x - 1 } = \\frac { m + 1 } { x - 1 }", "m", "( x - 1 )", "x + 2 = m + 1", "x - 1 = 0", "x = 1", "m = 2"], "exprs": ["x + 2 = m + 1", "x - 1 = 0", "x = 1", "m = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x + 2 } { x - 1 } = \\frac { m + 1 } { x - 1 }"}, {"id": "x + 2 = m + 1"}, {"id": "x - 1 = 0"}, {"id": "解关于 $x$ 的方程 $\\frac { x + 2 } { x - 1 } = \\frac { m + 1 } { x - 1 }$ 产生增根"}, {"id": "分式有增根，则分母为0"}, {"id": "x = 1"}, {"id": "m = 2"}], "links": [{"rel": "同乘除", "source": "\\frac { x + 2 } { x - 1 } = \\frac { m + 1 } { x - 1 }", "target": "x + 2 = m + 1"}, {"rel": "被描述", "source": "\\frac { x + 2 } { x - 1 } = \\frac { m + 1 } { x - 1 }", "target": "x - 1 = 0"}, {"rel": "联立", "source": "x + 2 = m + 1", "target": "m = 2"}, {"rel": "等式方程求解", "source": "x - 1 = 0", "target": "x = 1"}, {"rel": "限制性描述", "source": "解关于 $x$ 的方程 $\\frac { x + 2 } { x - 1 } = \\frac { m + 1 } { x - 1 }$ 产生增根", "target": "x - 1 = 0"}, {"rel": "属性描述", "source": "分式有增根，则分母为0", "target": "x - 1 = 0"}, {"rel": "联立", "source": "x = 1", "target": "m = 2"}]}}
{"content": "The solution to the equation $5 x - 6 = 0$ is $x$ = ____ ?", "answer": "\\frac { 6 } { 5 }", "steps": "From the original equation, we have $5 x = 6$. Dividing both sides by $5$, we get $x = \\frac { 6 } { 5 }$.", "expr_cands": ["5 x - 6 = 0", "x", "5 x = 6", "x = \\frac { 6 } { 5 }", "1"], "exprs": ["x = \\frac { 6 } { 5 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5 x - 6 = 0"}, {"id": "x = \\frac { 6 } { 5 }"}], "links": [{"rel": "等式方程求解", "source": "5 x - 6 = 0", "target": "x = \\frac { 6 } { 5 }"}]}}
{"content": "If the solution to the inequality $( a + 1 ) x > a + 1$ is $x < 1$, then $a$ satisfies ____?", "answer": "a < - 1", "steps": "$\\because$ The solution to the inequality $( a + 1 ) x > a + 1$ is $x < 1$, $\\therefore$ $a + 1 < 0$, and solving for $a$ gives $a < - 1$.", "expr_cands": ["( a + 1 ) x > a + 1", "a", "x", "x < 1", "a + 1 < 0", "a < - 1"], "exprs": ["a + 1 < 0", "a < - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( a + 1 ) x > a + 1"}, {"id": "a + 1 < 0"}, {"id": "x < 1"}, {"id": "不等式 $( a + 1 ) x > a + 1$ 的解是 $x < 1$"}, {"id": "不等式两边都乘或除同一个负数，不等号的方向改变"}, {"id": "a < - 1"}], "links": [{"rel": "被描述", "source": "( a + 1 ) x > a + 1", "target": "a + 1 < 0"}, {"rel": "不等式方程求解", "source": "a + 1 < 0", "target": "a < - 1"}, {"rel": "被描述", "source": "x < 1", "target": "a + 1 < 0"}, {"rel": "限制性描述", "source": "不等式 $( a + 1 ) x > a + 1$ 的解是 $x < 1$", "target": "a + 1 < 0"}, {"rel": "属性描述", "source": "不等式两边都乘或除同一个负数，不等号的方向改变", "target": "a + 1 < 0"}]}}
{"content": "If ${ x } ^ { 3 } + a { x } ^ { 2 } + bx + 8$ has two factors $x + 1$ and $x + 2$, then $a + b$ = ____ ?", "answer": "21", "steps": "Let $x ^ { 3 } + ax ^ { 2 } + bx + 8 = ( x + 1 ) ( x + 2 ) ( x + c ) = x ^ { 3 } + ( 3 + c ) x ^ { 2 } + ( 2 + 3 c ) x + 2 c$ , $\\therefore c = 4$ , so $a = 7$ , $b = 14$ , $\\therefore a + b = 21$ .", "expr_cands": ["{ x } ^ { 3 } + a { x } ^ { 2 } + bx + 8", "x", "a", "b", "x + 1", "x + 2", "a + b", "x ^ { 3 } + ax ^ { 2 } + bx + 8 = x ^ { 3 } + ( 3 + c ) x ^ { 2 } + ( 2 + 3 c ) x + 2 c", "c", "c = 4", "a = 7", "b = 14", "21"], "exprs": ["x ^ { 3 } + ax ^ { 2 } + bx + 8 = x ^ { 3 } + ( 3 + c ) x ^ { 2 } + ( 2 + 3 c ) x + 2 c", "c = 4", "a = 7", "b = 14", "21"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "设 $x ^ { 3 } + ax ^ { 2 } + bx + 8 = ( x + 1 ) ( x + 2 ) ( x + c ) = x ^ { 3 } + ( 3 + c ) x ^ { 2 } + ( 2 + 3 c ) x + 2 c$"}, {"id": "x ^ { 3 } + ax ^ { 2 } + bx + 8 = x ^ { 3 } + ( 3 + c ) x ^ { 2 } + ( 2 + 3 c ) x + 2 c"}, {"id": "c = 4"}, {"id": "a = 7"}, {"id": "b = 14"}, {"id": "a + b"}, {"id": "21"}], "links": [{"rel": "假设描述", "source": "设 $x ^ { 3 } + ax ^ { 2 } + bx + 8 = ( x + 1 ) ( x + 2 ) ( x + c ) = x ^ { 3 } + ( 3 + c ) x ^ { 2 } + ( 2 + 3 c ) x + 2 c$", "target": "x ^ { 3 } + ax ^ { 2 } + bx + 8 = x ^ { 3 } + ( 3 + c ) x ^ { 2 } + ( 2 + 3 c ) x + 2 c"}, {"rel": "移项", "source": "x ^ { 3 } + ax ^ { 2 } + bx + 8 = x ^ { 3 } + ( 3 + c ) x ^ { 2 } + ( 2 + 3 c ) x + 2 c", "target": "c = 4"}, {"rel": "移项", "source": "x ^ { 3 } + ax ^ { 2 } + bx + 8 = x ^ { 3 } + ( 3 + c ) x ^ { 2 } + ( 2 + 3 c ) x + 2 c", "target": "a = 7"}, {"rel": "移项", "source": "x ^ { 3 } + ax ^ { 2 } + bx + 8 = x ^ { 3 } + ( 3 + c ) x ^ { 2 } + ( 2 + 3 c ) x + 2 c", "target": "b = 14"}, {"rel": "代入", "source": "a = 7", "target": "21"}, {"rel": "代入", "source": "b = 14", "target": "21"}, {"rel": "被代入", "source": "a + b", "target": "21"}]}}
{"content": "If two different square roots of a positive number are $5$ and $3 m + 1$, then $m$ = ____ ?", "answer": "- 2", "steps": "From the given information, we can obtain that $5 + 3 m + 1 = 0$, which leads to the solution $m = - 2$.", "expr_cands": ["5", "3 m + 1", "m", "5 + 3 m + 1 = 0", "m = - 2"], "exprs": ["5 + 3 m + 1 = 0", "m = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5"}, {"id": "5 + 3 m + 1 = 0"}, {"id": "3 m + 1"}, {"id": "一个正数的两个不同的平方根分别是 $5$ 和 $3 m + 1$"}, {"id": "平方根互为相反数"}, {"id": "m = - 2"}], "links": [{"rel": "被描述", "source": "5", "target": "5 + 3 m + 1 = 0"}, {"rel": "等式方程求解", "source": "5 + 3 m + 1 = 0", "target": "m = - 2"}, {"rel": "被描述", "source": "3 m + 1", "target": "5 + 3 m + 1 = 0"}, {"rel": "限制性描述", "source": "一个正数的两个不同的平方根分别是 $5$ 和 $3 m + 1$", "target": "5 + 3 m + 1 = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "5 + 3 m + 1 = 0"}]}}
{"content": "$14$. If the expansion of $( 2 x + m ) ( x - 5 )$ does not contain a linear term in $x$, then $m$ = ____?", "answer": "10", "steps": "$\\because$ $( 2 x + m ) ( x - 5 ) = 2 x ^ 2 - 10 x + mx - 5 m = 2 x ^ 2 + ( m - 10 ) x - 5 m$, and $\\because$ there is no linear term in $x$ in the result, $\\therefore$ $m - 10 = 0$, $\\therefore$ $m = 10$.", "expr_cands": ["14", "( 2 x + m ) ( x - 5 )", "x", "m", "2 x ^ { 2 } + ( m - 10 ) x - 5 m", "m - 10 = 0", "m = 10"], "exprs": ["2 x ^ { 2 } + ( m - 10 ) x - 5 m", "m - 10 = 0", "m = 10"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( 2 x + m ) ( x - 5 )"}, {"id": "2 x ^ { 2 } + ( m - 10 ) x - 5 m"}, {"id": "x"}, {"id": "m - 10 = 0"}, {"id": "$( 2 x + m ) ( x - 5 )$ 的展开式中不含 $x$ 的一次项"}, {"id": "m = 10"}], "links": [{"rel": "提取因式", "source": "( 2 x + m ) ( x - 5 )", "target": "2 x ^ { 2 } + ( m - 10 ) x - 5 m"}, {"rel": "被描述", "source": "2 x ^ { 2 } + ( m - 10 ) x - 5 m", "target": "m - 10 = 0"}, {"rel": "提取因式参考", "source": "x", "target": "2 x ^ { 2 } + ( m - 10 ) x - 5 m"}, {"rel": "等式方程求解", "source": "m - 10 = 0", "target": "m = 10"}, {"rel": "限制性描述", "source": "$( 2 x + m ) ( x - 5 )$ 的展开式中不含 $x$ 的一次项", "target": "m - 10 = 0"}]}}
{"content": "Given that $- 25 a ^ { 2 m } b$ and $7 b ^ { 3 - n } a ^ 4$ are like terms, what is the value of $m + n$?", "answer": "4", "steps": "According to the problem, we have $2 m = 4$ and $3 - n = 1$. Solving for $m$ and $n$, we get $m = 2$ and $n = 2$. Therefore, $m + n = 4$.", "expr_cands": ["- 25 a ^ { 2 m } b", "b", "m", "a", "7 b ^ { 3 - n } a ^ { 4 }", "n", "m + n", "2 m = 4", "m = 2", "3 - n = 1", "n = 2", "4"], "exprs": ["2 m = 4", "3 - n = 1", "m = 2", "n = 2", "4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 25 a ^ { 2 m } b"}, {"id": "2 m = 4"}, {"id": "7 b ^ { 3 - n } a ^ { 4 }"}, {"id": "$- 25 a ^ { 2 m } b$ 和 $7 b ^ { 3 - n } a ^ { 4 }$ 是同类项"}, {"id": "3 - n = 1"}, {"id": "m = 2"}, {"id": "n = 2"}, {"id": "m + n"}, {"id": "4"}], "links": [{"rel": "被描述", "source": "- 25 a ^ { 2 m } b", "target": "2 m = 4"}, {"rel": "被描述", "source": "- 25 a ^ { 2 m } b", "target": "3 - n = 1"}, {"rel": "等式方程求解", "source": "2 m = 4", "target": "m = 2"}, {"rel": "被描述", "source": "7 b ^ { 3 - n } a ^ { 4 }", "target": "2 m = 4"}, {"rel": "被描述", "source": "7 b ^ { 3 - n } a ^ { 4 }", "target": "3 - n = 1"}, {"rel": "限制性描述", "source": "$- 25 a ^ { 2 m } b$ 和 $7 b ^ { 3 - n } a ^ { 4 }$ 是同类项", "target": "2 m = 4"}, {"rel": "限制性描述", "source": "$- 25 a ^ { 2 m } b$ 和 $7 b ^ { 3 - n } a ^ { 4 }$ 是同类项", "target": "3 - n = 1"}, {"rel": "等式方程求解", "source": "3 - n = 1", "target": "n = 2"}, {"rel": "代入", "source": "m = 2", "target": "4"}, {"rel": "代入", "source": "n = 2", "target": "4"}, {"rel": "被代入", "source": "m + n", "target": "4"}]}}
{"content": "If $a - b = 3$, $a - c = 1$, then the value of $( 2 a - b - c ) ^ 2 + ( b - c ) ^ 2$ is ____?", "answer": "20", "steps": "$\\because a - b = 3$, $a - c = 1$, $\\therefore 2 a - b - c = 4$, $b - c = - 2$, then the original expression $= 16 + 4 = 20$.", "expr_cands": ["a - b = 3", "b", "a", "a - c = 1", "c", "( 2 a - b - c ) ^ { 2 } + ( b - c ) ^ { 2 }", "2 a - b - c = 4", "b - c = - 2", "16 + 4", "20"], "exprs": ["2 a - b - c = 4", "b - c = - 2", "20"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a - b = 3"}, {"id": "2 a - b - c = 4"}, {"id": "a - c = 1"}, {"id": "b - c = - 2"}, {"id": "( 2 a - b - c ) ^ { 2 } + ( b - c ) ^ { 2 }"}, {"id": "20"}], "links": [{"rel": "联立", "source": "a - b = 3", "target": "2 a - b - c = 4"}, {"rel": "联立", "source": "a - b = 3", "target": "b - c = - 2"}, {"rel": "代入", "source": "2 a - b - c = 4", "target": "20"}, {"rel": "联立", "source": "a - c = 1", "target": "2 a - b - c = 4"}, {"rel": "联立", "source": "a - c = 1", "target": "b - c = - 2"}, {"rel": "代入", "source": "b - c = - 2", "target": "20"}, {"rel": "被代入", "source": "( 2 a - b - c ) ^ { 2 } + ( b - c ) ^ { 2 }", "target": "20"}]}}
{"content": "If the value of the fraction $\\frac { x ^ 2 + 1 } { 2 x + 3 }$ is negative, then the range of possible values for $x$ is ____?", "answer": "x < - \\frac { 3 } { 2 }", "steps": "$\\because$ The value of the fraction $\\frac { x ^ 2 + 1 } { 2 x + 3 }$ is negative, $\\therefore$ $x ^ 2 + 1$ must be greater than $0$, which means $2 x + 3 < 0$. Solving for $x$, we get $x < - \\frac { 3 } { 2 }$.", "expr_cands": ["\\frac { x ^ { 2 } + 1 } { 2 x + 3 }", "x", "x ^ { 2 } + 1", "0", "2 x + 3 < 0", "x < - \\frac { 3 } { 2 }"], "exprs": ["2 x + 3 < 0", "x < - \\frac { 3 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x ^ { 2 } + 1 } { 2 x + 3 }"}, {"id": "2 x + 3 < 0"}, {"id": "分式 $\\frac { x ^ { 2 } + 1 } { 2 x + 3 }$ 的值是负数"}, {"id": "分式为负数，则分子分母异号"}, {"id": "多项式偶次方项恒大于等于0"}, {"id": "x < - \\frac { 3 } { 2 }"}], "links": [{"rel": "被描述", "source": "\\frac { x ^ { 2 } + 1 } { 2 x + 3 }", "target": "2 x + 3 < 0"}, {"rel": "不等式方程求解", "source": "2 x + 3 < 0", "target": "x < - \\frac { 3 } { 2 }"}, {"rel": "限制性描述", "source": "分式 $\\frac { x ^ { 2 } + 1 } { 2 x + 3 }$ 的值是负数", "target": "2 x + 3 < 0"}, {"rel": "属性描述", "source": "分式为负数，则分子分母异号", "target": "2 x + 3 < 0"}, {"rel": "属性描述", "source": "多项式偶次方项恒大于等于0", "target": "2 x + 3 < 0"}]}}
{"content": "If all the solutions of the inequality $2 x < 4$ can make the linear inequality $3 x < a + 5$ hold true, then the possible range of $a$ is ____?", "answer": "a \\ge 1", "steps": "The solution set of the inequality $2 x < 4$ is $x < 2$, and the solution set of the inequality $3 x < a + 5$ is $x < \\frac { a + 5 } { 3 }$. Since all solutions of the equation $2 x < 4$ also satisfy the linear inequality $3 x < a + 5$, we have $\\frac { a + 5 } { 3 } \\geq 2$. Solving for $a$, we get $a \\geq 1$.", "expr_cands": ["2 x < 4", "x", "3 x < a + 5", "a", "x < 2", "x < \\frac { a + 5 } { 3 }", "\\frac { a + 5 } { 3 } \\ge 2", "1 \\le a", "a \\ge 1"], "exprs": ["x < 2", "x < \\frac { a + 5 } { 3 }", "\\frac { a + 5 } { 3 } \\ge 2", "a \\ge 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x < 4"}, {"id": "x < 2"}, {"id": "3 x < a + 5"}, {"id": "x < \\frac { a + 5 } { 3 }"}, {"id": "\\frac { a + 5 } { 3 } \\ge 2"}, {"id": "不等式 $2 x < 4$ 的解都能使关于 $x$ 的一次不等式 $3 x < a + 5$ 成立"}, {"id": "a \\ge 1"}], "links": [{"rel": "不等式方程求解", "source": "2 x < 4", "target": "x < 2"}, {"rel": "被描述", "source": "x < 2", "target": "\\frac { a + 5 } { 3 } \\ge 2"}, {"rel": "不等式方程部分求解", "source": "3 x < a + 5", "target": "x < \\frac { a + 5 } { 3 }"}, {"rel": "被描述", "source": "x < \\frac { a + 5 } { 3 }", "target": "\\frac { a + 5 } { 3 } \\ge 2"}, {"rel": "不等式方程求解", "source": "\\frac { a + 5 } { 3 } \\ge 2", "target": "a \\ge 1"}, {"rel": "限制性描述", "source": "不等式 $2 x < 4$ 的解都能使关于 $x$ 的一次不等式 $3 x < a + 5$ 成立", "target": "\\frac { a + 5 } { 3 } \\ge 2"}]}}
{"content": "The range of values of $x$ that make $\\frac { 2 } { x + 1 }$ meaningful is ____?", "answer": "x \\neq - 1", "steps": "To make $\\frac { 2 } { x + 1 }$ meaningful, we have $x + 1 \\neq 0$. Solving for $x$, we get $x \\neq - 1$.", "expr_cands": ["\\frac { 2 } { x + 1 }", "x", "x + 1 \\neq 0", "x \\neq - 1"], "exprs": ["x + 1 \\neq 0", "x \\neq - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 2 } { x + 1 }"}, {"id": "x + 1 \\neq 0"}, {"id": "使 $\\frac { 2 } { x + 1 }$ 有意义"}, {"id": "分式有意义，则分母不为0"}, {"id": "x \\neq - 1"}], "links": [{"rel": "被描述", "source": "\\frac { 2 } { x + 1 }", "target": "x + 1 \\neq 0"}, {"rel": "不等式方程求解", "source": "x + 1 \\neq 0", "target": "x \\neq - 1"}, {"rel": "限制性描述", "source": "使 $\\frac { 2 } { x + 1 }$ 有意义", "target": "x + 1 \\neq 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "x + 1 \\neq 0"}]}}
{"content": "The condition for transforming the fraction $\\frac { | x - 2 | } { x ^ 2 - 4 x + 4 } = \\frac { 1 } { 2 - x }$ from left to right is ____?", "answer": "x < 2", "steps": "Since $\\frac { | x - 2 | } { x ^ { 2 } - 4 x + 4 } = \\frac { | x - 2 | } { ( x - 2 ) ^ { 2 } } = \\frac { 1 } { 2 - x }$, it follows that $x - 2 < 0$, which implies that $x < 2$.", "expr_cands": ["\\frac { | x - 2 | } { x ^ { 2 } - 4 x + 4 } = \\frac { 1 } { 2 - x }", "x", "x - 2 < 0", "x < 2"], "exprs": ["x - 2 < 0", "x < 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { | x - 2 | } { x ^ { 2 } - 4 x + 4 } = \\frac { 1 } { 2 - x }"}, {"id": "x - 2 < 0"}, {"id": "使分式 $\\frac { | x - 2 | } { x ^ { 2 } - 4 x + 4 } = \\frac { 1 } { 2 - x }$ 自左向右变形成立的条件"}, {"id": "绝对值恒大于等于0"}, {"id": "x < 2"}], "links": [{"rel": "被描述", "source": "\\frac { | x - 2 | } { x ^ { 2 } - 4 x + 4 } = \\frac { 1 } { 2 - x }", "target": "x - 2 < 0"}, {"rel": "不等式方程求解", "source": "x - 2 < 0", "target": "x < 2"}, {"rel": "限制性描述", "source": "使分式 $\\frac { | x - 2 | } { x ^ { 2 } - 4 x + 4 } = \\frac { 1 } { 2 - x }$ 自左向右变形成立的条件", "target": "x - 2 < 0"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "x - 2 < 0"}]}}
{"content": "To make $- 2 x ^ { 2 - n }$ and $x ^ 4$ like terms, what is the value of $n$?", "answer": "- 2", "steps": "Because $- 2 { x } ^ { 2 - n }$ and ${ x } ^ { 4 }$ are like terms, then $2 - n = 4$, so $n = - 2$.", "expr_cands": ["- 2 x ^ { 2 - n }", "n", "x", "x ^ { 4 }", "- 2 { x } ^ { 2 - n }", "{ x } ^ { 4 }", "2 - n = 4", "n = - 2"], "exprs": ["2 - n = 4", "n = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 2 x ^ { 2 - n }"}, {"id": "2 - n = 4"}, {"id": "x ^ { 4 }"}, {"id": "要使 $- 2 x ^ { 2 - n }$ 与 $x ^ { 4 }$ 是同类项"}, {"id": "n = - 2"}], "links": [{"rel": "被描述", "source": "- 2 x ^ { 2 - n }", "target": "2 - n = 4"}, {"rel": "等式方程求解", "source": "2 - n = 4", "target": "n = - 2"}, {"rel": "被描述", "source": "x ^ { 4 }", "target": "2 - n = 4"}, {"rel": "限制性描述", "source": "要使 $- 2 x ^ { 2 - n }$ 与 $x ^ { 4 }$ 是同类项", "target": "2 - n = 4"}]}}
{"content": "Given the algebraic expressions $8 x - 7$ and $6 - 2 x$ are opposite in value, what is the value of $x$?", "answer": "\\frac { 1 } { 6 }", "steps": "According to the problem, we have $( 8 x - 7 ) + ( 6 - 2 x ) = 0$, which simplifies to $8 x - 7 + 6 - 2 x = 0$. Combining like terms and moving variables to one side, we get $6 x = 1$. Solving for $x$, we get $x = \\frac { 1 } { 6 }$.", "expr_cands": ["8 x - 7", "x", "6 - 2 x", "( 8 x - 7 ) + ( 6 - 2 x ) = 0", "x = \\frac { 1 } { 6 }", "8 x - 7 + 6 - 2 x = 0", "6 x = 1"], "exprs": ["( 8 x - 7 ) + ( 6 - 2 x ) = 0", "x = \\frac { 1 } { 6 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "8 x - 7"}, {"id": "( 8 x - 7 ) + ( 6 - 2 x ) = 0"}, {"id": "6 - 2 x"}, {"id": "代数式 $8 x - 7$ 与 $6 - 2 x$ 的值互为相反数"}, {"id": "x = \\frac { 1 } { 6 }"}], "links": [{"rel": "被描述", "source": "8 x - 7", "target": "( 8 x - 7 ) + ( 6 - 2 x ) = 0"}, {"rel": "等式方程求解", "source": "( 8 x - 7 ) + ( 6 - 2 x ) = 0", "target": "x = \\frac { 1 } { 6 }"}, {"rel": "被描述", "source": "6 - 2 x", "target": "( 8 x - 7 ) + ( 6 - 2 x ) = 0"}, {"rel": "限制性描述", "source": "代数式 $8 x - 7$ 与 $6 - 2 x$ 的值互为相反数", "target": "( 8 x - 7 ) + ( 6 - 2 x ) = 0"}]}}
{"content": "Given the monomial $- 3 { x } ^ { n } { y } ^ { 2 }$ is a 5th degree monomial, what is the value of $n$?", "answer": "3", "steps": "By the definition of a monomial, we know that $n + 2 = 5$, solving for $n$ gives us $n = 3$.", "expr_cands": ["- 3 { x } ^ { n } { y } ^ { 2 }", "y", "n", "x", "5", "n + 2 = 5", "n = 3"], "exprs": ["n + 2 = 5", "n = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- 3 { x } ^ { n } { y } ^ { 2 }"}, {"id": "n + 2 = 5"}, {"id": "单项式 $- 3 { x } ^ { n } { y } ^ { 2 }$ 是 $5$ 次单项式"}, {"id": "n = 3"}], "links": [{"rel": "被描述", "source": "- 3 { x } ^ { n } { y } ^ { 2 }", "target": "n + 2 = 5"}, {"rel": "等式方程求解", "source": "n + 2 = 5", "target": "n = 3"}, {"rel": "限制性描述", "source": "单项式 $- 3 { x } ^ { n } { y } ^ { 2 }$ 是 $5$ 次单项式", "target": "n + 2 = 5"}]}}
{"content": "What makes $\\sqrt { x - 5 }$ meaningful in the real number range?", "answer": "x \\ge 5", "steps": "From the given condition, we have $x - 5 \\ge 0$, which implies $x \\ge 5$.", "expr_cands": ["\\sqrt { x - 5 }", "x", "x - 5 \\ge 0", "5 \\le x", "x \\ge 5"], "exprs": ["x - 5 \\ge 0", "x \\ge 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x - 5 }"}, {"id": "x - 5 \\ge 0"}, {"id": "能使 $\\sqrt { x - 5 }$ 在实数范围内有意义"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x \\ge 5"}], "links": [{"rel": "被描述", "source": "\\sqrt { x - 5 }", "target": "x - 5 \\ge 0"}, {"rel": "不等式方程求解", "source": "x - 5 \\ge 0", "target": "x \\ge 5"}, {"rel": "限制性描述", "source": "能使 $\\sqrt { x - 5 }$ 在实数范围内有意义", "target": "x - 5 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 5 \\ge 0"}]}}
{"content": "$10$, $3 x - 2$, and $5 x - 6$ are two square roots of a positive number. What is the number?", "answer": "1", "steps": "$\\because$ The two square roots of a positive number are $3 x - 2$ and $5 x - 6$. $\\therefore$ $3 x - 2 + 5 x - 6 = 0$. Solving for $x$, we get $x = 1$ and $3 x - 2 = 1$. This positive number is $1 ^ 2 = 1$.", "expr_cands": ["10", "3 x - 2", "x", "5 x - 6", "3 x - 2 + 5 x - 6 = 0", "x = 1", "1", "1 ^ { 2 }"], "exprs": ["3 x - 2 + 5 x - 6 = 0", "x = 1", "1 ^ { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 x - 2"}, {"id": "3 x - 2 + 5 x - 6 = 0"}, {"id": "5 x - 6"}, {"id": "$10$ , $3 x - 2$ 和 $5 x - 6$ 是一个正数的两个平方根"}, {"id": "平方根互为相反数"}, {"id": "x = 1"}, {"id": "1 ^ { 2 }"}, {"id": "平方"}], "links": [{"rel": "被描述", "source": "3 x - 2", "target": "3 x - 2 + 5 x - 6 = 0"}, {"rel": "被描述", "source": "3 x - 2", "target": "1 ^ { 2 }"}, {"rel": "等式方程求解", "source": "3 x - 2 + 5 x - 6 = 0", "target": "x = 1"}, {"rel": "被描述", "source": "5 x - 6", "target": "3 x - 2 + 5 x - 6 = 0"}, {"rel": "限制性描述", "source": "$10$ , $3 x - 2$ 和 $5 x - 6$ 是一个正数的两个平方根", "target": "3 x - 2 + 5 x - 6 = 0"}, {"rel": "属性描述", "source": "平方根互为相反数", "target": "3 x - 2 + 5 x - 6 = 0"}, {"rel": "被描述", "source": "x = 1", "target": "1 ^ { 2 }"}, {"rel": "限制性描述", "source": "平方", "target": "1 ^ { 2 }"}]}}
{"content": "If $a$ and $b$ are opposite numbers, and $c$ and $d$ are reciprocal numbers, then the value of $\\frac { a + b } { 2010 } - \\frac { 2011 } { cd }$ is ____?", "answer": "- 2011", "steps": "According to the problem, we have $a + b = 0$ and $cd = 1$. Therefore, the original expression is equal to $0 - 2011 = - 2011$.", "expr_cands": ["a", "b", "c", "d", "\\frac { a + b } { 2010 } - \\frac { 2011 } { cd }", "a + b = 0", "cd = 1", "0 - 2011", "- 2011"], "exprs": ["a + b = 0", "cd = 1", "- 2011"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a + b = 0"}, {"id": "b"}, {"id": "$a$ 和 $b$ 互为相反数"}, {"id": "c"}, {"id": "cd = 1"}, {"id": "d"}, {"id": "$c$ 和 $d$ 互为倒数"}, {"id": "\\frac { a + b } { 2010 } - \\frac { 2011 } { cd }"}, {"id": "- 2011"}], "links": [{"rel": "被描述", "source": "a", "target": "a + b = 0"}, {"rel": "代入", "source": "a + b = 0", "target": "- 2011"}, {"rel": "被描述", "source": "b", "target": "a + b = 0"}, {"rel": "限制性描述", "source": "$a$ 和 $b$ 互为相反数", "target": "a + b = 0"}, {"rel": "被描述", "source": "c", "target": "cd = 1"}, {"rel": "代入", "source": "cd = 1", "target": "- 2011"}, {"rel": "被描述", "source": "d", "target": "cd = 1"}, {"rel": "限制性描述", "source": "$c$ 和 $d$ 互为倒数", "target": "cd = 1"}, {"rel": "被代入", "source": "\\frac { a + b } { 2010 } - \\frac { 2011 } { cd }", "target": "- 2011"}]}}
{"content": "Given the equation $2 xk - 4 x - k - 3 = 0$ with respect to $x$, the sum of all integers $x$ that satisfy $k$ as an integer is ____?", "answer": "2", "steps": "$\\because k = \\frac { 4 x + 3 } { 2 x - 1 } = \\frac { 4 x - 2 + 5 } { 2 x - 1 } = \\frac { 2 ( 2 x - 1 ) + 5 } { 2 x - 1 } = 2 + \\frac { 5 } { 2 x - 1 }$, $\\therefore$ when $2 x - 1 = 1$ or $2 x - 1 = - 1$ or $2 x - 1 = 5$ or $2 x - 1 = - 5$, $k$ is an integer. Solving for $x$, we get $x = 1$ or $x = 0$ or $x = 3$ or $x = - 2$. Therefore, the sum of all integers $x$ that satisfy $k$ being an integer is $1 + 0 + 3 - 2 = 2$.", "expr_cands": ["x", "2 xk - 4 x - k - 3 = 0", "k", "k = 2 + \\frac { 5 } { 2 x - 1 }", "2 x - 1 = 1", "x = 1", "2 x - 1 = - 1", "x = 0", "2 x - 1 = 5", "x = 3", "2 x - 1 = - 5", "x = - 2", "1 + 0 + 3 - 2", "2"], "exprs": ["k = 2 + \\frac { 5 } { 2 x - 1 }", "2 x - 1 = 1", "2 x - 1 = - 1", "2 x - 1 = 5", "2 x - 1 = - 5", "x = 1", "x = 0", "x = 3", "x = - 2", "1 + 0 + 3 - 2", "2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 xk - 4 x - k - 3 = 0"}, {"id": "k = 2 + \\frac { 5 } { 2 x - 1 }"}, {"id": "2 x - 1 = 1"}, {"id": "$k$ 为整数"}, {"id": "2 x - 1 = - 1"}, {"id": "2 x - 1 = 5"}, {"id": "2 x - 1 = - 5"}, {"id": "x = 1"}, {"id": "x = 0"}, {"id": "x = 3"}, {"id": "x = - 2"}, {"id": "k"}, {"id": "1 + 0 + 3 - 2"}, {"id": "满足 $k$ 为整数的所有整数 $x$ 的和"}, {"id": "2"}], "links": [{"rel": "等式方程部分求解", "source": "2 xk - 4 x - k - 3 = 0", "target": "k = 2 + \\frac { 5 } { 2 x - 1 }"}, {"rel": "被描述", "source": "k = 2 + \\frac { 5 } { 2 x - 1 }", "target": "2 x - 1 = 1"}, {"rel": "被描述", "source": "k = 2 + \\frac { 5 } { 2 x - 1 }", "target": "2 x - 1 = - 1"}, {"rel": "被描述", "source": "k = 2 + \\frac { 5 } { 2 x - 1 }", "target": "2 x - 1 = 5"}, {"rel": "被描述", "source": "k = 2 + \\frac { 5 } { 2 x - 1 }", "target": "2 x - 1 = - 5"}, {"rel": "等式方程求解", "source": "2 x - 1 = 1", "target": "x = 1"}, {"rel": "限制性描述", "source": "$k$ 为整数", "target": "2 x - 1 = 1"}, {"rel": "限制性描述", "source": "$k$ 为整数", "target": "2 x - 1 = - 1"}, {"rel": "限制性描述", "source": "$k$ 为整数", "target": "2 x - 1 = 5"}, {"rel": "限制性描述", "source": "$k$ 为整数", "target": "2 x - 1 = - 5"}, {"rel": "等式方程求解", "source": "2 x - 1 = - 1", "target": "x = 0"}, {"rel": "等式方程求解", "source": "2 x - 1 = 5", "target": "x = 3"}, {"rel": "等式方程求解", "source": "2 x - 1 = - 5", "target": "x = - 2"}, {"rel": "被描述", "source": "x = 1", "target": "1 + 0 + 3 - 2"}, {"rel": "被描述", "source": "x = 0", "target": "1 + 0 + 3 - 2"}, {"rel": "被描述", "source": "x = 3", "target": "1 + 0 + 3 - 2"}, {"rel": "被描述", "source": "x = - 2", "target": "1 + 0 + 3 - 2"}, {"rel": "被描述", "source": "k", "target": "1 + 0 + 3 - 2"}, {"rel": "计算", "source": "1 + 0 + 3 - 2", "target": "2"}, {"rel": "限制性描述", "source": "满足 $k$ 为整数的所有整数 $x$ 的和", "target": "1 + 0 + 3 - 2"}]}}
{"content": "The parabola $y = \\frac { 1 } { 2 } { x } ^ { 2 }$ is shifted $3$ units to the left, resulting in the parabola _____.", "answer": "y = \\frac { 1 } { 2 } ( x + 3 )  ^ { 2 }", "steps": "The equation of the parabola $y = \\frac { 1 } { 2 } x ^ { 2 }$ after shifting it $3$ units to the left is: $y = \\frac { 1 } { 2 } ( x + 3 ) ^ { 2 }$.", "expr_cands": ["y = \\frac { 1 } { 2 } { x } ^ { 2 }", "y", "x", "3", "y = \\frac { 1 } { 2 } x ^ { 2 }", "y = \\frac { 1 } { 2 } ( x + 3 ) ^ { 2 }", "\\frac { x ^ { 2 }} { 2 } = \\frac { 1 } { 2 } ( x + 3 ) ^ { 2 }", "\\frac { x ^ { 2 }} { 2 }", "y = \\frac { 1 } { 2 } ( x + 3 )  ^ { 2 }"], "exprs": ["y = \\frac { 1 } { 2 } ( x + 3 ) ^ { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3"}, {"id": "y = \\frac { 1 } { 2 } ( x + 3 ) ^ { 2 }"}, {"id": "y = \\frac { 1 } { 2 } { x } ^ { 2 }"}, {"id": "抛物线 $y = \\frac { 1 } { 2 } { x } ^ { 2 }$ 向左平移 $3$ 个单位"}, {"id": "就得到抛物线"}], "links": [{"rel": "被描述", "source": "3", "target": "y = \\frac { 1 } { 2 } ( x + 3 ) ^ { 2 }"}, {"rel": "被描述", "source": "y = \\frac { 1 } { 2 } { x } ^ { 2 }", "target": "y = \\frac { 1 } { 2 } ( x + 3 ) ^ { 2 }"}, {"rel": "限制性描述", "source": "抛物线 $y = \\frac { 1 } { 2 } { x } ^ { 2 }$ 向左平移 $3$ 个单位", "target": "y = \\frac { 1 } { 2 } ( x + 3 ) ^ { 2 }"}, {"rel": "限制性描述", "source": "就得到抛物线", "target": "y = \\frac { 1 } { 2 } ( x + 3 ) ^ { 2 }"}]}}
{"content": "Given $3 m - 2 x ^ { 2 - m } < 2$ is a one-variable linear inequality in $x$, then $m$ = ____ ?", "answer": "1", "steps": "According to the problem, we have $2 - m = 1$. Solving for $m$, we get $m = 1$.", "expr_cands": ["3 m - 2 x ^ { 2 - m } < 2", "x", "m", "2 - m = 1", "m = 1"], "exprs": ["2 - m = 1", "m = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 m - 2 x ^ { 2 - m } < 2"}, {"id": "2 - m = 1"}, {"id": "x"}, {"id": "$3 m - 2 x ^ { 2 - m } < 2$ 是关于 $x$ 的一元一次不等式"}, {"id": "m = 1"}], "links": [{"rel": "被描述", "source": "3 m - 2 x ^ { 2 - m } < 2", "target": "2 - m = 1"}, {"rel": "等式方程求解", "source": "2 - m = 1", "target": "m = 1"}, {"rel": "被描述", "source": "x", "target": "2 - m = 1"}, {"rel": "限制性描述", "source": "$3 m - 2 x ^ { 2 - m } < 2$ 是关于 $x$ 的一元一次不等式", "target": "2 - m = 1"}]}}
{"content": "Given real numbers $x$ and $y$ satisfying $| x - 3 | + ( y + 4 ) ^ { 2 } = 0$, what is the value of the algebraic expression $x + y$?", "answer": "- 1", "steps": "Since $| x - 3 | + ( y + 4 ) ^ { 2 } = 0$, it follows that $x - 3 = 0$ and $y + 4 = 0$. Solving for $x$ and $y$, we get $x = 3$ and $y = - 4$. Therefore, $x + y = 3 - 4 = - 1$.", "expr_cands": ["x", "y", "| x - 3 | + ( y + 4 ) ^ { 2 } = 0", "x + y", "x - 3 = 0", "x = 3", "y + 4 = 0", "y = - 4", "- 1"], "exprs": ["x - 3 = 0", "y + 4 = 0", "x = 3", "y = - 4", "- 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| x - 3 | + ( y + 4 ) ^ { 2 } = 0"}, {"id": "x - 3 = 0"}, {"id": "实数 $x$ , $y$ 满足 $| x - 3 | + ( y + 4 ) ^ { 2 } = 0$"}, {"id": "绝对值恒大于等于0"}, {"id": "y + 4 = 0"}, {"id": "多项式偶次方项恒大于等于0"}, {"id": "x = 3"}, {"id": "y = - 4"}, {"id": "x + y"}, {"id": "- 1"}], "links": [{"rel": "被描述", "source": "| x - 3 | + ( y + 4 ) ^ { 2 } = 0", "target": "x - 3 = 0"}, {"rel": "被描述", "source": "| x - 3 | + ( y + 4 ) ^ { 2 } = 0", "target": "y + 4 = 0"}, {"rel": "等式方程求解", "source": "x - 3 = 0", "target": "x = 3"}, {"rel": "限制性描述", "source": "实数 $x$ , $y$ 满足 $| x - 3 | + ( y + 4 ) ^ { 2 } = 0$", "target": "x - 3 = 0"}, {"rel": "限制性描述", "source": "实数 $x$ , $y$ 满足 $| x - 3 | + ( y + 4 ) ^ { 2 } = 0$", "target": "y + 4 = 0"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "x - 3 = 0"}, {"rel": "等式方程求解", "source": "y + 4 = 0", "target": "y = - 4"}, {"rel": "属性描述", "source": "多项式偶次方项恒大于等于0", "target": "y + 4 = 0"}, {"rel": "代入", "source": "x = 3", "target": "- 1"}, {"rel": "代入", "source": "y = - 4", "target": "- 1"}, {"rel": "被代入", "source": "x + y", "target": "- 1"}]}}
{"content": "If $x = 1$ is a root of the quadratic equation $x ^ 2 + mx + 1 = 0$, then $m$ = ____ ?", "answer": "- 2", "steps": "Substituting $x = 1$ into the equation $x ^ 2 + mx + 1 = 0$ yields $1 + m + 1 = 0$, which gives the solution $m = - 2$.", "expr_cands": ["x = 1", "x", "x ^ { 2 } + mx + 1 = 0", "m", "m + 2 = 0", "1 + m + 1 = 0", "m = - 2"], "exprs": ["1 + m + 1 = 0", "m = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } + mx + 1 = 0"}, {"id": "1 + m + 1 = 0"}, {"id": "x = 1"}, {"id": "m = - 2"}], "links": [{"rel": "被代入", "source": "x ^ { 2 } + mx + 1 = 0", "target": "1 + m + 1 = 0"}, {"rel": "等式方程求解", "source": "1 + m + 1 = 0", "target": "m = - 2"}, {"rel": "代入", "source": "x = 1", "target": "1 + m + 1 = 0"}]}}
{"content": "Given a one-variable linear equation in $x$, $( 3 - a ) x - x + 2 - a = 0$, with a solution that is the reciprocal of $\\frac { 1 } { 3 }$, what is the value of $a$?", "answer": "2", "steps": "The reciprocal of $\\frac { 1 } { 3 }$ is $3$. Substituting $x = 3$ into the equation $( 3 - a ) x - x + 2 - a = 0$, we get $3 ( 3 - a ) - 3 + 2 - a = 0$. Solving for $a$, we get $a = 2$.", "expr_cands": ["x", "( 3 - a ) x - x + 2 - a = 0", "a", "\\frac { 1 } { 3 }", "3", "x = 3", "8 - 4 a = 0", "3 ( 3 - a ) - 3 + 2 - a = 0", "a = 2"], "exprs": ["x = 3", "3 ( 3 - a ) - 3 + 2 - a = 0", "a = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( 3 - a ) x - x + 2 - a = 0"}, {"id": "x = 3"}, {"id": "\\frac { 1 } { 3 }"}, {"id": "x"}, {"id": "关于 $x$ 的一元一次方程 $( 3 - a ) x - x + 2 - a = 0$ 的解是 $\\frac { 1 } { 3 }$ 的倒数"}, {"id": "3 ( 3 - a ) - 3 + 2 - a = 0"}, {"id": "a = 2"}], "links": [{"rel": "被描述", "source": "( 3 - a ) x - x + 2 - a = 0", "target": "x = 3"}, {"rel": "被代入", "source": "( 3 - a ) x - x + 2 - a = 0", "target": "3 ( 3 - a ) - 3 + 2 - a = 0"}, {"rel": "代入", "source": "x = 3", "target": "3 ( 3 - a ) - 3 + 2 - a = 0"}, {"rel": "被描述", "source": "\\frac { 1 } { 3 }", "target": "x = 3"}, {"rel": "被描述", "source": "x", "target": "x = 3"}, {"rel": "限制性描述", "source": "关于 $x$ 的一元一次方程 $( 3 - a ) x - x + 2 - a = 0$ 的解是 $\\frac { 1 } { 3 }$ 的倒数", "target": "x = 3"}, {"rel": "等式方程求解", "source": "3 ( 3 - a ) - 3 + 2 - a = 0", "target": "a = 2"}]}}
{"content": "Given $| m | = 4$, $| n | = 5$, and $n < 0$, $m < 0$, what is $m + n$?", "answer": "- 9", "steps": "Since $| m | = 4$ and $m < 0$, we know that $m = - 4$. Also, since $| n | = 5$ and $n < 0$, we know that $n = - 5$. Therefore, $m + n = - 4 - 5 = - 9$.", "expr_cands": ["| m | = 4", "m", "| n | = 5", "n", "n < 0", "m < 0", "m + n", "m = - 4", "m = 4", "n = - 5", "n = 5", "- 9"], "exprs": ["m = - 4", "n = - 5", "- 9"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "| m | = 4"}, {"id": "m = - 4"}, {"id": "m < 0"}, {"id": "| n | = 5"}, {"id": "n = - 5"}, {"id": "n < 0"}, {"id": "m + n"}, {"id": "- 9"}], "links": [{"rel": "联立", "source": "| m | = 4", "target": "m = - 4"}, {"rel": "代入", "source": "m = - 4", "target": "- 9"}, {"rel": "联立", "source": "m < 0", "target": "m = - 4"}, {"rel": "联立", "source": "| n | = 5", "target": "n = - 5"}, {"rel": "代入", "source": "n = - 5", "target": "- 9"}, {"rel": "联立", "source": "n < 0", "target": "n = - 5"}, {"rel": "被代入", "source": "m + n", "target": "- 9"}]}}
{"content": "If the value of the polynomial $2 y ^ 2 + 3 y + 7$ is $9$, then the value of $4 y ^ 2 + 6 y - 7$ is ____?", "answer": "- 3", "steps": "Since $2 y ^ 2 + 3 y + 7 = 9$, therefore $2 y ^ 2 + 3 y = 2$, therefore $4 y ^ 2 + 6 y = 4$, therefore $4 y ^ 2 + 6 y - 7 = 4 - 7 = - 3$.", "expr_cands": ["2 y ^ { 2 } + 3 y + 7", "y", "9", "4 y ^ { 2 } + 6 y - 7", "2 y ^ { 2 } + 3 y + 7 = 9", "y = - 2", "y = \\frac { 1 } { 2 }", "2 y ^ { 2 } + 3 y = 2", "4 y ^ { 2 } + 6 y = 4", "- 3"], "exprs": ["2 y ^ { 2 } + 3 y + 7 = 9", "2 y ^ { 2 } + 3 y = 2", "4 y ^ { 2 } + 6 y = 4", "- 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 y ^ { 2 } + 3 y + 7"}, {"id": "2 y ^ { 2 } + 3 y + 7 = 9"}, {"id": "9"}, {"id": "多项式 $2 y ^ { 2 } + 3 y + 7$ 的值为 $9$"}, {"id": "2 y ^ { 2 } + 3 y = 2"}, {"id": "4 y ^ { 2 } + 6 y = 4"}, {"id": "4 y ^ { 2 } + 6 y - 7"}, {"id": "- 3"}], "links": [{"rel": "被描述", "source": "2 y ^ { 2 } + 3 y + 7", "target": "2 y ^ { 2 } + 3 y + 7 = 9"}, {"rel": "移项", "source": "2 y ^ { 2 } + 3 y + 7 = 9", "target": "2 y ^ { 2 } + 3 y = 2"}, {"rel": "被描述", "source": "9", "target": "2 y ^ { 2 } + 3 y + 7 = 9"}, {"rel": "限制性描述", "source": "多项式 $2 y ^ { 2 } + 3 y + 7$ 的值为 $9$", "target": "2 y ^ { 2 } + 3 y + 7 = 9"}, {"rel": "同乘除", "source": "2 y ^ { 2 } + 3 y = 2", "target": "4 y ^ { 2 } + 6 y = 4"}, {"rel": "代入", "source": "4 y ^ { 2 } + 6 y = 4", "target": "- 3"}, {"rel": "被代入", "source": "4 y ^ { 2 } + 6 y - 7", "target": "- 3"}]}}
{"content": "If the monomial $3 ab ^ { 2 x - 1 }$ and $ab ^ { x + 1 }$ have a sum that is also a monomial, then $x$ = ____ ?", "answer": "2", "steps": "$\\because$ The sum of the monomials $3 ab ^ { 2 x - 1 }$ and $ab ^ { x + 1 }$ is also a monomial, $\\therefore$ $2 x - 1 = x + 1$, solving for $x$ gives $x = 2$.", "expr_cands": ["3 ab ^ { 2 x - 1 }", "b", "a", "x", "ab ^ { x + 1 }", "2 x - 1 = x + 1", "x = 2"], "exprs": ["2 x - 1 = x + 1", "x = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 ab ^ { 2 x - 1 }"}, {"id": "2 x - 1 = x + 1"}, {"id": "ab ^ { x + 1 }"}, {"id": "单项式 $3 ab ^ { 2 x - 1 }$ 与 $ab ^ { x + 1 }$ 的和也是单項式"}, {"id": "x = 2"}], "links": [{"rel": "被描述", "source": "3 ab ^ { 2 x - 1 }", "target": "2 x - 1 = x + 1"}, {"rel": "等式方程求解", "source": "2 x - 1 = x + 1", "target": "x = 2"}, {"rel": "被描述", "source": "ab ^ { x + 1 }", "target": "2 x - 1 = x + 1"}, {"rel": "限制性描述", "source": "单项式 $3 ab ^ { 2 x - 1 }$ 与 $ab ^ { x + 1 }$ 的和也是单項式", "target": "2 x - 1 = x + 1"}]}}
{"content": "If $2 ^ { x + 1 } = 16$, then $x$ = ____ ?", "answer": "3", "steps": "Since ${ 2 } ^ { x + 1 } = 16$, it follows that ${ 2 } ^ { x + 1 } = { 2 } ^ { 4 }$, which implies that $x + 1 = 4$. Therefore, $x = 3$.", "expr_cands": ["2 ^ { x + 1 } = 16", "x", "{ 2 } ^ { x + 1 } = 16", "x = 3", "{ 2 } ^ { x + 1 }", "16", "x + 1 = 4"], "exprs": ["x = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 ^ { x + 1 } = 16"}, {"id": "x = 3"}], "links": [{"rel": "等式方程求解", "source": "2 ^ { x + 1 } = 16", "target": "x = 3"}]}}
{"content": "Given that $m$ and $n$ are two solutions of the quadratic equation $x ^ 2 - 3 x + a = 0$, if $( m - 1 ) ( n - 1 ) = - 6$, then the value of $a$ is ____?", "answer": "- 4", "steps": "According to the problem, we have $m + n = 3$ and $mn = a$. Since $( m - 1 ) ( n - 1 ) = mn - ( m + n ) + 1 = - 6$, we have $a - 3 + 1 = - 6$. Solving for $a$, we get $a = - 4$.", "expr_cands": ["m", "n", "x", "x ^ { 2 } - 3 x + a = 0", "a", "( m - 1 ) ( n - 1 ) = - 6", "m + n = 3", "mn = a", "mn - ( m + n ) + 1 = - 6", "a - 3 + 1 = - 6", "a = - 4"], "exprs": ["m + n = 3", "mn = a", "a - 3 + 1 = - 6", "a = - 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "m"}, {"id": "m + n = 3"}, {"id": "n"}, {"id": "x ^ { 2 } - 3 x + a = 0"}, {"id": "$m$ , $n$ 是关于 $x$ 的一元二次方程 $x ^ { 2 } - 3 x + a = 0$ 的两个解"}, {"id": "一元二次方程根与系数关系，两根之和"}, {"id": "mn = a"}, {"id": "一元二次方程根与系数关系，两根之积"}, {"id": "( m - 1 ) ( n - 1 ) = - 6"}, {"id": "a - 3 + 1 = - 6"}, {"id": "a = - 4"}], "links": [{"rel": "被描述", "source": "m", "target": "m + n = 3"}, {"rel": "被描述", "source": "m", "target": "mn = a"}, {"rel": "联立", "source": "m + n = 3", "target": "a - 3 + 1 = - 6"}, {"rel": "被描述", "source": "n", "target": "m + n = 3"}, {"rel": "被描述", "source": "n", "target": "mn = a"}, {"rel": "被描述", "source": "x ^ { 2 } - 3 x + a = 0", "target": "m + n = 3"}, {"rel": "被描述", "source": "x ^ { 2 } - 3 x + a = 0", "target": "mn = a"}, {"rel": "限制性描述", "source": "$m$ , $n$ 是关于 $x$ 的一元二次方程 $x ^ { 2 } - 3 x + a = 0$ 的两个解", "target": "m + n = 3"}, {"rel": "限制性描述", "source": "$m$ , $n$ 是关于 $x$ 的一元二次方程 $x ^ { 2 } - 3 x + a = 0$ 的两个解", "target": "mn = a"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "m + n = 3"}, {"rel": "联立", "source": "mn = a", "target": "a - 3 + 1 = - 6"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之积", "target": "mn = a"}, {"rel": "联立", "source": "( m - 1 ) ( n - 1 ) = - 6", "target": "a - 3 + 1 = - 6"}, {"rel": "等式方程求解", "source": "a - 3 + 1 = - 6", "target": "a = - 4"}]}}
{"content": "The polynomial $x ^ { 2 } - 3 mxy - 3 y ^ { 2 } + 6 xy - 8$ does not contain the term $xy$. What is the value of the constant $m$?", "answer": "2", "steps": "$x ^ { 2 } - 3 mxy - 3 y ^ { 2 } + 6 xy - 8 = x ^ { 2 } - 3 mxy + 6 xy - 3 y ^ { 2 } - 8 = x ^ { 2 } + ( - 3 m + 6 ) xy - 3 y ^ { 2 } - 8$, because there is no $xy$ term in the polynomial, therefore $- 3 m + 6 = 0$, which gives $m = 2$.", "expr_cands": ["x ^ { 2 } - 3 mxy - 3 y ^ { 2 } + 6 xy - 8", "y", "x", "m", "xy", "x ^ { 2 } + ( - 3 m + 6 ) xy - 3 y ^ { 2 } - 8", "- 3 m + 6 = 0", "m = 2"], "exprs": ["x ^ { 2 } + ( - 3 m + 6 ) xy - 3 y ^ { 2 } - 8", "- 3 m + 6 = 0", "m = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } - 3 mxy - 3 y ^ { 2 } + 6 xy - 8"}, {"id": "x ^ { 2 } + ( - 3 m + 6 ) xy - 3 y ^ { 2 } - 8"}, {"id": "xy"}, {"id": "- 3 m + 6 = 0"}, {"id": "多项式 $x ^ { 2 } - 3 mxy - 3 y ^ { 2 } + 6 xy - 8$ 中不含 $xy$ 项"}, {"id": "m = 2"}], "links": [{"rel": "提取因式", "source": "x ^ { 2 } - 3 mxy - 3 y ^ { 2 } + 6 xy - 8", "target": "x ^ { 2 } + ( - 3 m + 6 ) xy - 3 y ^ { 2 } - 8"}, {"rel": "被描述", "source": "x ^ { 2 } + ( - 3 m + 6 ) xy - 3 y ^ { 2 } - 8", "target": "- 3 m + 6 = 0"}, {"rel": "提取因式参考", "source": "xy", "target": "x ^ { 2 } + ( - 3 m + 6 ) xy - 3 y ^ { 2 } - 8"}, {"rel": "等式方程求解", "source": "- 3 m + 6 = 0", "target": "m = 2"}, {"rel": "限制性描述", "source": "多项式 $x ^ { 2 } - 3 mxy - 3 y ^ { 2 } + 6 xy - 8$ 中不含 $xy$ 项", "target": "- 3 m + 6 = 0"}]}}
{"content": "One day in math class, the students learned about polynomial division. After school, Xiao Ming went home and reviewed the notes from class. He suddenly noticed a problem involving polynomial division: $( 21 x ^ 4 y ^ 3 - + 7 x ^ 2 y ^ 2 ) { \\div } ( - 7 x ^ 2 y ) = - 3 x ^ 2 y ^ 2 + 5 { xy } - y$. The second term of the dividend was smudged by pen ink, can you calculate what the smudged content is?", "answer": "35 x ^ { 3 } y ^ { 2 }", "steps": "\\because $- 7 x ^ { 2 } y \\times 5 xy = - 35 x ^ { 3 } y ^ { 2 }$ , \\therefore the contaminated content is $35 x ^ { 3 } y ^ { 2 }$.", "expr_cands": ["( 21 x ^ { 4 } y ^ { 3 } - + 7 x ^ { 2 } y ^ { 2 } ) { \\div } ( - 7 x ^ { 2 } y ) = - 3 x ^ { 2 } y ^ { 2 } + 5 { xy } - y", "y", "x", "- 7 x ^ { 2 } y \\times 5 xy", "- 35 x ^ { 3 } y ^ { 2 }", "35 x ^ { 3 } y ^ { 2 }"], "exprs": ["- 7 x ^ { 2 } y \\times 5 xy", "- 35 x ^ { 3 } y ^ { 2 }", "35 x ^ { 3 } y ^ { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( 21 x ^ { 4 } y ^ { 3 } - + 7 x ^ { 2 } y ^ { 2 } ) { \\div } ( - 7 x ^ { 2 } y ) = - 3 x ^ { 2 } y ^ { 2 } + 5 { xy } - y"}, {"id": "- 7 x ^ { 2 } y \\times 5 xy"}, {"id": "他突然发现一道三项式除法运算题 : $( 21 x ^ { 4 } y ^ { 3 } - + 7 x ^ { 2 } y ^ { 2 } ) { \\div } ( - 7 x ^ { 2 } y ) = - 3 x ^ { 2 } y ^ { 2 } + 5 { xy } - y$"}, {"id": "被除式的第二项中被钢笔水弄污了"}, {"id": "你能算出被污染的内容"}, {"id": "- 35 x ^ { 3 } y ^ { 2 }"}, {"id": "35 x ^ { 3 } y ^ { 2 }"}], "links": [{"rel": "被描述", "source": "( 21 x ^ { 4 } y ^ { 3 } - + 7 x ^ { 2 } y ^ { 2 } ) { \\div } ( - 7 x ^ { 2 } y ) = - 3 x ^ { 2 } y ^ { 2 } + 5 { xy } - y", "target": "- 7 x ^ { 2 } y \\times 5 xy"}, {"rel": "被描述", "source": "( 21 x ^ { 4 } y ^ { 3 } - + 7 x ^ { 2 } y ^ { 2 } ) { \\div } ( - 7 x ^ { 2 } y ) = - 3 x ^ { 2 } y ^ { 2 } + 5 { xy } - y", "target": "35 x ^ { 3 } y ^ { 2 }"}, {"rel": "计算", "source": "- 7 x ^ { 2 } y \\times 5 xy", "target": "- 35 x ^ { 3 } y ^ { 2 }"}, {"rel": "限制性描述", "source": "他突然发现一道三项式除法运算题 : $( 21 x ^ { 4 } y ^ { 3 } - + 7 x ^ { 2 } y ^ { 2 } ) { \\div } ( - 7 x ^ { 2 } y ) = - 3 x ^ { 2 } y ^ { 2 } + 5 { xy } - y$", "target": "- 7 x ^ { 2 } y \\times 5 xy"}, {"rel": "限制性描述", "source": "他突然发现一道三项式除法运算题 : $( 21 x ^ { 4 } y ^ { 3 } - + 7 x ^ { 2 } y ^ { 2 } ) { \\div } ( - 7 x ^ { 2 } y ) = - 3 x ^ { 2 } y ^ { 2 } + 5 { xy } - y$", "target": "35 x ^ { 3 } y ^ { 2 }"}, {"rel": "限制性描述", "source": "被除式的第二项中被钢笔水弄污了", "target": "- 7 x ^ { 2 } y \\times 5 xy"}, {"rel": "限制性描述", "source": "被除式的第二项中被钢笔水弄污了", "target": "35 x ^ { 3 } y ^ { 2 }"}, {"rel": "限制性描述", "source": "你能算出被污染的内容", "target": "- 7 x ^ { 2 } y \\times 5 xy"}, {"rel": "限制性描述", "source": "你能算出被污染的内容", "target": "35 x ^ { 3 } y ^ { 2 }"}, {"rel": "被描述", "source": "- 35 x ^ { 3 } y ^ { 2 }", "target": "35 x ^ { 3 } y ^ { 2 }"}]}}
{"content": "If $m$ and $n$ are opposite numbers, what is the value of $| m + n - 2016 |$?", "answer": "2016", "steps": "\\because $m$ and $n$ are opposite numbers, \\therefore $m + n = 0$, \\therefore $| m + n - 2016 | = | - 2016 | = 2016$.", "expr_cands": ["m", "n", "| m + n - 2016 |", "m + n = 0", "2016"], "exprs": ["m + n = 0", "2016"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "m"}, {"id": "m + n = 0"}, {"id": "n"}, {"id": "$m$ , $n$ 互为相反数"}, {"id": "| m + n - 2016 |"}, {"id": "2016"}], "links": [{"rel": "被描述", "source": "m", "target": "m + n = 0"}, {"rel": "代入", "source": "m + n = 0", "target": "2016"}, {"rel": "被描述", "source": "n", "target": "m + n = 0"}, {"rel": "限制性描述", "source": "$m$ , $n$ 互为相反数", "target": "m + n = 0"}, {"rel": "被代入", "source": "| m + n - 2016 |", "target": "2016"}]}}
{"content": "If the equation $( 2 m - 6 ) x ^ { | m - 2 | } - ( n + 2 ) y ^ { | n + 3 | } = 16$ is a linear equation in $x$ and $y$, then $m + n$ = ____?", "answer": "- 3", "steps": "From the given conditions, we have $| m - 2 | = 1$, $| n + 3 | = 1$, and $2 m - 6 \\neq 0$, $n + 2 \\neq 0$. Solving for $m$ and $n$, we get $m = 1$, $n = - 4$, and $m + n = - 3$.", "expr_cands": ["( 2 m - 6 ) x ^ { | m - 2 | } - ( n + 2 ) y ^ { | n + 3 | } = 16", "n", "y", "m", "x", "m + n", "| m - 2 | = 1", "m = 1", "m = 3", "| n + 3 | = 1", "n = - 4", "n = - 2", "2 m - 6 \\neq 0", "m \\neq 3", "n + 2 \\neq 0", "n \\neq - 2", "- 3"], "exprs": ["| m - 2 | = 1", "| n + 3 | = 1", "2 m - 6 \\neq 0", "n + 2 \\neq 0", "m = 1", "n = - 4", "- 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( 2 m - 6 ) x ^ { | m - 2 | } - ( n + 2 ) y ^ { | n + 3 | } = 16"}, {"id": "| m - 2 | = 1"}, {"id": "方程 $( 2 m - 6 ) x ^ { | m - 2 | } - ( n + 2 ) y ^ { | n + 3 | } = 16$ 是关于 $x$ , $y$ 的二元一次方程"}, {"id": "| n + 3 | = 1"}, {"id": "2 m - 6 \\neq 0"}, {"id": "n + 2 \\neq 0"}, {"id": "m = 1"}, {"id": "n = - 4"}, {"id": "m + n"}, {"id": "- 3"}], "links": [{"rel": "被描述", "source": "( 2 m - 6 ) x ^ { | m - 2 | } - ( n + 2 ) y ^ { | n + 3 | } = 16", "target": "| m - 2 | = 1"}, {"rel": "被描述", "source": "( 2 m - 6 ) x ^ { | m - 2 | } - ( n + 2 ) y ^ { | n + 3 | } = 16", "target": "| n + 3 | = 1"}, {"rel": "被描述", "source": "( 2 m - 6 ) x ^ { | m - 2 | } - ( n + 2 ) y ^ { | n + 3 | } = 16", "target": "2 m - 6 \\neq 0"}, {"rel": "被描述", "source": "( 2 m - 6 ) x ^ { | m - 2 | } - ( n + 2 ) y ^ { | n + 3 | } = 16", "target": "n + 2 \\neq 0"}, {"rel": "联立", "source": "| m - 2 | = 1", "target": "m = 1"}, {"rel": "限制性描述", "source": "方程 $( 2 m - 6 ) x ^ { | m - 2 | } - ( n + 2 ) y ^ { | n + 3 | } = 16$ 是关于 $x$ , $y$ 的二元一次方程", "target": "| m - 2 | = 1"}, {"rel": "限制性描述", "source": "方程 $( 2 m - 6 ) x ^ { | m - 2 | } - ( n + 2 ) y ^ { | n + 3 | } = 16$ 是关于 $x$ , $y$ 的二元一次方程", "target": "| n + 3 | = 1"}, {"rel": "限制性描述", "source": "方程 $( 2 m - 6 ) x ^ { | m - 2 | } - ( n + 2 ) y ^ { | n + 3 | } = 16$ 是关于 $x$ , $y$ 的二元一次方程", "target": "2 m - 6 \\neq 0"}, {"rel": "限制性描述", "source": "方程 $( 2 m - 6 ) x ^ { | m - 2 | } - ( n + 2 ) y ^ { | n + 3 | } = 16$ 是关于 $x$ , $y$ 的二元一次方程", "target": "n + 2 \\neq 0"}, {"rel": "联立", "source": "| n + 3 | = 1", "target": "n = - 4"}, {"rel": "联立", "source": "2 m - 6 \\neq 0", "target": "m = 1"}, {"rel": "联立", "source": "n + 2 \\neq 0", "target": "n = - 4"}, {"rel": "代入", "source": "m = 1", "target": "- 3"}, {"rel": "代入", "source": "n = - 4", "target": "- 3"}, {"rel": "被代入", "source": "m + n", "target": "- 3"}]}}
{"content": "Given $a = 2 + \\sqrt { 5 }$, what is the value of $( a - 1 ) ( a - 3 )$?", "answer": "4", "steps": "Because $a = 2 + \\sqrt { 5 }$, therefore $( a - 1 ) ( a - 3 ) = a ^ 2 - 4 a + 3 = ( a - 2 ) ^ 2 - 1 = ( 2 + \\sqrt { 5 } - 2 ) ^ 2 - 1 = 5 - 1 = 4$.", "expr_cands": ["a = 2 + \\sqrt { 5 }", "a", "( a - 1 ) ( a - 3 )", "4"], "exprs": ["4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( a - 1 ) ( a - 3 )"}, {"id": "4"}, {"id": "a = 2 + \\sqrt { 5 }"}], "links": [{"rel": "被代入", "source": "( a - 1 ) ( a - 3 )", "target": "4"}, {"rel": "代入", "source": "a = 2 + \\sqrt { 5 }", "target": "4"}]}}
{"content": "The sum of the monomials $9 x ^ { m } y ^ { 3 }$ and $4 x ^ { 2 } y ^ { n }$ is a monomial. What is the value of $m + n$?", "answer": "5", "steps": "From the given information, we have: $m = 2$, $n = 3$. Therefore, $m + n = 2 + 3 = 5$.", "expr_cands": ["9 x ^ { m } y ^ { 3 }", "m", "y", "x", "4 x ^ { 2 } y ^ { n }", "n", "m + n", "m = 2", "n = 3", "5"], "exprs": ["m = 2", "n = 3", "5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "9 x ^ { m } y ^ { 3 }"}, {"id": "m = 2"}, {"id": "4 x ^ { 2 } y ^ { n }"}, {"id": "单项式 $9 x ^ { m } y ^ { 3 }$ 与单项式 $4 x ^ { 2 } y ^ { n }$ 的和是单项式"}, {"id": "n = 3"}, {"id": "m + n"}, {"id": "5"}], "links": [{"rel": "被描述", "source": "9 x ^ { m } y ^ { 3 }", "target": "m = 2"}, {"rel": "被描述", "source": "9 x ^ { m } y ^ { 3 }", "target": "n = 3"}, {"rel": "代入", "source": "m = 2", "target": "5"}, {"rel": "被描述", "source": "4 x ^ { 2 } y ^ { n }", "target": "m = 2"}, {"rel": "被描述", "source": "4 x ^ { 2 } y ^ { n }", "target": "n = 3"}, {"rel": "限制性描述", "source": "单项式 $9 x ^ { m } y ^ { 3 }$ 与单项式 $4 x ^ { 2 } y ^ { n }$ 的和是单项式", "target": "m = 2"}, {"rel": "限制性描述", "source": "单项式 $9 x ^ { m } y ^ { 3 }$ 与单项式 $4 x ^ { 2 } y ^ { n }$ 的和是单项式", "target": "n = 3"}, {"rel": "代入", "source": "n = 3", "target": "5"}, {"rel": "被代入", "source": "m + n", "target": "5"}]}}
{"content": "The expression of the parabola obtained by shifting the parabola $y = { x } ^ { 2 } - 3$ 2 units to the left is:", "answer": "y = { ( x + 2 ) } ^ { 2 } - 3", "steps": "According to the principle of adding to the left and subtracting from the right, we know that the expression of the parabola $y = x ^ 2 - 3$ after shifting it 2 units to the left is $y = ( x + 2 ) ^ 2 - 3$.", "expr_cands": ["y = { x } ^ { 2 } - 3", "y", "x", "2", "y = x ^ { 2 } - 3", "y = ( x + 2 ) ^ { 2 } - 3", "x ^ { 2 } - 3 = ( x + 2 ) ^ { 2 } - 3", "x ^ { 2 } - 3", "y = { ( x + 2 ) } ^ { 2 } - 3"], "exprs": ["y = { ( x + 2 ) } ^ { 2 } - 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = { x } ^ { 2 } - 3"}, {"id": "y = { ( x + 2 ) } ^ { 2 } - 3"}, {"id": "2"}, {"id": "将抛物线 $y = { x } ^ { 2 } - 3$ 向左平移 $2$ 个单位后得到的抛物线表达式"}], "links": [{"rel": "被描述", "source": "y = { x } ^ { 2 } - 3", "target": "y = { ( x + 2 ) } ^ { 2 } - 3"}, {"rel": "被描述", "source": "2", "target": "y = { ( x + 2 ) } ^ { 2 } - 3"}, {"rel": "限制性描述", "source": "将抛物线 $y = { x } ^ { 2 } - 3$ 向左平移 $2$ 个单位后得到的抛物线表达式", "target": "y = { ( x + 2 ) } ^ { 2 } - 3"}]}}
{"content": "If the monomial $2 x { y } ^ { m }$ and $- 5 { x } ^ { 2 } { y } ^ { 2 }$ have the same degree with respect to $x$ and $y$, then the value of $m$ is ____?", "answer": "3", "steps": "From the given information, we have $1 + m = 2 + 2$, so $m = 3$.", "expr_cands": ["x", "y", "2 x { y } ^ { m }", "m", "- 5 { x } ^ { 2 } { y } ^ { 2 }", "1 + m = 2 + 2", "m = 3"], "exprs": ["1 + m = 2 + 2", "m = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 x { y } ^ { m }"}, {"id": "1 + m = 2 + 2"}, {"id": "- 5 { x } ^ { 2 } { y } ^ { 2 }"}, {"id": "关于 $x$ , $y$ 的单项式 $2 x { y } ^ { m }$ 与 $- 5 { x } ^ { 2 } { y } ^ { 2 }$ 次数相同"}, {"id": "m = 3"}], "links": [{"rel": "被描述", "source": "2 x { y } ^ { m }", "target": "1 + m = 2 + 2"}, {"rel": "等式方程求解", "source": "1 + m = 2 + 2", "target": "m = 3"}, {"rel": "被描述", "source": "- 5 { x } ^ { 2 } { y } ^ { 2 }", "target": "1 + m = 2 + 2"}, {"rel": "限制性描述", "source": "关于 $x$ , $y$ 的单项式 $2 x { y } ^ { m }$ 与 $- 5 { x } ^ { 2 } { y } ^ { 2 }$ 次数相同", "target": "1 + m = 2 + 2"}]}}
{"content": "If 3 times $x - 1$ is 2 more than 2 times $x - 3$, then $x$ = ____ ?", "answer": "- 1", "steps": "According to the problem, we have $3 ( x - 1 ) = 2 ( x - 3 ) + 2$. Expanding the brackets gives $3 x - 3 = 2 x - 6 + 2$. Rearranging terms gives $3 x - 2 x = - 6 + 2 + 3$. Combining like terms gives $x = - 1$.", "expr_cands": ["x - 1", "x", "3", "x - 3", "2", "3 ( x - 1 ) = 2 ( x - 3 ) + 2", "x = - 1", "3 x - 3 = 2 x - 6 + 2", "3 x - 2 x = - 6 + 2 + 3"], "exprs": ["3 ( x - 1 ) = 2 ( x - 3 ) + 2", "x = - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x - 1"}, {"id": "3 ( x - 1 ) = 2 ( x - 3 ) + 2"}, {"id": "3"}, {"id": "x - 3"}, {"id": "2"}, {"id": "$x - 1$ 的 $3$ 倍比 $x - 3$ 的 $2$ 倍大 $2$"}, {"id": "x = - 1"}], "links": [{"rel": "被描述", "source": "x - 1", "target": "3 ( x - 1 ) = 2 ( x - 3 ) + 2"}, {"rel": "等式方程求解", "source": "3 ( x - 1 ) = 2 ( x - 3 ) + 2", "target": "x = - 1"}, {"rel": "被描述", "source": "3", "target": "3 ( x - 1 ) = 2 ( x - 3 ) + 2"}, {"rel": "被描述", "source": "x - 3", "target": "3 ( x - 1 ) = 2 ( x - 3 ) + 2"}, {"rel": "被描述", "source": "2", "target": "3 ( x - 1 ) = 2 ( x - 3 ) + 2"}, {"rel": "限制性描述", "source": "$x - 1$ 的 $3$ 倍比 $x - 3$ 的 $2$ 倍大 $2$", "target": "3 ( x - 1 ) = 2 ( x - 3 ) + 2"}]}}
{"content": "The algebraic expression $\\frac { 1 } { \\sqrt { 2 x - 1 } }$ is meaningful, then the range of values for $x$ is ____?", "answer": "x > \\frac { 1 } { 2 }", "steps": "From the given condition, we have $2 x - 1 > 0$, which implies $x > \\frac { 1 } { 2 }$.", "expr_cands": ["\\frac { 1 } { \\sqrt { 2 x - 1 } }", "x", "2 x - 1 > 0", "\\frac { 1 } { 2 } < x", "x > \\frac { 1 } { 2 }"], "exprs": ["2 x - 1 > 0", "x > \\frac { 1 } { 2 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { 1 } { \\sqrt { 2 x - 1 } }"}, {"id": "2 x - 1 > 0"}, {"id": "代数式 $\\frac { 1 } { \\sqrt { 2 x - 1 } }$ 有意义"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "分式有意义，则分母不为0"}, {"id": "x > \\frac { 1 } { 2 }"}], "links": [{"rel": "被描述", "source": "\\frac { 1 } { \\sqrt { 2 x - 1 } }", "target": "2 x - 1 > 0"}, {"rel": "不等式方程求解", "source": "2 x - 1 > 0", "target": "x > \\frac { 1 } { 2 }"}, {"rel": "限制性描述", "source": "代数式 $\\frac { 1 } { \\sqrt { 2 x - 1 } }$ 有意义", "target": "2 x - 1 > 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "2 x - 1 > 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "2 x - 1 > 0"}]}}
{"content": "The range of $k$ for the one-variable linear equation $3 x - k + 1 = 2 x - 1$ with a negative root for $x$ is _____.", "answer": "k < 2", "steps": "The equation $3 x - k + 1 = 2 x - 1$ can be rearranged as $3 x - 2 x = k - 2$, which simplifies to $x = k - 2$. Since the root of the equation is negative, we have $k - 2 < 0$, which implies $k < 2$.", "expr_cands": ["x", "3 x - k + 1 = 2 x - 1", "k", "3 x - 2 x = k - 2", "x = k - 2", "k - 2 < 0", "k < 2"], "exprs": ["3 x - 2 x = k - 2", "x = k - 2", "k - 2 < 0", "k < 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 x - k + 1 = 2 x - 1"}, {"id": "3 x - 2 x = k - 2"}, {"id": "x = k - 2"}, {"id": "k - 2 < 0"}, {"id": "关于 $x$ 的一元一次方程 $3 x - k + 1 = 2 x - 1$ 的根是负数"}, {"id": "k < 2"}], "links": [{"rel": "移项", "source": "3 x - k + 1 = 2 x - 1", "target": "3 x - 2 x = k - 2"}, {"rel": "等式方程部分求解", "source": "3 x - 2 x = k - 2", "target": "x = k - 2"}, {"rel": "被描述", "source": "x = k - 2", "target": "k - 2 < 0"}, {"rel": "不等式方程求解", "source": "k - 2 < 0", "target": "k < 2"}, {"rel": "限制性描述", "source": "关于 $x$ 的一元一次方程 $3 x - k + 1 = 2 x - 1$ 的根是负数", "target": "k - 2 < 0"}]}}
{"content": "To make the value of the fraction $\\frac { x - 4 } { 2 x + 6 }$ equal to $0$, what is the value of $x$?", "answer": "x = 4", "steps": "$\\because$ To make the value of the fraction $\\frac { x - 4 } { 2 x + 6 }$ equal to $0$, $\\therefore$ $x - 4 = 0$, which gives $x = 4$.", "expr_cands": ["\\frac { x - 4 } { 2 x + 6 }", "x", "0", "x - 4 = 0", "x = 4"], "exprs": ["x - 4 = 0", "x = 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x - 4 } { 2 x + 6 }"}, {"id": "x - 4 = 0"}, {"id": "0"}, {"id": "要使分式 $\\frac { x - 4 } { 2 x + 6 }$ 的值等于 $0$"}, {"id": "分式有意义，则分母不为0"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "x = 4"}], "links": [{"rel": "被描述", "source": "\\frac { x - 4 } { 2 x + 6 }", "target": "x - 4 = 0"}, {"rel": "等式方程求解", "source": "x - 4 = 0", "target": "x = 4"}, {"rel": "被描述", "source": "0", "target": "x - 4 = 0"}, {"rel": "限制性描述", "source": "要使分式 $\\frac { x - 4 } { 2 x + 6 }$ 的值等于 $0$", "target": "x - 4 = 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "x - 4 = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x - 4 = 0"}]}}
{"content": "If $x = - 1$ is a solution of the one-variable linear equation $ax + 3 = 0$, then the value of $a$ is ____?", "answer": "3", "steps": "$\\because x = - 1$ is a solution to the one-variable linear equation $ax + 3 = 0$, $\\therefore - a + 3 = 0$, and solving for $a$ gives $a = 3$.", "expr_cands": ["x = - 1", "x", "ax + 3 = 0", "a", "3 - a = 0", "- a + 3 = 0", "a = 3"], "exprs": ["- a + 3 = 0", "a = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "ax + 3 = 0"}, {"id": "- a + 3 = 0"}, {"id": "x = - 1"}, {"id": "x"}, {"id": "$x = - 1$ 是关于 $x$ 的一元一次方程 $ax + 3 = 0$ 的解"}, {"id": "a = 3"}], "links": [{"rel": "被描述", "source": "ax + 3 = 0", "target": "- a + 3 = 0"}, {"rel": "等式方程求解", "source": "- a + 3 = 0", "target": "a = 3"}, {"rel": "被描述", "source": "x = - 1", "target": "- a + 3 = 0"}, {"rel": "被描述", "source": "x", "target": "- a + 3 = 0"}, {"rel": "限制性描述", "source": "$x = - 1$ 是关于 $x$ 的一元一次方程 $ax + 3 = 0$ 的解", "target": "- a + 3 = 0"}]}}
{"content": "The equation $( 2 x - 1 ) ^ 3 = 27$ has a value of $x$ equal to _____.", "answer": "2", "steps": "Because $( 2 x - 1 ) ^ 3 = 27$, therefore $2 x - 1 = 3$, therefore $2 x = 4$, which leads to the solution: $x = 2$.", "expr_cands": ["( 2 x - 1 ) ^ { 3 } = 27", "x", "x = 2", "2 x - 1 = 3", "2 x = 4"], "exprs": ["x = 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( 2 x - 1 ) ^ { 3 } = 27"}, {"id": "x = 2"}], "links": [{"rel": "等式方程求解", "source": "( 2 x - 1 ) ^ { 3 } = 27", "target": "x = 2"}]}}
{"content": "If $a ^ { 7 } \\times a ^ { m } = a ^ { 2 } \\times a ^ { 8 }$, then $m$ = ____ ?", "answer": "3", "steps": "Because $a ^ { 7 } \\times a ^ { m } = a ^ { 2 } \\times a ^ { 8 }$, therefore $a ^ { 7 + m } = a ^ { 10 }$, therefore $7 + m = 10$, therefore $m = 3$.", "expr_cands": ["a ^ { 7 } \\times a ^ { m } = a ^ { 2 } \\times a ^ { 8 }", "a", "m", "a ^ { 7 + m } = a ^ { 10 }", "7 + m = 10", "m = 3"], "exprs": ["7 + m = 10", "m = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a ^ { 7 } \\times a ^ { m } = a ^ { 2 } \\times a ^ { 8 }"}, {"id": "7 + m = 10"}, {"id": "m = 3"}], "links": [{"rel": "同取对数", "source": "a ^ { 7 } \\times a ^ { m } = a ^ { 2 } \\times a ^ { 8 }", "target": "7 + m = 10"}, {"rel": "等式方程求解", "source": "7 + m = 10", "target": "m = 3"}]}}
{"content": "If the solution set of the inequality $x - a < 2$ with respect to $x$ is $x < 1$, then $a$ = ____ ?", "answer": "- 1", "steps": "From the inequality $x - a < 2$, we have $x < 2 + a$. Since the solution set of the inequality $x - a < 2$ with respect to $x$ is $x < 1$, we have $2 + a = 1$. Therefore, $a = - 1$.", "expr_cands": ["x", "x - a < 2", "a", "x < 1", "x < 2 + a", "2 + a = 1", "a = - 1"], "exprs": ["x < 2 + a", "2 + a = 1", "a = - 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x - a < 2"}, {"id": "x < 2 + a"}, {"id": "x < 1"}, {"id": "2 + a = 1"}, {"id": "关于 $x$ 的不等式 $x - a < 2$ 的解集是 $x < 1$"}, {"id": "a = - 1"}], "links": [{"rel": "不等式方程部分求解", "source": "x - a < 2", "target": "x < 2 + a"}, {"rel": "被描述", "source": "x < 2 + a", "target": "2 + a = 1"}, {"rel": "被描述", "source": "x < 1", "target": "2 + a = 1"}, {"rel": "等式方程求解", "source": "2 + a = 1", "target": "a = - 1"}, {"rel": "限制性描述", "source": "关于 $x$ 的不等式 $x - a < 2$ 的解集是 $x < 1$", "target": "2 + a = 1"}]}}
{"content": "If the value of $4 x + 2$ is the opposite of $3 x - 9$, then the value of $x$ is ____?", "answer": "1", "steps": "According to the problem, we have $4 x + 2 + 3 x - 9 = 0$. By rearranging and combining terms, we get $7 x = 7$. Solving for $x$, we get $x = 1$.", "expr_cands": ["4 x + 2", "x", "3 x - 9", "4 x + 2 + 3 x - 9 = 0", "x = 1", "7 x = 7"], "exprs": ["4 x + 2 + 3 x - 9 = 0", "x = 1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 x + 2"}, {"id": "4 x + 2 + 3 x - 9 = 0"}, {"id": "3 x - 9"}, {"id": "$4 x + 2$ 与 $3 x - 9$ 的值互为相反数"}, {"id": "x = 1"}], "links": [{"rel": "被描述", "source": "4 x + 2", "target": "4 x + 2 + 3 x - 9 = 0"}, {"rel": "等式方程求解", "source": "4 x + 2 + 3 x - 9 = 0", "target": "x = 1"}, {"rel": "被描述", "source": "3 x - 9", "target": "4 x + 2 + 3 x - 9 = 0"}, {"rel": "限制性描述", "source": "$4 x + 2$ 与 $3 x - 9$ 的值互为相反数", "target": "4 x + 2 + 3 x - 9 = 0"}]}}
{"content": "If the product of $( x + p ) ( x + \\frac { 1 } { 7 })$ does not contain a linear term in $x$, then the value of $p$ is ____?", "answer": "- \\frac { 1 } { 7 }", "steps": "$( x + p ) ( x + \\frac { 1 } { 7 } ) = x ^ { 2 } + ( p + \\frac { 1 } { 7 } ) x + \\frac { p } { 7 }$ , from the fact that there is no linear term in $x$ in the result, we get $p + \\frac { 1 } { 7 } = 0$ , solving for $p$ gives: $p = - \\frac { 1 } { 7 }$ . ", "expr_cands": ["( x + p ) ( x + \\frac { 1 } { 7 } )", "p", "x", "x ^ { 2 } + ( p + \\frac { 1 } { 7 } ) x + \\frac { p } { 7 }", "p + \\frac { 1 } { 7 } = 0", "p = - \\frac { 1 } { 7 }"], "exprs": ["x ^ { 2 } + ( p + \\frac { 1 } { 7 } ) x + \\frac { p } { 7 }", "p + \\frac { 1 } { 7 } = 0", "p = - \\frac { 1 } { 7 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( x + p ) ( x + \\frac { 1 } { 7 } )"}, {"id": "x ^ { 2 } + ( p + \\frac { 1 } { 7 } ) x + \\frac { p } { 7 }"}, {"id": "x"}, {"id": "p + \\frac { 1 } { 7 } = 0"}, {"id": "$( x + p ) ( x + \\frac { 1 } { 7 } )$ 的积中不含 $x$ 的一次项"}, {"id": "p = - \\frac { 1 } { 7 }"}], "links": [{"rel": "提取因式", "source": "( x + p ) ( x + \\frac { 1 } { 7 } )", "target": "x ^ { 2 } + ( p + \\frac { 1 } { 7 } ) x + \\frac { p } { 7 }"}, {"rel": "被描述", "source": "x ^ { 2 } + ( p + \\frac { 1 } { 7 } ) x + \\frac { p } { 7 }", "target": "p + \\frac { 1 } { 7 } = 0"}, {"rel": "提取因式参考", "source": "x", "target": "x ^ { 2 } + ( p + \\frac { 1 } { 7 } ) x + \\frac { p } { 7 }"}, {"rel": "等式方程求解", "source": "p + \\frac { 1 } { 7 } = 0", "target": "p = - \\frac { 1 } { 7 }"}, {"rel": "限制性描述", "source": "$( x + p ) ( x + \\frac { 1 } { 7 } )$ 的积中不含 $x$ 的一次项", "target": "p + \\frac { 1 } { 7 } = 0"}]}}
{"content": "For solving the inequality $- \\frac { 2 } { 3 } x > \\frac { 3 } { 2 }$, the correct result is ____ ?", "answer": "x < - \\frac { 9 } { 4 }", "steps": "To eliminate the denominator, we get $- 4 x > 9$. By making the coefficient of $x$ equal to $1$, we have $x < - \\frac { 9 } { 4 }$.", "expr_cands": ["- \\frac { 2 } { 3 } x > \\frac { 3 } { 2 }", "x", "- 4 x > 9", "x < - \\frac { 9 } { 4 }", "1"], "exprs": ["x < - \\frac { 9 } { 4 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "- \\frac { 2 } { 3 } x > \\frac { 3 } { 2 }"}, {"id": "x < - \\frac { 9 } { 4 }"}], "links": [{"rel": "不等式方程求解", "source": "- \\frac { 2 } { 3 } x > \\frac { 3 } { 2 }", "target": "x < - \\frac { 9 } { 4 }"}]}}
{"content": "The value of the algebraic expression $3 ( x - 2 ) + 1$ is greater than $\\frac { 1 } { 2 }$. What is the range of possible values for $x$?", "answer": "x > \\frac { 11 } { 6 }", "steps": "From the given information, we can derive that $3 ( x - 2 ) + 1 > \\frac { 1 } { 2 }$. Solving the inequality, we get $x > \\frac { 11 } { 6 }$. Therefore, the possible values of $x$ are $x > \\frac { 11 } { 6 }$.", "expr_cands": ["3 ( x - 2 ) + 1", "x", "\\frac { 1 } { 2 }", "3 ( x - 2 ) + 1 > \\frac { 1 } { 2 }", "\\frac { 11 } { 6 } < x", "x > \\frac { 11 } { 6 }"], "exprs": ["3 ( x - 2 ) + 1 > \\frac { 1 } { 2 }", "x > \\frac { 11 } { 6 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 ( x - 2 ) + 1"}, {"id": "3 ( x - 2 ) + 1 > \\frac { 1 } { 2 }"}, {"id": "\\frac { 1 } { 2 }"}, {"id": "代数式 $3 ( x - 2 ) + 1$ 的值大于 $\\frac { 1 } { 2 }$"}, {"id": "x > \\frac { 11 } { 6 }"}], "links": [{"rel": "被描述", "source": "3 ( x - 2 ) + 1", "target": "3 ( x - 2 ) + 1 > \\frac { 1 } { 2 }"}, {"rel": "不等式方程求解", "source": "3 ( x - 2 ) + 1 > \\frac { 1 } { 2 }", "target": "x > \\frac { 11 } { 6 }"}, {"rel": "被描述", "source": "\\frac { 1 } { 2 }", "target": "3 ( x - 2 ) + 1 > \\frac { 1 } { 2 }"}, {"rel": "限制性描述", "source": "代数式 $3 ( x - 2 ) + 1$ 的值大于 $\\frac { 1 } { 2 }$", "target": "3 ( x - 2 ) + 1 > \\frac { 1 } { 2 }"}]}}
{"content": "If $x = m$ is a solution to the equation $\\frac { 1 } { 2 } x - m = 1$ in terms of $x$, then the value of $m$ is ____?", "answer": "- 2", "steps": "Substituting $x = m$ into the equation, we get $\\frac { m } { 2 } - m = 1$. Solving for $m$, we get $m = - 2$.", "expr_cands": ["x = m", "m", "x", "\\frac { 1 } { 2 } x - m = 1", "\\frac { m } { 2 } - m = 1", "m = - 2"], "exprs": ["\\frac { m } { 2 } - m = 1", "m = - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x = m"}, {"id": "\\frac { m } { 2 } - m = 1"}, {"id": "x"}, {"id": "\\frac { 1 } { 2 } x - m = 1"}, {"id": "$x = m$ 是关于 $x$ 的方程 $\\frac { 1 } { 2 } x - m = 1$ 的解"}, {"id": "m = - 2"}], "links": [{"rel": "被描述", "source": "x = m", "target": "\\frac { m } { 2 } - m = 1"}, {"rel": "等式方程求解", "source": "\\frac { m } { 2 } - m = 1", "target": "m = - 2"}, {"rel": "被描述", "source": "x", "target": "\\frac { m } { 2 } - m = 1"}, {"rel": "被描述", "source": "\\frac { 1 } { 2 } x - m = 1", "target": "\\frac { m } { 2 } - m = 1"}, {"rel": "限制性描述", "source": "$x = m$ 是关于 $x$ 的方程 $\\frac { 1 } { 2 } x - m = 1$ 的解", "target": "\\frac { m } { 2 } - m = 1"}]}}
{"content": "$8$. The sum of the two roots of the equation $x ^ 2 - 3 x - 4 = 0$ is ____?", "answer": "3", "steps": "If the roots of the equation $x ^ 2 - 3 x - 4 = 0$ are $x _ 1$ and $x _ 2$, then $x _ 1 + x _ 2 = 3$.", "expr_cands": ["8", "x ^ { 2 } - 3 x - 4 = 0", "x", "x = - 1", "x = 4", "x _ { 1 }", "x _ { 2 }", "x _ { 1 } + x _ { 2 } = 3", "x _ { 1 } + x _ { 2 }", "3"], "exprs": ["3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "x ^ { 2 } - 3 x - 4 = 0"}, {"id": "3"}, {"id": "方程 $x ^ { 2 } - 3 x - 4 = 0$ 的两根之和"}, {"id": "一元二次方程根与系数关系，两根之和"}], "links": [{"rel": "被描述", "source": "x ^ { 2 } - 3 x - 4 = 0", "target": "3"}, {"rel": "限制性描述", "source": "方程 $x ^ { 2 } - 3 x - 4 = 0$ 的两根之和", "target": "3"}, {"rel": "属性描述", "source": "一元二次方程根与系数关系，两根之和", "target": "3"}]}}
{"content": "For the linear function $y = ( 3 k + 6 ) x - k$, if the function value $y$ decreases as $x$ increases, then the range of possible values for $k$ is ____?", "answer": "k < - 2", "steps": "$\\because$ The linear function $y = ( 3 k + 6 ) x - k$ decreases as $x$ increases. $\\therefore$ $3 k + 6 < 0$, and solving for $k$ gives $k < - 2$.", "expr_cands": ["y = ( 3 k + 6 ) x - k", "k", "y", "x", "3 k + 6 < 0", "k < - 2"], "exprs": ["3 k + 6 < 0", "k < - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = ( 3 k + 6 ) x - k"}, {"id": "3 k + 6 < 0"}, {"id": "对于一次函数 $y = ( 3 k + 6 ) x - k$"}, {"id": "函数值 $y$ 随 $x$ 的增大而减小"}, {"id": "k < - 2"}], "links": [{"rel": "被描述", "source": "y = ( 3 k + 6 ) x - k", "target": "3 k + 6 < 0"}, {"rel": "不等式方程求解", "source": "3 k + 6 < 0", "target": "k < - 2"}, {"rel": "限制性描述", "source": "对于一次函数 $y = ( 3 k + 6 ) x - k$", "target": "3 k + 6 < 0"}, {"rel": "限制性描述", "source": "函数值 $y$ 随 $x$ 的增大而减小", "target": "3 k + 6 < 0"}]}}
{"content": "The range of values of $x$ that make the algebraic expression $\\sqrt { \\frac { 1 } { 5 - x }}$ meaningful is _____.", "answer": "x < 5", "steps": "$\\because$ The algebraic expression $\\sqrt { \\frac { 1 } { 5 - x }}$ is meaningful, $\\therefore \\frac { 1 } { 5 - x } > 0$, which implies $x < 5$.", "expr_cands": ["\\sqrt { \\frac { 1 } { 5 - x } }", "x", "\\frac { 1 } { 5 - x } > 0", "x < 5"], "exprs": ["\\frac { 1 } { 5 - x } > 0", "x < 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { \\frac { 1 } { 5 - x } }"}, {"id": "\\frac { 1 } { 5 - x } > 0"}, {"id": "代数式 $\\sqrt { \\frac { 1 } { 5 - x } }$ 有意义"}, {"id": "使得代数式 $\\sqrt { \\frac { 1 } { 5 - x } }$ 有意义的 $x$ 的取值范围"}, {"id": "分式有意义，则分母不为0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x < 5"}], "links": [{"rel": "被描述", "source": "\\sqrt { \\frac { 1 } { 5 - x } }", "target": "\\frac { 1 } { 5 - x } > 0"}, {"rel": "不等式方程求解", "source": "\\frac { 1 } { 5 - x } > 0", "target": "x < 5"}, {"rel": "限制性描述", "source": "代数式 $\\sqrt { \\frac { 1 } { 5 - x } }$ 有意义", "target": "\\frac { 1 } { 5 - x } > 0"}, {"rel": "限制性描述", "source": "使得代数式 $\\sqrt { \\frac { 1 } { 5 - x } }$ 有意义的 $x$ 的取值范围", "target": "\\frac { 1 } { 5 - x } > 0"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "\\frac { 1 } { 5 - x } > 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "\\frac { 1 } { 5 - x } > 0"}]}}
{"content": "$5 m + 8$ is the opposite of $16 - 9 m$, then $m$ = ____ ?", "answer": "6", "steps": "According to the problem, we have $5 m + 8 + 16 - 9 m = 0$. Moving terms around, we get $5 m - 9 m = - 8 - 16$. Combining like terms, we get $- 4 m = - 24$. Dividing both sides by $- 4$, we get $m = 6$.", "expr_cands": ["5 m + 8", "m", "16 - 9 m", "5 m + 8 + 16 - 9 m = 0", "m = 6", "5 m - 9 m = - 8 - 16", "- 4 m = - 24", "1"], "exprs": ["5 m + 8 + 16 - 9 m = 0", "m = 6"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5 m + 8"}, {"id": "5 m + 8 + 16 - 9 m = 0"}, {"id": "16 - 9 m"}, {"id": "$5 m + 8$ 与 $16 - 9 m$ 互为相反数"}, {"id": "m = 6"}], "links": [{"rel": "被描述", "source": "5 m + 8", "target": "5 m + 8 + 16 - 9 m = 0"}, {"rel": "等式方程求解", "source": "5 m + 8 + 16 - 9 m = 0", "target": "m = 6"}, {"rel": "被描述", "source": "16 - 9 m", "target": "5 m + 8 + 16 - 9 m = 0"}, {"rel": "限制性描述", "source": "$5 m + 8$ 与 $16 - 9 m$ 互为相反数", "target": "5 m + 8 + 16 - 9 m = 0"}]}}
{"content": "If the value of $2 ( x + 3 )$ is the opposite of the value of $3 ( 1 - x )$, then $x$ equals ____?", "answer": "9", "steps": "According to the problem, we have $2 ( x + 3 ) + 3 ( 1 - x ) = 0$. Expanding the brackets, we get $2 x + 6 + 3 - 3 x = 0$. Rearranging and simplifying, we get $- x = - 9$. Solving for $x$, we get $x = 9$.", "expr_cands": ["2 ( x + 3 )", "x", "3 ( 1 - x )", "2 ( x + 3 ) + 3 ( 1 - x ) = 0", "x = 9", "2 x + 6 + 3 - 3 x = 0", "- x = - 9"], "exprs": ["2 ( x + 3 ) + 3 ( 1 - x ) = 0", "x = 9"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "2 ( x + 3 )"}, {"id": "2 ( x + 3 ) + 3 ( 1 - x ) = 0"}, {"id": "3 ( 1 - x )"}, {"id": "$2 ( x + 3 )$ 的值与 $3 ( 1 - x )$ 的值互为相反数"}, {"id": "x = 9"}], "links": [{"rel": "被描述", "source": "2 ( x + 3 )", "target": "2 ( x + 3 ) + 3 ( 1 - x ) = 0"}, {"rel": "等式方程求解", "source": "2 ( x + 3 ) + 3 ( 1 - x ) = 0", "target": "x = 9"}, {"rel": "被描述", "source": "3 ( 1 - x )", "target": "2 ( x + 3 ) + 3 ( 1 - x ) = 0"}, {"rel": "限制性描述", "source": "$2 ( x + 3 )$ 的值与 $3 ( 1 - x )$ 的值互为相反数", "target": "2 ( x + 3 ) + 3 ( 1 - x ) = 0"}]}}
{"content": "If $\\frac { a } { b } = \\frac { 3 } { 4 }$ and $a + b = 14$, then the value of $2 a - b$ is ____?", "answer": "4", "steps": "Since $\\frac { a } { b } = \\frac { 3 } { 4 }$, we have $a = \\frac { 3 } { 4 } b$. Since $a + b = 14$, we have $\\frac { 3 } { 4 } b + b = 14$. Solving for $b$, we get $b = 8$, and thus $a = 6$. Therefore, $2 a - b = 2 ( 6 ) - 8 = 4$.", "expr_cands": ["\\frac { a } { b } = \\frac { 3 } { 4 }", "b", "a", "a + b = 14", "2 a - b", "a = \\frac { 3 } { 4 } b", "\\frac { 7 b } { 4 } = 14", "\\frac { 3 } { 4 } b + b = 14", "b = 8", "a = 6", "4"], "exprs": ["a = \\frac { 3 } { 4 } b", "\\frac { 3 } { 4 } b + b = 14", "b = 8", "a = 6", "4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { a } { b } = \\frac { 3 } { 4 }"}, {"id": "a = \\frac { 3 } { 4 } b"}, {"id": "a + b = 14"}, {"id": "\\frac { 3 } { 4 } b + b = 14"}, {"id": "b = 8"}, {"id": "a = 6"}, {"id": "2 a - b"}, {"id": "4"}], "links": [{"rel": "等式方程部分求解", "source": "\\frac { a } { b } = \\frac { 3 } { 4 }", "target": "a = \\frac { 3 } { 4 } b"}, {"rel": "代入", "source": "a = \\frac { 3 } { 4 } b", "target": "\\frac { 3 } { 4 } b + b = 14"}, {"rel": "被代入", "source": "a = \\frac { 3 } { 4 } b", "target": "a = 6"}, {"rel": "被代入", "source": "a + b = 14", "target": "\\frac { 3 } { 4 } b + b = 14"}, {"rel": "等式方程求解", "source": "\\frac { 3 } { 4 } b + b = 14", "target": "b = 8"}, {"rel": "代入", "source": "b = 8", "target": "a = 6"}, {"rel": "代入", "source": "b = 8", "target": "4"}, {"rel": "代入", "source": "a = 6", "target": "4"}, {"rel": "被代入", "source": "2 a - b", "target": "4"}]}}
{"content": "For the linear function $y = ( k - 3 ) x + 2$ with respect to $x$, $y$ increases as $x$ increases. The range of values for $k$ is _____.", "answer": "k > 3", "steps": "$\\because$ The linear function $y = ( k - 3 ) x + 2$ increases as $x$ increases. $\\therefore$ $k - 3 > 0$, which implies $k > 3$.", "expr_cands": ["x", "y = ( k - 3 ) x + 2", "k", "y", "k - 3 > 0", "3 < k", "k > 3"], "exprs": ["k - 3 > 0", "k > 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = ( k - 3 ) x + 2"}, {"id": "k - 3 > 0"}, {"id": "对于关于 $x$ 的一次函数 $y = ( k - 3 ) x + 2$ , $y$ 随 $x$ 的增大而增大"}, {"id": "k > 3"}], "links": [{"rel": "被描述", "source": "y = ( k - 3 ) x + 2", "target": "k - 3 > 0"}, {"rel": "不等式方程求解", "source": "k - 3 > 0", "target": "k > 3"}, {"rel": "限制性描述", "source": "对于关于 $x$ 的一次函数 $y = ( k - 3 ) x + 2$ , $y$ 随 $x$ 的增大而增大", "target": "k - 3 > 0"}]}}
{"content": "If $a$ and $b$ are opposite numbers, and $c$ and $d$ are reciprocal numbers, then the value of $\\frac { a + b } { 3 } + 3 cd$ is ____?", "answer": "3", "steps": "According to the problem, we have $a + b = 0$ and $cd = 1$. Therefore, the original expression is equal to $0 + 3 = 3$.", "expr_cands": ["a", "b", "c", "d", "\\frac { a + b } { 3 } + 3 cd", "a + b = 0", "cd = 1", "0 + 3", "3"], "exprs": ["a + b = 0", "cd = 1", "0 + 3", "3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a + b = 0"}, {"id": "b"}, {"id": "$a$ , $b$ 互为相反数"}, {"id": "c"}, {"id": "cd = 1"}, {"id": "d"}, {"id": "$c$ , $d$ 互为倒数"}, {"id": "\\frac { a + b } { 3 } + 3 cd"}, {"id": "0 + 3"}, {"id": "3"}], "links": [{"rel": "被描述", "source": "a", "target": "a + b = 0"}, {"rel": "代入", "source": "a + b = 0", "target": "0 + 3"}, {"rel": "被描述", "source": "b", "target": "a + b = 0"}, {"rel": "限制性描述", "source": "$a$ , $b$ 互为相反数", "target": "a + b = 0"}, {"rel": "被描述", "source": "c", "target": "cd = 1"}, {"rel": "代入", "source": "cd = 1", "target": "0 + 3"}, {"rel": "被描述", "source": "d", "target": "cd = 1"}, {"rel": "限制性描述", "source": "$c$ , $d$ 互为倒数", "target": "cd = 1"}, {"rel": "被代入", "source": "\\frac { a + b } { 3 } + 3 cd", "target": "0 + 3"}, {"rel": "计算", "source": "0 + 3", "target": "3"}]}}
{"content": "Regarding the algebraic expression of $x$ { , } $y$ in $( - 3 { kxy } + 3 y ) + ( 9 { xy } - 8 x + 1 )$, which does not contain quadratic terms, $k$ = ____?", "answer": "3", "steps": "$\\because$ The algebraic expression in terms of $x$ and $y$, $( - 3 kxy + 3 y ) + ( 9 xy - 8 x + 1 )$, does not contain any quadratic terms. $\\therefore$ $- 3 k + 9 = 0$, which solves for $k = 3$.", "expr_cands": ["x", "y", "( - 3 { kxy } + 3 y ) + ( 9 { xy } - 8 x + 1 )", "k", "( - 3 kxy + 3 y ) + ( 9 xy - 8 x + 1 )", "- 3 k + 9 = 0", "k = 3"], "exprs": ["- 3 k + 9 = 0", "k = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "( - 3 { kxy } + 3 y ) + ( 9 { xy } - 8 x + 1 )"}, {"id": "- 3 k + 9 = 0"}, {"id": "关于 $x$ , $y$ 的代数式 $( - 3 kxy + 3 y ) + ( 9 xy - 8 x + 1 )$ 中不含二次项"}, {"id": "k = 3"}], "links": [{"rel": "被描述", "source": "( - 3 { kxy } + 3 y ) + ( 9 { xy } - 8 x + 1 )", "target": "- 3 k + 9 = 0"}, {"rel": "等式方程求解", "source": "- 3 k + 9 = 0", "target": "k = 3"}, {"rel": "限制性描述", "source": "关于 $x$ , $y$ 的代数式 $( - 3 kxy + 3 y ) + ( 9 xy - 8 x + 1 )$ 中不含二次项", "target": "- 3 k + 9 = 0"}]}}
{"content": "Given $5 x - 3 * \\frac { 1 } { 5 } = 0.8$, what is the value of $x$?", "answer": "\\frac { 7 } { 25 }", "steps": "$5 x - 3 * \\frac { 1 } { 5 } = 0.8$ is rearranged to $5 x - 0.6 = 0.8$. Then, moving the constant term to the right side gives $5 x = 0.8 + 0.6$. Combining like terms yields $5 x = 1.4$. Dividing both sides by the coefficient of $x$ gives $x = \\frac { 7 } { 25 }$.", "expr_cands": ["5 x - 3 * \\frac { 1 } { 5 } = 0.8", "x", "x = 0.28", "5 x - 0.6 = 0.8", "5 x = 0.8 + 0.6", "5 x", "1.4", "1", "x = \\frac { 7 } { 25 }"], "exprs": ["5 x = 0.8 + 0.6", "x = \\frac { 7 } { 25 }"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "5 x - 3 * \\frac { 1 } { 5 } = 0.8"}, {"id": "5 x = 0.8 + 0.6"}, {"id": "x = \\frac { 7 } { 25 }"}], "links": [{"rel": "移项", "source": "5 x - 3 * \\frac { 1 } { 5 } = 0.8", "target": "5 x = 0.8 + 0.6"}, {"rel": "等式方程求解", "source": "5 x = 0.8 + 0.6", "target": "x = \\frac { 7 } { 25 }"}]}}
{"content": "Given that $y$ is inversely proportional to $x$, and when $x = 2$, $y = - 1$, what is the value of $x$ when $y = \\frac { 1 } { 2 }$?", "answer": "- 4", "steps": "$\\because$ $y$ is inversely proportional to $x$, $\\therefore$ let the analytic expression of the inverse proportion function be $y = \\frac { k } { x }$ ($k \\neq 0$). $\\because$ when $x = 2$, $y = - 1$, that is $- 1 = \\frac { k } { 2 }$, $k = - 2$. Therefore, the analytic expression of the inverse proportion function is $y = - \\frac { 2 } { x }$. Then when $y = \\frac { 1 } { 2 }$, that is $\\frac { 1 } { 2 } = - \\frac { 2 } { x }$, $x = - 4$.", "expr_cands": ["y", "x", "x = 2", "y = - 1", "y = \\frac { 1 } { 2 }", "y = \\frac { k } { x } ( k \\neq 0 )", "k", "- 1 = \\frac { k } { 2 }", "k = - 2", "y = - \\frac { 2 } { x }", "\\frac { 1 } { 2 } = - \\frac { 2 } { x }", "x = - 4"], "exprs": ["y = \\frac { k } { x } ( k \\neq 0 )", "- 1 = \\frac { k } { 2 }", "k = - 2", "y = - \\frac { 2 } { x }", "\\frac { 1 } { 2 } = - \\frac { 2 } { x }", "x = - 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "设反比例函数的解析式为 $y = \\frac { k } { x } ( k \\neq 0 )$"}, {"id": "y = \\frac { k } { x } ( k \\neq 0 )"}, {"id": "x = 2"}, {"id": "- 1 = \\frac { k } { 2 }"}, {"id": "y = - 1"}, {"id": "k = - 2"}, {"id": "y = - \\frac { 2 } { x }"}, {"id": "y = \\frac { 1 } { 2 }"}, {"id": "\\frac { 1 } { 2 } = - \\frac { 2 } { x }"}, {"id": "x = - 4"}], "links": [{"rel": "假设描述", "source": "设反比例函数的解析式为 $y = \\frac { k } { x } ( k \\neq 0 )$", "target": "y = \\frac { k } { x } ( k \\neq 0 )"}, {"rel": "被代入", "source": "y = \\frac { k } { x } ( k \\neq 0 )", "target": "- 1 = \\frac { k } { 2 }"}, {"rel": "被代入", "source": "y = \\frac { k } { x } ( k \\neq 0 )", "target": "y = - \\frac { 2 } { x }"}, {"rel": "代入", "source": "x = 2", "target": "- 1 = \\frac { k } { 2 }"}, {"rel": "等式方程求解", "source": "- 1 = \\frac { k } { 2 }", "target": "k = - 2"}, {"rel": "代入", "source": "y = - 1", "target": "- 1 = \\frac { k } { 2 }"}, {"rel": "代入", "source": "k = - 2", "target": "y = - \\frac { 2 } { x }"}, {"rel": "被代入", "source": "y = - \\frac { 2 } { x }", "target": "\\frac { 1 } { 2 } = - \\frac { 2 } { x }"}, {"rel": "代入", "source": "y = \\frac { 1 } { 2 }", "target": "\\frac { 1 } { 2 } = - \\frac { 2 } { x }"}, {"rel": "等式方程求解", "source": "\\frac { 1 } { 2 } = - \\frac { 2 } { x }", "target": "x = - 4"}]}}
{"content": "When using the method of completing the square to solve the equation $4 x ^ { 2 } - 4 x = 3$, both sides of the equation should be added with ____?", "answer": "1", "steps": "When using the quadratic formula to solve the equation $4 x ^ { 2 } - 4 x = 3$, both sides of the equation should be added by $1$.", "expr_cands": ["4 x ^ { 2 } - 4 x = 3", "x", "x = - \\frac { 1 } { 2 }", "x = \\frac { 3 } { 2 }", "1"], "exprs": ["1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "4 x ^ { 2 } - 4 x = 3"}, {"id": "1"}, {"id": "方程的两边都应加上 $1$"}], "links": [{"rel": "被描述", "source": "4 x ^ { 2 } - 4 x = 3", "target": "1"}, {"rel": "限制性描述", "source": "方程的两边都应加上 $1$", "target": "1"}]}}
{"content": "When $x$ = ____ ?, the value of the fraction $\\frac { x - 3 } { x + 2 }$ is equal to $0$.", "answer": "3", "steps": "From the given information, we have $x - 3 = 0$ and $x + 2 \\neq 0$, which implies $x = 3$.", "expr_cands": ["x", "\\frac { x - 3 } { x + 2 }", "0", "x - 3 = 0", "x = 3", "x + 2 \\neq 0", "x \\neq - 2"], "exprs": ["x - 3 = 0", "x + 2 \\neq 0", "x = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { x - 3 } { x + 2 }"}, {"id": "x - 3 = 0"}, {"id": "0"}, {"id": "分式 $\\frac { x - 3 } { x + 2 }$ 的值等于 $0$"}, {"id": "分式为0，则分子为0，分母不为0"}, {"id": "x + 2 \\neq 0"}, {"id": "x = 3"}], "links": [{"rel": "被描述", "source": "\\frac { x - 3 } { x + 2 }", "target": "x - 3 = 0"}, {"rel": "被描述", "source": "\\frac { x - 3 } { x + 2 }", "target": "x + 2 \\neq 0"}, {"rel": "联立", "source": "x - 3 = 0", "target": "x = 3"}, {"rel": "被描述", "source": "0", "target": "x - 3 = 0"}, {"rel": "被描述", "source": "0", "target": "x + 2 \\neq 0"}, {"rel": "限制性描述", "source": "分式 $\\frac { x - 3 } { x + 2 }$ 的值等于 $0$", "target": "x - 3 = 0"}, {"rel": "限制性描述", "source": "分式 $\\frac { x - 3 } { x + 2 }$ 的值等于 $0$", "target": "x + 2 \\neq 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x - 3 = 0"}, {"rel": "属性描述", "source": "分式为0，则分子为0，分母不为0", "target": "x + 2 \\neq 0"}, {"rel": "联立", "source": "x + 2 \\neq 0", "target": "x = 3"}]}}
{"content": "If the sum of $x$ and twice 8 is equal to three times the difference between $x$ and 1, then $x$ = ____ ?", "answer": "19", "steps": "According to the problem, we have $2 ( x + 8 ) = 3 ( x - 1 )$. Expanding the brackets, we get $2 x + 16 = 3 x - 3$. Rearranging and combining like terms, we have $- x = - 19$. Solving for $x$, we get $x = 19$.", "expr_cands": ["x", "8", "1", "2 ( x + 8 ) = 3 ( x - 1 )", "x = 19", "2 x + 16 = 3 x - 3", "- x = - 19"], "exprs": ["2 ( x + 8 ) = 3 ( x - 1 )", "x = 19"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "8"}, {"id": "2 ( x + 8 ) = 3 ( x - 1 )"}, {"id": "x"}, {"id": "1"}, {"id": "$x$ 与 $8$ 之和的两倍等于 $x$ 与 $1$ 之差的三倍"}, {"id": "x = 19"}], "links": [{"rel": "被描述", "source": "8", "target": "2 ( x + 8 ) = 3 ( x - 1 )"}, {"rel": "等式方程求解", "source": "2 ( x + 8 ) = 3 ( x - 1 )", "target": "x = 19"}, {"rel": "被描述", "source": "x", "target": "2 ( x + 8 ) = 3 ( x - 1 )"}, {"rel": "被描述", "source": "1", "target": "2 ( x + 8 ) = 3 ( x - 1 )"}, {"rel": "限制性描述", "source": "$x$ 与 $8$ 之和的两倍等于 $x$ 与 $1$ 之差的三倍", "target": "2 ( x + 8 ) = 3 ( x - 1 )"}]}}
{"content": "If $\\frac { \\sqrt {( x - 3 ) ^ 2 }} { x - 3 } = 1$, then the range of possible values for $x$ is ____?", "answer": "x > 3", "steps": "Given the equation, we can simplify it to $\\sqrt {( x - 3 ) ^ 2 } = x - 3$. Therefore, we know that $x - 3 \\geq 0$ and $x - 3 \\neq 0$. Solving for $x$, we get $x > 3$.", "expr_cands": ["\\frac { \\sqrt { ( x - 3 ) ^ { 2 } } } { x - 3 } = 1", "x", "\\sqrt { ( x - 3 ) ^ { 2 } } = x - 3", "x - 3 \\ge 0", "3 \\le x", "x - 3 \\neq 0", "x \\neq 3", "x > 3"], "exprs": ["x - 3 \\ge 0", "x - 3 \\neq 0", "x > 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { \\sqrt { ( x - 3 ) ^ { 2 } } } { x - 3 } = 1"}, {"id": "x - 3 \\ge 0"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x - 3 \\neq 0"}, {"id": "分式有意义，则分母不为0"}, {"id": "x > 3"}], "links": [{"rel": "被描述", "source": "\\frac { \\sqrt { ( x - 3 ) ^ { 2 } } } { x - 3 } = 1", "target": "x - 3 \\ge 0"}, {"rel": "被描述", "source": "\\frac { \\sqrt { ( x - 3 ) ^ { 2 } } } { x - 3 } = 1", "target": "x - 3 \\neq 0"}, {"rel": "联立", "source": "x - 3 \\ge 0", "target": "x > 3"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 3 \\ge 0"}, {"rel": "联立", "source": "x - 3 \\neq 0", "target": "x > 3"}, {"rel": "属性描述", "source": "分式有意义，则分母不为0", "target": "x - 3 \\neq 0"}]}}
{"content": "If the square root of $x - 5$ is defined, then the possible values of $x$ are _____.", "answer": "x \\ge 5", "steps": "According to the problem, we have $x - 5 \\ge 0$, which implies $x \\ge 5$.", "expr_cands": ["\\sqrt { x - 5 }", "x", "x - 5 \\ge 0", "5 \\le x", "x \\ge 5"], "exprs": ["x - 5 \\ge 0", "x \\ge 5"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\sqrt { x - 5 }"}, {"id": "x - 5 \\ge 0"}, {"id": "二次根式 $\\sqrt { x - 5 }$ 有意义"}, {"id": "二次根式有意义，则根式恒大于等于0"}, {"id": "x \\ge 5"}], "links": [{"rel": "被描述", "source": "\\sqrt { x - 5 }", "target": "x - 5 \\ge 0"}, {"rel": "不等式方程求解", "source": "x - 5 \\ge 0", "target": "x \\ge 5"}, {"rel": "限制性描述", "source": "二次根式 $\\sqrt { x - 5 }$ 有意义", "target": "x - 5 \\ge 0"}, {"rel": "属性描述", "source": "二次根式有意义，则根式恒大于等于0", "target": "x - 5 \\ge 0"}]}}
{"content": "If $a > 2$, then $| a - 2 |$ = ____? (Remove the absolute value symbol)", "answer": "a - 2", "steps": "Since $a > 2$, it follows that $a - 2 > 0$. Therefore, $| a - 2 | = a - 2$.", "expr_cands": ["a > 2", "a", "| a - 2 |", "a - 2 > 0", "2 < a", "| a - 2 | = a - 2", "a - 2"], "exprs": ["a - 2 > 0", "a - 2"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a > 2"}, {"id": "a - 2 > 0"}, {"id": "| a - 2 |"}, {"id": "a - 2"}, {"id": "绝对值恒大于等于0"}], "links": [{"rel": "移项", "source": "a > 2", "target": "a - 2 > 0"}, {"rel": "被描述", "source": "| a - 2 |", "target": "a - 2"}, {"rel": "属性描述", "source": "绝对值恒大于等于0", "target": "a - 2"}]}}
{"content": "If $\\frac { x } { 2 } = \\frac { y } { 5 } = \\frac { z } { 7 }$, then the value of the fraction $\\frac { 3 x + 2 y - z } { 5 x - 3 y + 2 z }$ is ____?", "answer": "1", "steps": "Let $\\frac { x } { 2 } = \\frac { y } { 5 } = \\frac { z } { 7 } = k$. Therefore, $x = 2 k$, $y = 5 k$, and $z = 7 k$. Thus, $\\frac { 3 x + 2 y - z } { 5 x - 3 y + 2 z } = \\frac { 6 k + 10 k - 7 k } { 10 k - 15 k + 14 k } = \\frac { 9 k } { 9 k }$.", "expr_cands": ["\\frac { x } { 2 } = \\frac { y } { 5 } = \\frac { z } { 7 }", "\\frac { 3 x + 2 y - z } { 5 x - 3 y + 2 z }", "x", "z", "y", "\\frac { x } { 2 } = k", "k", "x = 2 k", "y = 5 k", "z = 7 k", "1"], "exprs": ["x = 2 k", "y = 5 k", "z = 7 k", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "设 $\\frac { x } { 2 } = \\frac { y } { 5 } = \\frac { z } { 7 } = k$"}, {"id": "x = 2 k"}, {"id": "y = 5 k"}, {"id": "z = 7 k"}, {"id": "\\frac { 3 x + 2 y - z } { 5 x - 3 y + 2 z }"}, {"id": "1"}], "links": [{"rel": "假设描述", "source": "设 $\\frac { x } { 2 } = \\frac { y } { 5 } = \\frac { z } { 7 } = k$", "target": "x = 2 k"}, {"rel": "假设描述", "source": "设 $\\frac { x } { 2 } = \\frac { y } { 5 } = \\frac { z } { 7 } = k$", "target": "y = 5 k"}, {"rel": "假设描述", "source": "设 $\\frac { x } { 2 } = \\frac { y } { 5 } = \\frac { z } { 7 } = k$", "target": "z = 7 k"}, {"rel": "代入", "source": "x = 2 k", "target": "1"}, {"rel": "代入", "source": "y = 5 k", "target": "1"}, {"rel": "代入", "source": "z = 7 k", "target": "1"}, {"rel": "被代入", "source": "\\frac { 3 x + 2 y - z } { 5 x - 3 y + 2 z }", "target": "1"}]}}
{"content": "Given that the reciprocal of $a$ is $- \\frac { 1 } { 2 }$, $b$ and $c$ are opposite in sign, and $m$ and $n$ are reciprocals, what is the value of $b - a + c - mn$?", "answer": "1", "steps": "$\\because$ The reciprocal of $a$ is $- \\frac { 1 } { 2 }$, $\\therefore$ $a = - 2$. $\\because$ $b$ and $c$ are opposite numbers, $\\therefore$ $b + c = 0$. $\\because$ $m$ and $n$ are reciprocal numbers, $\\therefore$ $mn = 1$. $\\therefore$ $b - a + c - mn = 0 - ( - 2 ) - 1 = 2 - 1 = 1$.", "expr_cands": ["a", "- \\frac { 1 } { 2 }", "b", "c", "m", "n", "b - a + c - mn", "a = - 2", "b + c = 0", "mn = 1", "1"], "exprs": ["a = - 2", "b + c = 0", "mn = 1", "1"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "a"}, {"id": "a = - 2"}, {"id": "- \\frac { 1 } { 2 }"}, {"id": "$a$ 的倒数是 $- \\frac { 1 } { 2 }$ , $b$ 与 $c$ 互为相反数"}, {"id": "b"}, {"id": "b + c = 0"}, {"id": "c"}, {"id": "m"}, {"id": "mn = 1"}, {"id": "n"}, {"id": "$m$ 与 $n$ 互为倒数"}, {"id": "b - a + c - mn"}, {"id": "1"}], "links": [{"rel": "被描述", "source": "a", "target": "a = - 2"}, {"rel": "代入", "source": "a = - 2", "target": "1"}, {"rel": "被描述", "source": "- \\frac { 1 } { 2 }", "target": "a = - 2"}, {"rel": "限制性描述", "source": "$a$ 的倒数是 $- \\frac { 1 } { 2 }$ , $b$ 与 $c$ 互为相反数", "target": "a = - 2"}, {"rel": "限制性描述", "source": "$a$ 的倒数是 $- \\frac { 1 } { 2 }$ , $b$ 与 $c$ 互为相反数", "target": "b + c = 0"}, {"rel": "被描述", "source": "b", "target": "b + c = 0"}, {"rel": "代入", "source": "b + c = 0", "target": "1"}, {"rel": "被描述", "source": "c", "target": "b + c = 0"}, {"rel": "被描述", "source": "m", "target": "mn = 1"}, {"rel": "代入", "source": "mn = 1", "target": "1"}, {"rel": "被描述", "source": "n", "target": "mn = 1"}, {"rel": "限制性描述", "source": "$m$ 与 $n$ 互为倒数", "target": "mn = 1"}, {"rel": "被代入", "source": "b - a + c - mn", "target": "1"}]}}
{"content": "The analytical expression of a quadratic function is $y = mx ^ { m ^ { 2 } - 3 m + 2 }$. What is the value of the constant $m$?", "answer": "3", "steps": "$\\because y = mx ^ { m ^ { 2 } - 3 m + 2 }$ is a quadratic function in terms of $x$, $\\therefore m ^ { 2 } - 3 m + 2 = 2$, and $m \\neq 0$. Solving for $m$, we get $m = 3$.", "expr_cands": ["y = mx ^ { m ^ { 2 } - 3 m + 2 }", "x", "m", "y", "m ^ { 2 } - 3 m + 2 = 2", "m = 0", "m = 3", "m \\neq 0"], "exprs": ["m ^ { 2 } - 3 m + 2 = 2", "m \\neq 0", "m = 3"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "y = mx ^ { m ^ { 2 } - 3 m + 2 }"}, {"id": "m ^ { 2 } - 3 m + 2 = 2"}, {"id": "二次函数的解析式为 $y = mx ^ { m ^ { 2 } - 3 m + 2 }$"}, {"id": ", $y = mx ^ { m ^ { 2 } - 3 m + 2 }$ 是关于 $x$ 的二次函数"}, {"id": "m = 3"}, {"id": "m \\neq 0"}], "links": [{"rel": "被描述", "source": "y = mx ^ { m ^ { 2 } - 3 m + 2 }", "target": "m ^ { 2 } - 3 m + 2 = 2"}, {"rel": "被描述", "source": "y = mx ^ { m ^ { 2 } - 3 m + 2 }", "target": "m \\neq 0"}, {"rel": "联立", "source": "m ^ { 2 } - 3 m + 2 = 2", "target": "m = 3"}, {"rel": "限制性描述", "source": "二次函数的解析式为 $y = mx ^ { m ^ { 2 } - 3 m + 2 }$", "target": "m ^ { 2 } - 3 m + 2 = 2"}, {"rel": "限制性描述", "source": "二次函数的解析式为 $y = mx ^ { m ^ { 2 } - 3 m + 2 }$", "target": "m \\neq 0"}, {"rel": "限制性描述", "source": ", $y = mx ^ { m ^ { 2 } - 3 m + 2 }$ 是关于 $x$ 的二次函数", "target": "m ^ { 2 } - 3 m + 2 = 2"}, {"rel": "限制性描述", "source": ", $y = mx ^ { m ^ { 2 } - 3 m + 2 }$ 是关于 $x$ 的二次函数", "target": "m \\neq 0"}, {"rel": "联立", "source": "m \\neq 0", "target": "m = 3"}]}}
{"content": "If the polynomial $3 { x } ^ 3 + 2 mx ^ 2 - 5 x + 8 { x } ^ 2 + 3$ in terms of $x$ does not contain a quadratic term, then $m$ is equal to ____?", "answer": "- 4", "steps": "$\\because$ The polynomial in $x$, $3 { x } ^ 3 + 2 mx ^ 2 - 5 x + 8 { x } ^ 2 + 3 = 3 { x } ^ 3 + ( 2 m + 8 ) { x } ^ 2 - 5 x + 3$, does not contain a quadratic term. $\\therefore$ $2 m + 8 = 0$, which solves for $m = - 4$.", "expr_cands": ["x", "3 { x } ^ { 3 } + 2 m { x } ^ { 2 } - 5 x + 8 { x } ^ { 2 } + 3", "m", "3 { x } ^ { 3 } + ( 2 m + 8 ) { x } ^ { 2 } - 5 x + 3", "2 m + 8 = 0", "m = - 4"], "exprs": ["3 { x } ^ { 3 } + ( 2 m + 8 ) { x } ^ { 2 } - 5 x + 3", "2 m + 8 = 0", "m = - 4"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "3 { x } ^ { 3 } + 2 m { x } ^ { 2 } - 5 x + 8 { x } ^ { 2 } + 3"}, {"id": "3 { x } ^ { 3 } + ( 2 m + 8 ) { x } ^ { 2 } - 5 x + 3"}, {"id": "x"}, {"id": "2 m + 8 = 0"}, {"id": "关于 $x$ 的多项式 $3 { x } ^ { 3 } + 2 m { x } ^ { 2 } - 5 x + 8 { x } ^ { 2 } + 3$ 中不含二次项"}, {"id": "m = - 4"}], "links": [{"rel": "提取因式", "source": "3 { x } ^ { 3 } + 2 m { x } ^ { 2 } - 5 x + 8 { x } ^ { 2 } + 3", "target": "3 { x } ^ { 3 } + ( 2 m + 8 ) { x } ^ { 2 } - 5 x + 3"}, {"rel": "被描述", "source": "3 { x } ^ { 3 } + ( 2 m + 8 ) { x } ^ { 2 } - 5 x + 3", "target": "2 m + 8 = 0"}, {"rel": "提取因式参考", "source": "x", "target": "3 { x } ^ { 3 } + ( 2 m + 8 ) { x } ^ { 2 } - 5 x + 3"}, {"rel": "等式方程求解", "source": "2 m + 8 = 0", "target": "m = - 4"}, {"rel": "限制性描述", "source": "关于 $x$ 的多项式 $3 { x } ^ { 3 } + 2 m { x } ^ { 2 } - 5 x + 8 { x } ^ { 2 } + 3$ 中不含二次项", "target": "2 m + 8 = 0"}]}}
{"content": "If the value of $\\frac { y - 3 } { 2 }$ is greater than the value of $\\frac { 2 y - 1 } { 3 }$ by $1$, then the value of $y$ is ____?", "answer": "- 13", "steps": "From the given problem, we have $\\frac { y - 3 } { 2 } - \\frac { 2 y - 1 } { 3 } = 1$. Multiplying both sides by $6$ to eliminate the denominators, we get $3 ( y - 3 ) - 2 ( 2 y - 1 ) = 6$. Expanding the brackets, we get $3 y - 9 - 4 y + 2 = 6$. Simplifying, we get $- x = 13$. Dividing both sides by $- 1$, we get $x = - 13$.", "expr_cands": ["\\frac { y - 3 } { 2 }", "y", "\\frac { 2 y - 1 } { 3 }", "1", "\\frac { y - 3 } { 2 } - \\frac { 2 y - 1 } { 3 } = 1", "y = - 13", "3 ( y - 3 ) - 2 ( 2 y - 1 ) = 6", "3 y - 9 - 4 y + 2 = 6", "3 y - 4 y = 6 - 2 + 9", "- x = 13", "x = - 13", "x"], "exprs": ["\\frac { y - 3 } { 2 } - \\frac { 2 y - 1 } { 3 } = 1", "y = - 13"], "edges": {"directed": true, "multigraph": false, "graph": {}, "nodes": [{"id": "\\frac { y - 3 } { 2 }"}, {"id": "\\frac { y - 3 } { 2 } - \\frac { 2 y - 1 } { 3 } = 1"}, {"id": "\\frac { 2 y - 1 } { 3 }"}, {"id": "1"}, {"id": "$\\frac { y - 3 } { 2 }$ 的值比 $\\frac { 2 y - 1 } { 3 }$ 的值大 $1$"}, {"id": "y = - 13"}], "links": [{"rel": "被描述", "source": "\\frac { y - 3 } { 2 }", "target": "\\frac { y - 3 } { 2 } - \\frac { 2 y - 1 } { 3 } = 1"}, {"rel": "等式方程求解", "source": "\\frac { y - 3 } { 2 } - \\frac { 2 y - 1 } { 3 } = 1", "target": "y = - 13"}, {"rel": "被描述", "source": "\\frac { 2 y - 1 } { 3 }", "target": "\\frac { y - 3 } { 2 } - \\frac { 2 y - 1 } { 3 } = 1"}, {"rel": "被描述", "source": "1", "target": "\\frac { y - 3 } { 2 } - \\frac { 2 y - 1 } { 3 } = 1"}, {"rel": "限制性描述", "source": "$\\frac { y - 3 } { 2 }$ 的值比 $\\frac { 2 y - 1 } { 3 }$ 的值大 $1$", "target": "\\frac { y - 3 } { 2 } - \\frac { 2 y - 1 } { 3 } = 1"}]}}