《2025年暑假总动员宁夏人民教育出版社八升九数学苏科版》

20. (6分)已知关于$ x,y $的二元一次方程组$ \begin{cases} 2x + y = 6m, \\ 3x - 2y = 2m \end{cases} $的解满足二元一次方程$ \frac{x}{3} - \frac{y}{5} = 4 $,求$ m $的值.

答案： $m = 15$

21. (6分)若$ \begin{cases} 4x + y = 3, \\ ax - 3y = -1 \end{cases} $与$ \begin{cases} 2x = 5 + 3y, \\ 2x + 1 = -by \end{cases} $有相同的解,求$ (a + b)^{1000} $的值.

答案： 根据题意,得$\left\{\begin{array}{l} 4x + y = 3,\\ 2x = 5 + 3y,\end{array}\right.$解得$\left\{\begin{array}{l} x = 1,\\ y = -1,\end{array}\right.$
代入$\left\{\begin{array}{l} ax - 3y = -1,\\ 2x + 1 = -by,\end{array}\right.$得$\left\{\begin{array}{l} a + 3 = -1,\\ 2 + 1 = b,\end{array}\right.$解得$\left\{\begin{array}{l} a = -4,\\ b = 3,\end{array}\right.$
所以$(a + b)^{1000}=(-4 + 3)^{1000}=(-1)^{1000}=1$.

22. (6分)某班同学参加运土劳动,女同学除一人请假外,全部分配去抬土(每两人用一根扁担抬一个筐),男生除3名体弱者跟女生一起抬土外,其余全部去挑土(每人用一根扁担挑两个筐),这样全班共需50个筐,36根扁担.问:该班男、女生各有多少人?
设该班女生有$ x $人,男生有$ y $人,先填表格,再列方程组求解.
| | 人数 | 需扁担数 | 需筐数 |
| --- | --- | --- | --- |
| 抬土 | | | |
| 挑土 | | | |

答案： 根据题意,得$\left\{\begin{array}{l} 0.5x + 1 + y - 3 = 36,\\ 0.5x + 1 + 2y - 6 = 50,\end{array}\right.$解得$\left\{\begin{array}{l} x = 42,\\ y = 17.\end{array}\right.$
答:该班男生有 17 人,女生有 42 人.