《2025年初中生暑假生活七年级综合全一册人教版》

10. 解下列方程组：
(1)$\left\{\begin{array}{l} x= y-50,\\ x+y= 180;\end{array} \right. $
(2)$\left\{\begin{array}{l} x+3y= 7,\\ y-x= 1;\end{array} \right. $
(3)$\left\{\begin{array}{l} y= 3x,\\ 7x-2y= 2;\end{array} \right. $
(4)$\left\{\begin{array}{l} 2x+y= 3,\\ 3x-5y= 11.\end{array} \right. $

答案： （1）解：将$x = y - 50$代入$x + y = 180$，得$y - 50 + y = 180$，$2y = 230$，$y = 115$。把$y = 115$代入$x = y - 50$，得$x = 65$。所以方程组的解为$\begin{cases}x = 65 \\ y = 115\end{cases}$。
（2）解：由$y - x = 1$得$y = x + 1$，代入$x + 3y = 7$，得$x + 3(x + 1) = 7$，$x + 3x + 3 = 7$，$4x = 4$，$x = 1$。把$x = 1$代入$y = x + 1$，得$y = 2$。所以方程组的解为$\begin{cases}x = 1 \\ y = 2\end{cases}$。
（3）解：将$y = 3x$代入$7x - 2y = 2$，得$7x - 2×3x = 2$，$7x - 6x = 2$，$x = 2$。把$x = 2$代入$y = 3x$，得$y = 6$。所以方程组的解为$\begin{cases}x = 2 \\ y = 6\end{cases}$。
（4）解：由$2x + y = 3$得$y = 3 - 2x$，代入$3x - 5y = 11$，得$3x - 5(3 - 2x) = 11$，$3x - 15 + 10x = 11$，$13x = 26$，$x = 2$。把$x = 2$代入$y = 3 - 2x$，得$y = -1$。所以方程组的解为$\begin{cases}x = 2 \\ y = -1\end{cases}$。

1. 已知$\left\{\begin{array}{l} m= 1,\\ n= 2\end{array} \right. 是方程组\left\{\begin{array}{l} am+bn= 2,\\ am-bn= 3\end{array} \right. $的解,求$a和b$的值.

答案： 解：
∵$\left\{\begin{array}{l} m= 1,\\ n= 2\end{array}\right. $是方程组$\left\{\begin{array}{l} am+bn= 2,\\ am-bn= 3\end{array}\right. $的解，
∴$\left\{\begin{array}{l} a+2b=2,\\ a-2b=3,\end{array}\right. $
解得$\left\{\begin{array}{l} a=\frac{5}{2},\\ b=-\frac{1}{4}.\end{array}\right. $

2. 若关于$x$,$y的方程组\left\{\begin{array}{l} x+y= 2,\\ (k-1)x+(k+1)y= 4\end{array} \right. 的解x与y$的值相等,求$k$的值.

答案： 解：
∵方程组的解$x$与$y$的值相等，
∴设$x = y = a$，
将$x = a$，$y = a$代入$x + y = 2$，得$a + a = 2$，
解得$a = 1$，
∴$x = y = 1$，
将$x = 1$，$y = 1$代入$(k - 1)x + (k + 1)y = 4$，得$(k - 1)×1 + (k + 1)×1 = 4$，
即$k - 1 + k + 1 = 4$，
$2k = 4$，
解得$k = 2$。
答：$k$的值为$2$。