《2025年课堂点睛八年级数学上册北师大版》

12. (朝阳市中考)已知关于$x$,$y的方程组\begin{cases}2x + y = 2a + 1, \\ x + 2y = a - 1\end{cases} 的解满足x - y = 4$,则$a$的值为2。

答案： 2

13. 解方程组：$\begin{cases}2x + 3y = 10, \\ \dfrac{2x + y}{2} - \dfrac{1 + y}{4} = 1.\end{cases} $

答案： 解:整理方程组得:$\left\{\begin{array}{l} 2x+3y=10,①\\ 12x+3y=15,②\end{array}\right. $②-①得:$x=0.5$.把$x=0.5$代入①得:$2×0.5+3y=10$,解得:$y=3$,
∴方程组的解为$\left\{\begin{array}{l} x=0.5,\\ y=3.\end{array}\right. $

14. (核心素养·运算能力)阅读下列解方程组的方法,然后回答问题。
解方程组：$\begin{cases}19x + 18y = 17,① \\ 17x + 16y = 15.②\end{cases} $
解：① - ②,得$2x + 2y = 2$,即$x + y = 1$.③
③×16,得$16x + 16y = 16$.④
② - ④,得$x = -1$,从而可得$y = 2$.
∴原方程组的解是$\begin{cases}x = -1, \\ y = 2.\end{cases} $
(1)请你仿照上面的解法解方程组：$\begin{cases}2025x + 2024y = 2023,① \\ 2023x + 2022y = 2021;②\end{cases} $
(2)请大胆猜测关于$x$,$y的方程组\begin{cases}(a + 2)x + (a + 1)y = a, \\ (b + 2)x + (b + 1)y = b\end{cases} $($a ≠ b$)的解是什么,并利用方程组的解加以验证。

答案：
(1)解:①-②,得$2x+2y=2$,即$x+y=1$,③,①-③×2024,得$x=-1$.把$x=-1$代入③,得$-1+y=1$,解得$y=2$.所以原方程组的解为$\left\{\begin{array}{l} x=-1,\\ y=2.\end{array}\right. $
(2)猜想:方程组$\left\{\begin{array}{l} (a+2)x+(a+1)y=a,\\ (b+2)x+(b+1)y=b\end{array}\right. (a≠b)$的解为$\left\{\begin{array}{l} x=-1,\\ y=2.\end{array}\right. $检验:把$x=-1,y$=2代入$(a+2)x+(a+1)y=a$,得左边=a,左边=右边;把$x=-1,y$=2代入$(b+2)x+(b+1)y=b$,得左边=b,左边=右边.
∴$\left\{\begin{array}{l} x=-1,\\ y=2\end{array}\right. $是方程组的解.

【变式1】已知方程组$\begin{cases}2m - 3n = -2, \\ 3m + 5n = 35\end{cases} 的解是\begin{cases}m = 5, \\ n = 4,\end{cases} 则方程组\begin{cases}2(x + 2) - 3(y - 1) = -2, \\ 3(x + 2) + 5(y - 1) = 35\end{cases} $的解是$\left\{\begin{array}{l} x=3,\\ y=5\end{array}\right. $。

答案： $\left\{\begin{array}{l} x=3,\\ y=5\end{array}\right. $

【变式2】若关于$x$,$y的方程组\begin{cases}3x - my = 5, \\ 2x + ny = 6\end{cases} 的解是\begin{cases}x = 2, \\ y = 4,\end{cases} 则关于a$,$b的方程组\begin{cases}3(a + b) - m(a - b) = 5, \\ 2(a + b) + n(a - b) = 6\end{cases} $的解是$\left\{\begin{array}{l} a=3,\\ b=-1\end{array}\right. $。

答案： $\left\{\begin{array}{l} a=3,\\ b=-1\end{array}\right. $

【拓展】方程组$\begin{cases}a_{1}x + b_{1}y = c_{1}, \\ a_{2}x + b_{2}y = c_{2}\end{cases} 的解是\begin{cases}x = 3, \\ y = 4,\end{cases} 则方程组\begin{cases}3a_{1}x + 2b_{1}y = 5c_{1}, \\ 3a_{2}x + 2b_{2}y = 5c_{2}\end{cases} $的解是$\left\{\begin{array}{l} x=5,\\ y=10\end{array}\right. $。

答案： $\left\{\begin{array}{l} x=5,\\ y=10\end{array}\right. $