《2025年名校课堂八年级数学上册北师大版》

8. 解方程组:$\left\{\begin{array}{l} x-2y= 1,\enclose{circle} {1}\\ 2x+3y= 16.\enclose{circle} {2}\end{array}\right.$
解:由①,得$x= 1+2y$.③
将③代入②,得$2(1 + 2y)+3y = 16$.解得$y = 2$.将$y = 2$代入③,得$x = 5$.
∴原方程组的解为$\left\{\begin{array}{l} x = 5,\\ y = 2.\end{array}\right.$

答案： 解：上述解答过程不正确。正确过程如下：由①，得$x = 1 + 2y$。③将③代入②，得$2(1 + 2y)+3y = 16$。解得$y = 2$。将$y = 2$代入③，得$x = 5$。
∴原方程组的解为$\left\{\begin{array}{l} x = 5,\\ y = 2.\end{array}\right.$

9. 由方程组$\left\{\begin{array}{l} 2x+m= 1,\\ y-3= m\end{array}\right. $可得出x与y的关系是 (A)
A. $2x+y= 4$
B. $2x-y= 4$
C. $2x+y= -4$
D. $2x-y= -4$

答案： A

10. 若$-3x^{a-2b}y^{7}与2x^{8}y^{5a+b}$是同类项,则$a+b= $-1.

答案： -1

11. 已知关于x,y的二元一次方程组$\left\{\begin{array}{l} ax+by= 7,\\ bx-ay= 5\end{array}\right. 的解为\left\{\begin{array}{l} x= 1,\\ y= -2,\end{array}\right. 则a+b$的值为$\frac{8}{5}$.

答案： $\frac{8}{5}$

12. 用代入消元法解方程组:$\left\{\begin{array}{l} 2(x-1)= y-1,\enclose{circle} {1}\\ 3(y-3)= x+3.\enclose{circle} {2}\end{array}\right. $
解：由①，得$y=$$2x - 1$。③ 将③代入②，得$3($$2x - 1$$- 3)=x+3$。解得$x=$$3$。将$x=$$3$代入③，得$y=$$5$。∴原方程组的解为$\left\{\begin{array}{l} x=$$3$,\\ y=$$5$.\end{array}\right.$

答案： 解：由①，得$y = 2x - 1$。③ 将③代入②，得$3(2x - 1 - 3)=x+3$。解得$x = 3$。将$x = 3$代入③，得$y = 5$。
∴原方程组的解为$\left\{\begin{array}{l} x = 3,\\ y = 5.\end{array}\right.$

13. 如果$|x-y+4|与\sqrt {x+3y+2}$互为相反数,求$x+y$的值.

答案： 解：$\because|x - y + 4|$和$\sqrt{x + 3y + 2}$互为相反数，$\therefore|x - y + 4|+\sqrt{x + 3y + 2}=0$。$\therefore\left\{\begin{array}{l} x - y + 4 = 0,①\\ x + 3y + 2 = 0.②\end{array}\right.$由①，得$y = x + 4$。③将③代入②，得$x + 3(x + 4)+2 = 0$，解得$x=-\frac{7}{2}$。将$x=-\frac{7}{2}$代入③，得$y=\frac{1}{2}$。
∴原方程组的解为$\left\{\begin{array}{l} x = -\frac{7}{2},y=\frac{1}{2}.\end{array}\right.$ $\therefore x + y=-\frac{7}{2}+\frac{1}{2}=-3$。

14. 观察发现:
解方程组:$\left\{\begin{array}{l} x+y= 4,①\\ 3(x+y)+y= 14.②\end{array}\right. $
将①整体代入②,得$3×4+y= 14$.
解得$y= 2$.
将$y= 2$代入①,得$x= 2$.
∴原方程组的解为$\left\{\begin{array}{l} x= 2,\\ y= 2.\end{array}\right. $
这种解法称为“整体代入法”.你若留心观察,有很多方程组可采用此方法解答.
(1)请直接写出方程组$\left\{\begin{array}{l} x-y-1= 0,\\ 4(x-y)-y= 5\end{array}\right. $的解:$\left\{\begin{array}{l} x = 0,\\ y = -1\end{array}\right.$.
实践运用:
(2)请用“整体代入法”解方程组:$\left\{\begin{array}{l} 2x-3y-2= 0,①\\ \frac {2x-3y+5}{7}+2y= 9.②\end{array}\right. $
解：由①，得$2x - 3y = 2$。③ 将③代入②，得$1 + 2y = 9$。解得$y = 4$。将$y = 4$代入③，得$2x - 12 = 2$。解得$x=7$。∴原方程组的解为$\left\{\begin{array}{l} x = 7,\\ y = 4.\end{array}\right.$

答案： 解：
(1)$\left\{\begin{array}{l} x = 0,\\ y = -1\end{array}\right.$
(2)由①，得$2x - 3y = 2$。③ 将③代入②，得$1 + 2y = 9$。解得$y = 4$。将$y = 4$代入③，得$2x - 12 = 2$。解得$x=7$。
∴原方程组的解为$\left\{\begin{array}{l} x = 7,\\ y = 4.\end{array}\right.$