《2025年名校题库八年级数学上册北师大版》

1.(七中育才)解方程组:
(1)$\left\{\begin{array}{l} x-y= 8,\\ 3x+y= 12;\end{array} \right. $ (2)$\left\{\begin{array}{l} 2x-y= 3,\\ 3x+2y= 8.\end{array} \right. $

答案：
(1)解:$\left\{\begin{array}{l} x-y=8①,\\ 3x+y=12②.\end{array}\right.$
①+②,得$4x=20$,解得$x=5$.
将$x=5$代入①,得$5-y=8$,解得$y=-3$.
$\therefore$原方程组的解为$\left\{\begin{array}{l} x=5,\\ y=-3.\end{array}\right.$
(2)解:$\left\{\begin{array}{l} 2x-y=3①,\\ 3x+2y=8②.\end{array}\right.$
由①得$y=2x-3③$.
将③代入②,得$3x+2(2x-3)=8$,
即$3x+4x-6=8$,解得$x=2$.
将$x=2$代入③,得$y=1$.
$\therefore$原方程组的解为$\left\{\begin{array}{l} x=2,\\ y=1.\end{array}\right.$

2.(嘉祥)解方程组:
(1)$\left\{\begin{array}{l} 4x-3y= 15,\\ 3x+2y= 7;\end{array} \right. $ (2)$\left\{\begin{array}{l} 5x-4y= 3,\\ 3x-y= 2.\end{array} \right. $

答案：
(1)解:$\left\{\begin{array}{l} 4x-3y=15①\\ 3x+2y=7②\end{array}\right.$
由$①×2$,得$8x-6y=30③$
由$②×3$,得$9x+6y=21④$
$③+④$,得$17x=51$
解得$x=3$
将$x=3$代入$②$,得$9+2y=7$
解得$y=-1$
$\therefore$原方程组的解为$\left\{\begin{array}{l} x=3\\ y=-1\end{array}\right.$
(2)解:$\left\{\begin{array}{l} 5x-4y=3①\\ 3x-y=2②\end{array}\right.$
由$②$得$y=3x-2③$
将$③$代入$①$,得$5x-4(3x-2)=3$
即$5x-12x+8=3$
$-7x=-5$
解得$x=\frac {5}{7}$
将$x=\frac {5}{7}$代入$③$,得$y=3×\frac {5}{7}-2=\frac {1}{7}$
$\therefore$原方程组的解为$\left\{\begin{array}{l} x=\frac {5}{7}\\ y=\frac {1}{7}\end{array}\right.$

3.(青羊区期末)解方程组:
(1)$\left\{\begin{array}{l} 3x+y= 10,\\ x-y= 6;\end{array} \right. $ (2)$\left\{\begin{array}{l} 3x-\frac {1}{2}y= 1,\\ 2x+y= 2.\end{array} \right. $

答案：
(1)解:$\left\{\begin{array}{l} 3x+y=10①,\\ x-y=6②.\end{array}\right.$
由$①+②$,得$4x=16$,解得$x=4$.
将$x=4$代入$①$,得$12+y=10$,解得$y=-2$.
$\therefore$原方程组的解为$\left\{\begin{array}{l} x=4,\\ y=-2.\end{array}\right.$
(2)解:$\left\{\begin{array}{l} 3x-\frac {1}{2}y=1①,\\ 2x+y=2②.\end{array}\right.$
由$①×2$,得$6x-y=2③$.
由$②+③$,得$8x=4$,解得$x=\frac {1}{2}$.
将$x=\frac {1}{2}$代入$②$,得$1+y=2$,解得$y=1$.
$\therefore$原方程组的解为$\left\{\begin{array}{l} x=\frac {1}{2},\\ y=1.\end{array}\right.$

4.(武侯区期末)解方程组:
(1)$\left\{\begin{array}{l} y= 2x-4,\\ 3x+y= 1;\end{array} \right. $ (2)$\left\{\begin{array}{l} \frac {x-1}{6}-\frac {2-y}{3}= 1,\\ 2(x-1)= 13-(y+2).\end{array} \right. $

答案：
(1)解:$\left\{\begin{array}{l} y=2x-4①\\ 3x+y=1②\end{array}\right.$
将①代入②,得$3x+2x-4=1$.
解得$x=1$.
将$x=1$代入①,得$y=2×1-4=-2$.
$\therefore$原方程组的解为$\left\{\begin{array}{l} x=1\\ y=-2\end{array}\right.$
(2)解:$\left\{\begin{array}{l} \frac{x-1}{6}-\frac{2-y}{3}=1①\\ 2(x-1)=13-(y+2)②\end{array}\right.$
①$×6$得$(x-1)-2(2-y)=6$,即$x+2y=11③$.
由②得$2x-2=11-y$,即$2x+y=13④$.
由③得$x=11-2y$,代入④得$2(11-2y)+y=13$.
解得$y=3$.
将$y=3$代入③得$x=11-2×3=5$.
$\therefore$原方程组的解为$\left\{\begin{array}{l} x=5\\ y=3\end{array}\right.$

5.(高新区期末)解方程组:
(1)$\left\{\begin{array}{l} 3(x-1)= y+5,\\ -3x+5y= 20;\end{array} \right. $ (2)$\left\{\begin{array}{l} x-y= 3,\\ 2x+y= 0.\end{array} \right. $

答案：
(1)解:$\left\{\begin{array}{l} 3(x-1)=y+5①,\\ -3x+5y=20②.\end{array}\right.$
由①得$3x - 3 = y + 5$，即$3x - y = 8③$
由③得$y = 3x - 8$，代入②得$-3x + 5(3x - 8) = 20$
$-3x + 15x - 40 = 20$
$12x = 60$
解得$x = 5$
则$y = 3×5 - 8 = 7$
$\therefore$原方程组的解为$\left\{\begin{array}{l} x = 5,\\ y = 7.\end{array}\right.$
(2)解:$\left\{\begin{array}{l} x - y = 3①,\\ 2x + y = 0②.\end{array}\right.$
①+②得$3x = 3$，解得$x = 1$
将$x = 1$代入②得$2×1 + y = 0$，解得$y = -2$
$\therefore$原方程组的解为$\left\{\begin{array}{l} x = 1,\\ y = -2.\end{array}\right.$