《2025年同步精练广东八年级数学上册北师大版》

1. 下列方程组中,是二元一次方程组的有(B)
①$\left\{\begin{array}{l} 2x + 3y = 7,\\ 4x - y = 5;\end{array} \right.$②$\left\{\begin{array}{l} 2x + y = 3,\\ x + z = 4;\end{array} \right.$③$\left\{\begin{array}{l} x = 2,\\ y + 3 = 6;\end{array} \right.$
④$\left\{\begin{array}{l} x^{2} - y = 2,\\ x + 4y = 5.\end{array} \right.$
A.1个
B.2个
C.3个
D.4个

答案： B

2. 如果方程$x - y = 3与下面方程中的一个组成的方程组的解为\left\{\begin{array}{l} x = 4,\\ y = 1,\end{array} \right.$那么这个方程是(A)
A.$2(x - y) = 6y$
B.$\frac{1}{4}x + 2y = 5$
C.$x + 2y = 9$
D.$3x - 4y = 16$

答案： A

3. 新考向 开放性问题 写出二元一次方程$2x - y = 5$的一个整数解：$\left\{\begin{array}{l} x=3,\\ y=1\end{array}\right. $(答案不唯一).

答案： $\left\{\begin{array}{l} x=3,\\ y=1\end{array}\right. $(答案不唯一)

4. 用加减消元法解二元一次方程组$\left\{\begin{array}{l} 3x - 2y = 5,①\\ 2x - 3y = 10,②\end{array} \right.$由①×3 - ②×2可得(D)
A.$12x - 13y = 40$
B.$5x - 12y = - 5$
C.$5y = - 20$
D.$5x = - 5$

答案： D

5. (2024·无锡)二元一次方程组$\left\{\begin{array}{l} 3x - y = 1,\\ 2x + 3y = 8\end{array} \right.$的解为$\left\{\begin{array}{l} x=1,\\ y=2\end{array}\right. $.

答案： $\left\{\begin{array}{l} x=1,\\ y=2\end{array}\right. $

6. 若关于$x$,$y的方程组\left\{\begin{array}{l} 2x - y = 5k + 6,\\ 4x + 7y = k\end{array} \right.的解满足x + y = 2025$,则$k$的值为2024.

答案： 2024

7. (2024·宿迁)若关于$x$,$y的二元一次方程组\left\{\begin{array}{l} ax + y = b,\\ cx - y = d\end{array} \right.的解是\left\{\begin{array}{l} x = 3,\\ y = - 2,\end{array} \right.则关于x$,$y的方程组\left\{\begin{array}{l} ax + 2y = 2a + b,\\ cx - 2y = 2c + d\end{array} \right.$的解是$\left\{\begin{array}{l} x=5,\\ y=-1\end{array}\right. $.

答案： $\left\{\begin{array}{l} x=5,\\ y=-1\end{array}\right. $

8. 解方程组：
(1)$\left\{\begin{array}{l} x - 2y = 3,①\\ \frac{1}{2}x + \frac{3}{4}y = \frac{13}{4}.②\end{array} \right.$
(2)$\left\{\begin{array}{l} x + y - z = 0,①\\ 2x - 3y + 5z = 5,②\\ 3x + y - z = 2.③\end{array} \right.$

答案： 解:
(1)②×2-①,得$\frac {7}{2}y=\frac {7}{2}$.解得$y=1$.将$y=1$代入①,得$x-2=3$.解得$x=5$.
∴原方程组的解为$\left\{\begin{array}{l} x=5,\\ y=1.\end{array}\right. $
(2)①×5+②,得$7x+2y=5$.④ ①-③,得$-2x=-2$.解得$x=1$.将$x=1$代入④,得$7+2y=5$.解得$y=-1$.将$x=1,y=-1$代入①,得$z=0$.
∴原方程组的解为$\left\{\begin{array}{l} x=1,\\ y=-1,\\ z=0.\end{array}\right. $

9. (2024·天津)《孙子算经》是我国古代著名的数学典籍,其中有一道题：“今有木,不知长短.引绳度之,余绳四尺五寸；屈绳量之,不足一尺.木长几何?”意思是：用一根绳子去量一根长木,绳子还剩余4.5尺；将绳子对折再量长木,长木还剩余1尺.问木长多少尺?设木长$x$尺,绳子长$y$尺,则可以列出的方程组为(A)
A.$\left\{\begin{array}{l} y - x = 4.5,\\ x - 0.5y = 1\end{array} \right.$
B.$\left\{\begin{array}{l} y - x = 4.5,\\ x + 0.5y = 1\end{array} \right.$
C.$\left\{\begin{array}{l} x + y = 4.5,\\ x - y = 1\end{array} \right.$
D.$\left\{\begin{array}{l} x + y = 4.5,\\ y - x = 1\end{array} \right.$

答案： A