《2025年四清导航七年级数学上册湘教版》

11. 已知$\begin{cases}x = 2,\\y = 1\end{cases} 是二元一次方程组\begin{cases}mx + ny = 7,\\x - my = 1\end{cases} $的解,则$m + 3n= $8。

答案： 8

12. 用加减消元法解下列方程组：
(1)$\begin{cases}4m + 5n = 460,①\\2m + 3n = 240;②\end{cases} $
(2)$\begin{cases}x - 2y = 2,①\\3x - y = 11;②\end{cases} $
(3)$\begin{cases}x + 4y = 14,①\frac{x - 3}{4} - \frac{y - 3}{3} = \frac{1}{12}.②\end{cases} $

答案：
(1)$\left\{\begin{array}{l} m=90\\ n=20\end{array}\right. $
(2)$\left\{\begin{array}{l} x=4\\ y=1\end{array}\right. $
(3)$\left\{\begin{array}{l} x=3\\ y=\frac {11}{4}\end{array}\right. $

13. 已知$-2x^{m - 1}y^{3}与\frac{1}{2}x^{n}y^{m + n}$的和是单项式,求$m$,$n$的值。

答案： 由题意得$\left\{\begin{array}{l} m-1=n,\\ 3=m+n,\end{array}\right. $解得$\left\{\begin{array}{l} m=2\\ n=1\end{array}\right. $

14. 在解方程组$\begin{cases}ax + by = 2,\\cx - 7y = 8\end{cases} $时,哥哥正确地解得$\begin{cases}x = 3,\\y = -2,\end{cases} 弟弟因把c写错而解得\begin{cases}x = -2,\\y = 2,\end{cases} 求a + b + c$的值。

答案： 由题意可知$\left\{\begin{array}{l} x=3,\\ y=-2\end{array}\right. $是方程$cx-7y=8$的解,所以$3c+14=8$,所以$c=-2$,由题意可得$\left\{\begin{array}{l} 3a-2b=2,\\ -2a+2b=2,\end{array}\right. $解得$\left\{\begin{array}{l} a=4,\\ b=5,\end{array}\right. $所以$a+b+c=4+5+(-2)=7$

15. 在解关于x,y的方程组$\begin{cases}ax + (b - 2)y = 1,①\2b - 1)x - ay = 4②\end{cases} $时,可以用①×2 - ②消去未知数x,也可以用①×4 + ②×3消去未知数y,试求a,b的值。

答案： 由题意可知$\left\{\begin{array}{l} 2a-(2b-1)=0,\\ 4(b-2)-3a=0,\end{array}\right. $解得$\left\{\begin{array}{l} a=6\\ b=\frac {13}{2}\end{array}\right. $