《2025年暑假衔接七年级数学浙教版延边人民出版社》

15. 如图,在大长方形$ABCD$中,放入$8$个小长方形,则图中阴影部分面积为______$cm^{2}$。

53

答案： 53 【解析】设小长方形宽为$xcm$，长为$ycm$，则有$BC = 4x + y = 15$，$CD = 2x + y$，$AB = 9 + x$，因为$AB = CD$，所以$2x + y = 9 + x$，即$x + y = 9$，$\left\{\begin{array}{l} 4x + y = 15,\\ x + y = 9,\end{array}\right.$解得$\left\{\begin{array}{l} x = 2,\\ y = 7,\end{array}\right.$故小长方形的长和宽分别是$7cm$和$2cm$。因为长方形$ABCD$面积为$15×11 = 165(cm^{2})$，每个小长方形的面积为$2×7 = 14(cm^{2})$，故$S_{阴影}=S_{大长方形ABCD}-8×S_{小长方形}=165 - 8×14 = 53(cm^{2})$。

16. 解二元一次方程组。
(1)$\left\{\begin{array}{l} 3x-2y= 9,\\ x+2y= 3;\end{array}\right.$
解：$\left\{\begin{array}{l} x = 3,\\ y = 0\end{array}\right.$
(2)$\left\{\begin{array}{l} x+3y= 14,\\ \frac {x-2}{3}-\frac {y-2}{2}= 1。\end{array}\right.$
解：$\left\{\begin{array}{l} x = 6,\\ y = \frac{8}{3}\end{array}\right.$

答案：
(1)$\left\{\begin{array}{l} x = 3,\\ y = 0。\end{array}\right.$
(2)$\left\{\begin{array}{l} x = 6,\\ y = \frac{8}{3}。\end{array}\right.$

1，b=-10。

答案： 根据题意，$\left\{\begin{array}{l} x = -3,\\ y = -1\end{array}\right.$是方程②的解，$\left\{\begin{array}{l} x = 5,\\ y = 2\end{array}\right.$是方程①的解，所以$\left\{\begin{array}{l} 4×(-3)+b×(-1)=-2,\\ 5a + 5×2 = 15,\end{array}\right.$解得$\left\{\begin{array}{l} a = 1,\\ b = -10。\end{array}\right.$

18. 阅读下列材料,解答问题。
材料：解方程组$\left\{\begin{array}{l} 5(x+y)-3(x-y)= 2,\\ 2(x+y)+4(x-y)= 6,\end{array}\right. 若设(x+y)= m$,$(x-y)= n$,则原方程组可变形为$\left\{\begin{array}{l} 5m-3n= 2,\\ 2m+4n= 6,\end{array}\right. 用加减消元法解得\left\{\begin{array}{l} m= 1,\\ n= 1,\end{array}\right. 所以\left\{\begin{array}{l} x+y= 1,\\ x-y= 1,\end{array}\right. 再解这个方程组得\left\{\begin{array}{l} x= 1,\\ y= 0。\end{array}\right. $由此可以看出,在上述解方程组过程中,把某个式子看成一个整体,用一个字母去代替它,我们把这种解方程组的方法叫换元法。
问题：请你用上述方法解方程组$\left\{\begin{array}{l} \frac {x+y}{2}= \frac {x-y}{3},\\ 2(x+y)-3x+3y= 24。\end{array}\right. $
解：设$x + y = A$，$x - y = B$，方程组变形得$\left\{\begin{array}{l} \frac{A}{2}=\frac{B}{3},\\ 2A - 3B = 24,\end{array}\right.$整理得$\left\{\begin{array}{l} 3A - 2B = 0,①\\ 2A - 3B = 24,②\end{array}\right.$①×3 - ②×2 得$5A = -48$，即$A = -9.6$，把$A = -9.6$代入①得$B = -14.4$，所以$\left\{\begin{array}{l} x + y = -9.6,\\ x - y = -14.4,\end{array}\right.$解得$\left\{\begin{array}{l} x = -12,\\ y = 2.4。\end{array}\right.$

答案： 设$x + y = A$，$x - y = B$，方程组变形得$\left\{\begin{array}{l} \frac{A}{2}=\frac{B}{3},\\ 2A - 3B = 24,\end{array}\right.$整理得$\left\{\begin{array}{l} 3A - 2B = 0,①\\ 2A - 3B = 24,②\end{array}\right.$①×3 - ②×2 得$5A = -48$，即$A = -9.6$，把$A = -9.6$代入①得$B = -14.4$，所以$\left\{\begin{array}{l} x + y = -9.6,\\ x - y = -14.4,\end{array}\right.$解得$\left\{\begin{array}{l} x = -12,\\ y = 2.4。\end{array}\right.$