《2025年暑假衔接七年级数学浙教版延边人民出版社》

6. 在迎宾晚宴上,若每桌坐$12$人,则空出$3$张桌子；若每桌坐$10$人,则还有$12$人不能就座。设有嘉宾$x$名,共准备了$y$张桌子。根据题意,下列方程组正确的是 ………………………………………………………………………(A)
A. $\left\{\begin{array}{l} x= 12(y-3),\\ x-12= 10y\end{array}\right. $
B. $\left\{\begin{array}{l} x= 12(y+3),\\ x-12= 10y\end{array}\right. $
C. $\left\{\begin{array}{l} x= 12(y+3),\\ x+12= 10y\end{array}\right. $
D. $\left\{\begin{array}{l} x= 12(y-3),\\ x+12= 10y\end{array}\right. $

答案： A

7. 如果方程组$\left\{\begin{array}{l} x+y= ★,\\ 2x+y= 16\end{array}\right. 的解为\left\{\begin{array}{l} x= 6,\\ y= ■\end{array}\right. $那么被“★、■”遮住的两个数分别为 ………………………………………………………………………(C)
A. $3$,$10$
B. $4$,$10$
C. $10$,$4$
D. $10$,$3$

答案： C

8. 我们知道二元一次方程组$\left\{\begin{array}{l} 2x-3y= 3,\\ 3x-4y= 5\end{array}\right. 的解是\left\{\begin{array}{l} x= 3,\\ y= 1\end{array}\right. 现给出另一个二元一次方程组\left\{\begin{array}{l} 2(2x+1)-3(3y-1)= 3,\\ 3(2x+1)-4(3y-1)= 5,\end{array}\right. $它的解是 ……………………………(C)
A. $\left\{\begin{array}{l} x= -1,\\ y= \frac {2}{3}\end{array}\right. $
B. $\left\{\begin{array}{l} x= -1,\\ y= -\frac {2}{3}\end{array}\right. $
C. $\left\{\begin{array}{l} x= 1,\\ y= \frac {2}{3}\end{array}\right. $
D. $\left\{\begin{array}{l} x= 1,\\ y= -\frac {2}{3}\end{array}\right. $

答案： C 【解析】在二元一次方程组$\left\{\begin{array}{l} 2(2x + 1)-3(3y - 1)=3,\\ 3(2x + 1)-4(3y - 1)=5\end{array}\right.$中，令$\left\{\begin{array}{l} 2x + 1 = X,\\ 3y - 1 = Y,\end{array}\right.$则$\left\{\begin{array}{l} 2X - 3Y = 3,\\ 3X - 4Y = 5,\end{array}\right.$因为二元一次方程组$\left\{\begin{array}{l} 2x - 3y = 3,\\ 2x - 4y = 5\end{array}\right.$的解是$\left\{\begin{array}{l} x = 3,\\ y = 1,\end{array}\right.$所以$\left\{\begin{array}{l} X = 3,\\ Y = 1,\end{array}\right.$所以$\left\{\begin{array}{l} 2x + 1 = 3,\\ 3y - 1 = 1,\end{array}\right.$解得$\left\{\begin{array}{l} x = 1,\\ y = \frac{2}{3},\end{array}\right.$故选：C。

9. 方程$x+2y= 7$在自然数范围内的解为$\left\{\begin{array}{l} x = 7,\\ y = 0,\end{array}\right. \left\{\begin{array}{l} x = 5,\\ y = 1,\end{array}\right. \left\{\begin{array}{l} x = 3,\\ y = 2,\end{array}\right. \left\{\begin{array}{l} x = 1,\\ y = 3\end{array}\right.$。

答案： $\left\{\begin{array}{l} x = 7,\\ y = 0,\end{array}\right. \left\{\begin{array}{l} x = 5,\\ y = 1,\end{array}\right. \left\{\begin{array}{l} x = 3,\\ y = 2,\end{array}\right. \left\{\begin{array}{l} x = 1,\\ y = 3\end{array}\right.$

10. 已知$\left\{\begin{array}{l} 2x+y= 5,\\ x+2y= 6,\end{array}\right. 那么(x+y)(x-y)$的值为______$-\frac{11}{3}$。

答案： $-\frac{11}{3}$

11. 若$-x^{a+2}y^{2-b}与3x^{b-1}y^{a+1}$是同类项,则$b^{a}$的值是______$\frac{1}{2}$。

答案： $\frac{1}{2}$

12. 兄弟两人,弟弟五年后的年龄与哥哥五年前的年龄相等,$3年后兄弟两人的年龄和是他们年龄之差的3$倍,则兄弟两人今年的岁数分别是17 岁和 7 岁。

答案： 17 岁和 7 岁

13. 若关于$x$,$y的二元一次方程组\left\{\begin{array}{l} x-y= 3,\\ 2x+4y= 3k\end{array}\right. 的解满足x+y= 2025$,则$k$的值为______2024。

答案： 2024

14. 《九章算术》是我国古代数学的经典著作,书中记载了这样一个题目：今有黄金九枚,白银一十一枚,称之重适等,交易其一,金轻十三两,问金,银各重几何？其大意是：甲袋中装有黄金$9$枚(每枚黄金重量相同),乙袋中装有白银$11$枚(每枚白银重量相同),两袋重量相等,两袋互换一枚后,甲袋比乙袋轻了$13$两(袋子重量忽略不计),问黄金,白银各重几两？设每枚黄金重$x$两,每枚白银重$y$两,根据题意得方程组$\left\{\begin{array}{l} 9x = 11y,\\ (10y + x)-(8x + y)=13\end{array}\right.$。

答案： $\left\{\begin{array}{l} 9x = 11y,\\ (10y + x)-(8x + y)=13\end{array}\right.$