登录
7. 解二元一次方程组$\begin{cases}197x + 4y = 11,\\197x = 19 - 2y,\end{cases} $得 $y= $(
A.$-4$
B.$-\frac{4}{3}$
C.$\frac{5}{3}$
D.$5$
A
)。A.$-4$
B.$-\frac{4}{3}$
C.$\frac{5}{3}$
D.$5$
答案
A
解析
解：$\begin{cases}197x + 4y = 11, \quad①\\197x = 19 - 2y, \quad②\end{cases}$
将②代入①得：$19 - 2y + 4y = 11$
$2y = 11 - 19$
$2y = -8$
$y = -4$
A
将②代入①得：$19 - 2y + 4y = 11$
$2y = 11 - 19$
$2y = -8$
$y = -4$
A
8. 对于任意有理数 $a,b,c,d$,我们规定：$\begin{vmatrix}a&b\\c&d\end{vmatrix} = ad - bc$。已知 $x,y$ 同时满足$\begin{vmatrix}x&y\\-1&4\end{vmatrix} = 5$,$\begin{vmatrix}5&y\\-3&x\end{vmatrix} = 1$,则 $x + y= $
-1
。答案
-1
解析
由题意得：
$\begin{cases}4x - (-1)y = 5 \\5x - (-3)y = 1\end{cases}$
化简得：
$\begin{cases}4x + y = 5 \quad (1) \\5x + 3y = 1 \quad (2)\end{cases}$
由
(1)得：$y = 5 - 4x$，代入
(2)：
$5x + 3(5 - 4x) = 1$
$5x + 15 - 12x = 1$
$-7x = -14$
$x = 2$
将$x = 2$代入$y = 5 - 4x$得：$y = 5 - 8 = -3$
$x + y = 2 + (-3) = -1$
$-1$
$\begin{cases}4x - (-1)y = 5 \\5x - (-3)y = 1\end{cases}$
化简得：
$\begin{cases}4x + y = 5 \quad (1) \\5x + 3y = 1 \quad (2)\end{cases}$
由
(1)得：$y = 5 - 4x$，代入
(2)：
$5x + 3(5 - 4x) = 1$
$5x + 15 - 12x = 1$
$-7x = -14$
$x = 2$
将$x = 2$代入$y = 5 - 4x$得：$y = 5 - 8 = -3$
$x + y = 2 + (-3) = -1$
$-1$
9. 甲、乙两人同时解方程组$\begin{cases}Ax + By = 2,\\Cx + 3y = - 2,\end{cases} 甲正确解得\begin{cases}x = 1,\\y = - 1,\end{cases} $乙因抄错 $C$,解得$\begin{cases}x = 2,\\y = - 6,\end{cases} $求 $A + B - C$ 的值。
答案
9.解:把$\left\{\begin{array}{l} x=1,\\ y=-1\end{array}\right. $代入原方程组,得
$\left\{\begin{array}{l} A-B=2,\\ C-3=-2,\end{array}\right. $
解得$C=1$。
把$\left\{\begin{array}{l} x=2,\\ y=-6\end{array}\right. $代入$Ax+By=2$,得$2A-6B=2$。
可组成方程组$\left\{\begin{array}{l} A-B=2,\\ 2A-6B=2,\end{array}\right. $
解得$\left\{\begin{array}{l} A=2.5,\\ B=0.5,\end{array}\right. $
则$A+B-C=2$。
$\left\{\begin{array}{l} A-B=2,\\ C-3=-2,\end{array}\right. $
解得$C=1$。
把$\left\{\begin{array}{l} x=2,\\ y=-6\end{array}\right. $代入$Ax+By=2$,得$2A-6B=2$。
可组成方程组$\left\{\begin{array}{l} A-B=2,\\ 2A-6B=2,\end{array}\right. $
解得$\left\{\begin{array}{l} A=2.5,\\ B=0.5,\end{array}\right. $
则$A+B-C=2$。
10. 已知关于 $x,y$ 的方程组$\begin{cases}3x - y = 5,\\4ax + 5by = - 22\end{cases} 和\begin{cases}2x + 3y = - 4,\\ax - by = 8\end{cases} $有相同的解,求$( - a)^b$的值。
答案
10.解:因为题中两个方程组有相同的解,即
$\left\{\begin{array}{l} 3x-y=5,\\ 2x+3y=-4\end{array}\right. $
(1)和$\left\{\begin{array}{l} 4ax+5by=-22,\\ ax-by=8\end{array}\right. $
(2)有
相同的解。
解方程组
(1)得$\left\{\begin{array}{l} x=1,\\ y=-2,\end{array}\right. $
把$\left\{\begin{array}{l} x=1,\\ y=-2\end{array}\right. $代入
(2)得$\left\{\begin{array}{l} 4a-10b=-22,\\ a+2b=8,\end{array}\right. $
解得$\left\{\begin{array}{l} a=2,\\ b=3,\end{array}\right. $
所以$(-a)^{b}=(-2)^{3}=-8$。
$\left\{\begin{array}{l} 3x-y=5,\\ 2x+3y=-4\end{array}\right. $
(1)和$\left\{\begin{array}{l} 4ax+5by=-22,\\ ax-by=8\end{array}\right. $
(2)有
相同的解。
解方程组
(1)得$\left\{\begin{array}{l} x=1,\\ y=-2,\end{array}\right. $
把$\left\{\begin{array}{l} x=1,\\ y=-2\end{array}\right. $代入
(2)得$\left\{\begin{array}{l} 4a-10b=-22,\\ a+2b=8,\end{array}\right. $
解得$\left\{\begin{array}{l} a=2,\\ b=3,\end{array}\right. $
所以$(-a)^{b}=(-2)^{3}=-8$。
11. 【综合与实践】已知关于 $x,y$ 的二元一次方程组$\begin{cases}3x - ay = 16,\\2x + by = 15\end{cases} 的解是\begin{cases}x = 7,\\y = 1,\end{cases} $求下列关于 $x,y$ 的方程组的解：
(1)$\begin{cases}3(x + y) - a(x - y) = 16,\\2(x + y) + b(x - y) = 15;\end{cases} $
(2)$\begin{cases}\frac{3(x - 2y)}{2}-\frac{a}{3}y = 16,\\x - 2y+\frac{b}{3}y = 15。\end{cases} $
(1)$\begin{cases}3(x + y) - a(x - y) = 16,\\2(x + y) + b(x - y) = 15;\end{cases} $
(2)$\begin{cases}\frac{3(x - 2y)}{2}-\frac{a}{3}y = 16,\\x - 2y+\frac{b}{3}y = 15。\end{cases} $
答案
11.解:
(1)由已知可得$\left\{\begin{array}{l} x+y=7,\\ x-y=1,\end{array}\right. $
解得$\left\{\begin{array}{l} x=4,\\ y=3。\end{array}\right. $
(2)由已知可得$\left\{\begin{array}{l} \frac {x-2y}{2}=7,\\ \frac {1}{3}y=1,\end{array}\right. $
解得$\left\{\begin{array}{l} x=20,\\ y=3。\end{array}\right. $
(1)由已知可得$\left\{\begin{array}{l} x+y=7,\\ x-y=1,\end{array}\right. $
解得$\left\{\begin{array}{l} x=4,\\ y=3。\end{array}\right. $
(2)由已知可得$\left\{\begin{array}{l} \frac {x-2y}{2}=7,\\ \frac {1}{3}y=1,\end{array}\right. $
解得$\left\{\begin{array}{l} x=20,\\ y=3。\end{array}\right. $
