使用替换法求解 \left\{ \begin{array} { c } { \sqrt{x} + \sqrt{y} = 6 } \\ { \sqrt{x y} = 8 } \end{array} \right.

\begin{align}&\left(x_{1},y_{1}\right) = \left(16,4\right)\\&\left(x_{2},y_{2}\right) = \left(4,16\right)\end{align}

问题 :

begin{array c sqrt{x + sqrty = 6 } sqrt{x y = 8 } endarray .

(x_{1}, y_{1}) = (16, 4) (x_{2}, y_{2}) = (4, 16)

求值

{x + y = 6 x y = 8

求解 x 的方程

更多步骤

{x = 36 - 12 y + y x y = 8

将 x 的给定值代入方程 x y = 8

(36 - 12 y + y) y = 8

简化

更多步骤

6 y - y = 8

将变量移到右侧并更改其符号

6 y = 8 + y

重写表达式

y = \frac{8 + y}{6}

求值

y = \frac{8 + y}{6}, \frac{8 + y}{6} \geq 0

求值

更多步骤

y = \frac{8 + y}{6}, y \geq - 8

求解 y 的方程

更多步骤

y = 16 \cup y = 4, y \geq - 8

找到交点

y = 16 \cup y = 4

重新排列各项

{x = 36 - 12 y + y y = 16 \cup {x = 36 - 12 y + y y = 4

计算

更多步骤

{x = 4 y = 16 \cup {x = 36 - 12 y + y y = 4

计算

更多步骤

{x = 4 y = 16 \cup {x = 16 y = 4

计算

{x = 16 y = 4 \cup {x = 4 y = 16

检查解决方案

更多步骤

{x = 16 y = 4 \cup {x = 4 y = 16

检查解决方案

更多步骤

{x = 16 y = 4 \cup {x = 4 y = 16

解题方案

(x_{1}, y_{1}) = (16, 4) (x_{2}, y_{2}) = (4, 16)

显示解题方案