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You asked:
Solve the system of equations:
x y = 1 2 x + 1 y + 1 = − 1 \begin{aligned}\frac{x}{y} &= \frac{1}{2}\\x + \frac{1}{y} + 1 &= -1\end{aligned} y x x + y 1 + 1 = 2 1 = − 1
MathBot Answer:
The system of equations has 2 2 2
x = − 1 + 2 2 , y = − 2 + 2 x = -1 + \frac{\sqrt{2}}{2}, y = -2 + \sqrt{2} x = − 1 + 2 2 , y = − 2 + 2 x = − 1 − 2 2 , y = − 2 − 2 x = -1 - \frac{\sqrt{2}}{2}, y = -2 - \sqrt{2} x = − 1 − 2 2 , y = − 2 − 2
Solve x y = 1 2 \frac{x}{y} = \frac{1}{2} y x = 2 1 x x x x = y 2 x = \frac{y}{2} x = 2 y y 2 \frac{y}{2} 2 y x x x x + 1 + 1 y = − 1 x + 1 + \frac{1}{y} = -1 x + 1 + y 1 = − 1 x + 1 + 1 y = − 1 ( y 2 ) + 1 + 1 y = − 1 y 2 + 1 + 1 y = − 1 \begin{aligned}x + 1 + \frac{1}{y} &= -1 \\ \left(\frac{y}{2}\right) + 1 + \frac{1}{y} &= -1 \\ \frac{y}{2} + 1 + \frac{1}{y} &= -1 \end{aligned} x + 1 + y 1 ( 2 y ) + 1 + y 1 2 y + 1 + y 1 = − 1 = − 1 = − 1 − 2 − 2 -2 - \sqrt{2} − 2 − 2 x y = 1 2 \frac{x}{y} = \frac{1}{2} y x = 2 1 x x x x − 2 − 2 = 1 2 x = − 1 − 2 2 \begin{aligned}\frac{x}{-2 - \sqrt{2}} &= \frac{1}{2}\\x &= -1 - \frac{\sqrt{2}}{2}\end{aligned} − 2 − 2 x x = 2 1 = − 1 − 2 2 x = − 1 − 2 2 , y = − 2 − 2 \begin{aligned}x = -1 - \frac{\sqrt{2}}{2},\,y = -2 - \sqrt{2}\end{aligned} x = − 1 − 2 2 , y = − 2 − 2 − 2 + 2 -2 + \sqrt{2} − 2 + 2 x y = 1 2 \frac{x}{y} = \frac{1}{2} y x = 2 1 x x x x − 2 + 2 = 1 2 x = − 1 + 2 2 \begin{aligned}\frac{x}{-2 + \sqrt{2}} &= \frac{1}{2}\\x &= -1 + \frac{\sqrt{2}}{2}\end{aligned} − 2 + 2 x x = 2 1 = − 1 + 2 2 x = − 1 + 2 2 , y = − 2 + 2 \begin{aligned}x = -1 + \frac{\sqrt{2}}{2},\,y = -2 + \sqrt{2}\end{aligned} x = − 1 + 2 2 , y = − 2 + 2
