x/y=1/2 x+1/y+1=-1

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on Mar 17, 2025 at 2:41 pm



You asked:

Solve the system of equations:xy=12x+1y+1=1\begin{aligned}\frac{x}{y} &= \frac{1}{2}\\x + \frac{1}{y} + 1 &= -1\end{aligned}

MathBot Answer:

The system of equations has 22 solutions.

x=1+22,y=2+2x = -1 + \frac{\sqrt{2}}{2}, y = -2 + \sqrt{2}x=122,y=22x = -1 - \frac{\sqrt{2}}{2}, y = -2 - \sqrt{2}


Solve xy=12\frac{x}{y} = \frac{1}{2} for xx. x=y2x = \frac{y}{2}Substitute y2\frac{y}{2} for xx in x+1+1y=1x + 1 + \frac{1}{y} = -1 and simplify. x+1+1y=1(y2)+1+1y=1y2+1+1y=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}Substitute 22-2 - \sqrt{2} into xy=12\frac{x}{y} = \frac{1}{2} to solve for xx. x22=12x=122\begin{aligned}\frac{x}{-2 - \sqrt{2}} &= \frac{1}{2}\\x &= -1 - \frac{\sqrt{2}}{2}\end{aligned}This yields the following solution. x=122,y=22\begin{aligned}x = -1 - \frac{\sqrt{2}}{2},\,y = -2 - \sqrt{2}\end{aligned}Substitute 2+2-2 + \sqrt{2} into xy=12\frac{x}{y} = \frac{1}{2} to solve for xx. x2+2=12x=1+22\begin{aligned}\frac{x}{-2 + \sqrt{2}} &= \frac{1}{2}\\x &= -1 + \frac{\sqrt{2}}{2}\end{aligned}This yields the following solution. x=1+22,y=2+2\begin{aligned}x = -1 + \frac{\sqrt{2}}{2},\,y = -2 + \sqrt{2}\end{aligned}

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