Math Problem Solver

The system of equations has \(2\) solutions.

\[x = \sqrt{2} + \frac{3}{2}, y = 4 - 2 \sqrt{2}\]\[x = \frac{3}{2} - \sqrt{2}, y = 2 \sqrt{2} + 4\]

Solve \(y = 7 - 2 x\) for \(x\). \[x = \frac{7}{2} - \frac{y}{2}\]Substitute \(\frac{7}{2} - \frac{y}{2}\) for \(x\) in \(y = 1 + \frac{1}{2 x}\) and simplify. $$\begin{aligned}y &amp= 1 + \frac{1}{2 x} \\ y &= 1 + \frac{1}{2 \left(\frac{7}{2} - \frac{y}{2}\right)} \\ y &= \frac{y - 8}{y - 7} \end{aligned}$$Substitute \(4 - 2 \sqrt{2}\) into \(y = 7 - 2 x\) to solve for \(x\). \[\begin{aligned}4 - 2 \sqrt{2} &= 7 - 2 x\\2 x &= 2 \sqrt{2} + 3\\x &= \sqrt{2} + \frac{3}{2}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = \sqrt{2} + \frac{3}{2},\,y = 4 - 2 \sqrt{2}\end{aligned}$$Substitute \(2 \sqrt{2} + 4\) into \(y = 7 - 2 x\) to solve for \(x\). \[\begin{aligned}2 \sqrt{2} + 4 &= 7 - 2 x\\2 x &= 3 - 2 \sqrt{2}\\x &= \frac{3}{2} - \sqrt{2}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = \frac{3}{2} - \sqrt{2},\,y = 2 \sqrt{2} + 4\end{aligned}$$