Math Problem Solver

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

\[x = \frac{45}{2} - \frac{\sqrt{2026}}{2}, y = \frac{\sqrt{2026}}{2} + \frac{195}{2}\]\[x = \frac{45}{2} + \frac{\sqrt{2026}}{2}, y = \frac{195}{2} - \frac{\sqrt{2026}}{2}\]

Solve \(x + y = 120\) for \(x\). \[x = 120 - y\]Substitute \(120 - y\) for \(x\) in \(2 y + \frac{1}{2 x} = 150\) and simplify. $$\begin{aligned}2 y + \frac{1}{2 x} &amp= 150 \\ 2 y + \frac{1}{2 \left(120 - y\right)} &= 150 \\ \frac{4 y \left(y - 120\right) - 1}{2 \left(y - 120\right)} &= 150 \end{aligned}$$Substitute \(\frac{195}{2} - \frac{\sqrt{2026}}{2}\) into \(x + y = 120\) to solve for \(x\). \[\begin{aligned}x - \frac{\sqrt{2026}}{2} + \frac{195}{2} &= 120\\x + \left(\frac{195}{2} - \frac{\sqrt{2026}}{2}\right) &= 120\\x &= \frac{45}{2} + \frac{\sqrt{2026}}{2}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = \frac{45}{2} + \frac{\sqrt{2026}}{2},\,y = \frac{195}{2} - \frac{\sqrt{2026}}{2}\end{aligned}$$Substitute \(\frac{\sqrt{2026}}{2} + \frac{195}{2}\) into \(x + y = 120\) to solve for \(x\). \[\begin{aligned}x + \frac{\sqrt{2026}}{2} + \frac{195}{2} &= 120\\x + \left(\frac{\sqrt{2026}}{2} + \frac{195}{2}\right) &= 120\\x &= \frac{45}{2} - \frac{\sqrt{2026}}{2}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = \frac{45}{2} - \frac{\sqrt{2026}}{2},\,y = \frac{\sqrt{2026}}{2} + \frac{195}{2}\end{aligned}$$