Math Problem Solver

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

\[x = 4, y = 15\]\[x = 6, y = 10\]

Solve \(\frac{x}{2} + \frac{y}{5} = 5\) for \(x\). \[x = 10 - \frac{2 y}{5}\]Substitute \(10 - \frac{2 y}{5}\) for \(x\) in \(\frac{5}{y} + \frac{2}{x} = \frac{5}{6}\) and simplify. $$\begin{aligned}\frac{5}{y} + \frac{2}{x} &= \frac{5}{6} \\ \frac{5}{y} + \frac{2}{\left(10 - \frac{2 y}{5}\right)} &= \frac{5}{6} \\ \frac{125}{y \left(y - 25\right)} &= - \frac{5}{6} \end{aligned}$$Substitute \(10\) into \(\frac{x}{2} + \frac{y}{5} = 5\) to solve for \(x\). \[\begin{aligned}\frac{x}{2} + 2 &= 5\\\frac{x}{2} &= 3\\x &= 6\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 6,\,y = 10\end{aligned}$$Substitute \(15\) into \(\frac{x}{2} + \frac{y}{5} = 5\) to solve for \(x\). \[\begin{aligned}\frac{x}{2} + 3 &= 5\\\frac{x}{2} &= 2\\x &= 4\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 4,\,y = 15\end{aligned}$$