Math Problem Solver

The system of equations has one solution.

$$x = 4, y = 2$$

Solve $- x + \frac{x}{y} = -2$ for $x$. $$x = \frac{2 y}{y - 1}$$Substitute $\frac{2 y}{y - 1}$ for $x$ in $y + \frac{y}{x} = \frac{5}{2}$ and simplify. $$\begin{aligned}y + \frac{y}{x} &= \frac{5}{2} \\ y + \frac{y}{\left(\frac{2 y}{y - 1}\right)} &= \frac{5}{2} \\ y &= 2 \end{aligned}$$Substitute $2$ into $- x + \frac{x}{y} = -2$ to solve for $x$. $$\begin{aligned}- \frac{x}{2} &= -2\\x &= 4\end{aligned}$$This yields the following solution. $$\begin{aligned}x = 4,\,y = 2\end{aligned}$$