algebra precalculus - Solve ......................................................................

This is what I did

Can anyone tell me what's wrong me or the question?

Answer

You did the case $y=4x$, and got answers $x=1$ and $x=-\frac{9}{4}$. Neither is a solution of the original equation. It is easy to verify that when $y=4x$ we have $\sqrt{\frac{x}{y}}-\sqrt{\frac{y}{x}}=-\frac{3}{2}$.

When you squared, you introduced extraneous roots. Only $x=4y$ leads to a solution of the system.