smc

From $\frac{x}{4}=4y$, we get $x=16y$. Plugging in the other equation, $\frac{16y}{2}=y^2$, so $y^2-8y=0$. Factoring, we get $y(y-8)=0$, so the solutions are $0$ and $8$. Since $x\ne 0$, we also have $y\ne 0$, so $y=8$. Hence $x=16\cdot 8=\boxed{128}$.