solve
$\displaystyle 12+4i=5x + \frac{5x}{x^2+y^2}+5 i y -\frac{5 i y}{x^2+y^2}=5z+\frac{5}{z}$
where
$\displaystyle z=x+i y$
This leads to a quadratic in z with two solutions $\displaystyle z=2+i$ and $\displaystyle z=\frac{2}{5}-\frac{1}{5}i$
So x = 2 y = 1 and x = 2/5 and y = -1/5 are the two solutions of the system