TITLE: Solving these two equations
QUESTION [7 upvotes]: Solve for $x$ and $y$
  $$5x(1+\frac{1}{x^2+y^2})=12$$ $$5y(1-\frac{1}{x^2+y^2})=4$$

Combining these two equations we get 
$$5x^3-15x^2y+5xy^2-15y^3+5x+15y=0$$
Which could be factored if it had $-15y$ instead of $+15y$ for the last term.
Generally I do not post questions involving solving simple equations, but I find it really hard!

REPLY [4 votes]: $\dfrac{1}{x^2+y^2}=A$ $$5x(1+A)=12\implies A=\frac{12}{5x}-1$$ $$5y(1-A)=4 \implies A=1-\frac{4}{5y}$$ we now know that bu $A=A$ $$\frac{12}{5x}-1=1-\frac{4}{5y} (*)$$ when we write $y$ in terms of $x$ or the other way around in either of these equations (above of course) we are done, a hint for the remaining multiply the system $(*)$ with $5xy$...