Solving these two equations

Question

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!

Answer 1

We have $$1+\frac{1}{x^2+y^2}=\frac{12}{5x}$$ and $$1-\frac{1}{x^2+y^2}=\frac{4}{5y},$$ which gives $$\frac{6}{x}+\frac{2}{y}=5$$ or $$y=\frac{2x}{5x-6},$$ which after substitution to the first equation gives

$$25x^4-120x^3+209x^2-156x+36=0$$ or $$(5x^2-12x+6.5)^2-2.5^2=0$$ or $$(x-2)(5x-2)(5x^2-12x+9)=0.$$ Id est, we got the answer: $$\{(2,1),(0.4,-0.2)\}$$

Answer 2

$\dfrac{1}{x^2+y^2}=A$ $$5x(1+A)=12\implies A=\frac{12}{5x}-1$$ $$5y(1-A)=4 \implies A=1-\frac{4}{5y}$$ 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 $5xy$...