jasperE3 wrote:
Solve the system
\begin{align*}
36x^2y-27y^3&~=~8,\\
4x^3-27xy^2&~=~4.\end{align*}
$(36x^2y-27y^3)^2+(2(4x^3-27xy^2))^2=8^2+(2(4))^2$
$(4x^2+9y^2)^3=128$
$4x^2+9y^2=2\sqrt[3]{2}$
After this it is pretty easy.
There are 3 real solution and 5 complex solution with $\text{cis}\frac{2\pi}{3}$ that is very long and I am too lazy to type.