So I have this 2x2 system with fixed $\beta$ and $\gamma$ , and I tried to solve it by expressing $p$ or $r$ in terms of the other but it's not working out for me, I see the symmetry of the equations but I don't know how to use it. Also this is not homework it's something that came up while I was playing with a cubic.
$\beta,\gamma,p,r \in \mathbb{C}$
$$\beta=\frac{1+\sqrt[3]{r^3-1}\cdot{\sqrt[3]{p^3-1}}^2}{rp^2}$$
$$\gamma=\frac{1+\sqrt[3]{p^3-1}\cdot{\sqrt[3]{r^3-1}}^2}{pr^2}$$
for simplifications set $$\sqrt[3]{r^3-1}=a,\sqrt[3]{p^3-1}=b$$