Once you get the system of two equations
for finding critical points,
solve one of them for y. Which one?
The one that's easier to solve for y.
Then substitute into the other equation
to get an equation involving only x.
This new equation should be
a polynomial of x alone equaling 0.
This polynomial has an easy factor
and a hard factor. The hard factor is
a cubic polynomial with one real root.
You could use the (complicated) Cardano formula,
but it's also fine to use your calculator's
solve command to estimate this root.
(If your calculator doesn't have a solve command,
you could still estimate the root
by graphing the polynomial or by Newton's method
or by the bisection method or...)
After you solve that x equation,
at least approximately, substitute your
solutions for x in your formula for y.
You should find two critical points this way.
Finally, classify them and find all local extrema.