<s>[INST] How many real solutions $(x,y)$ for $x + y^2 = y^3 $ and $y + x^2 = x^3$ I think the answer is that there are $3$ real solutions.


*

*When $x = y$

*When $x = -y$
If $x\neq y$ is there a real solution? If so, how do we find the real solutions?
 [/INST] Substituting $y=x^3-x^2$ into the other equation gives
\begin{eqnarray*}
x(x^8-3x^7+3x^6-2x^5+2x^4-x^3-1)=0 \\
x(x^2-x-1)( x^6-2x^5 +2x^4 -2x^3 +2x^2-x+1) =0 \\
x(x^2-x-1) \left( (x^4+x^2)(x-1)^2+ \frac{1}{4} (2x-1)^2 + \frac{3}{4} \right) =0. \\
\end{eqnarray*}
So you are indeed correct: there are $3$ real solutions.
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