How to solve this equation algebraically Solve the following simultaneous equations on the set of real numbers: 
\begin{cases}x^2 + y^3 = x+1 \\ x^3+y^2=y+1\end{cases}
Thanks for helping!

Hint:
Eliminate one of the unknowns in a way to get a polynomial.
From the second equation,
$$y^2-y+1=2-x^3$$ and multiplying by $y+1$ and using the first equation,
$$(2-x^3)(y+1)=(y^2-y+1)(y+1)=y^3+1=2+x-x^2,$$
and
$$y=\frac{2+x-x^2}{2-x^3}-1=\frac{x-x^2+x^3}{2-x^3}.$$
Then
$$\left(x-x^2+x^3\right)^3=(1+x-x^2)(2-x^3)^3.$$
