Excludе $x$ frоm this systеm оf equаtiоns.

Question

How do you go about excluding $x$ from the system of equations? \begin{cases}x^3-xy-y^3+y=0\\x^2+x-y^2=1\end{cases}

Answer 1

The resultant of the two polynomials $x^3 - xy - y^3 + y$ and $x^2 + x - y^2 - 1$ with respect to $x$ is $5\,{y}^{5}-7\,{y}^{4}+6\,{y}^{3}-2\,{y}^{2}-y-1$. That is, the equation with $x$ removed is $5\,{y}^{5}-7\,{y}^{4}+6\,{y}^{3}-2\,{y}^{2}-y-1=0$.

Answer 2

Hint: $x^2+x+1/4=(x+1/2)^2$. Use that and your second equation to express $x$ as a function of $y$.