Solving a system of equation:

Question

Solve for $x,y$: \begin{align} x^3 + y^3&=2\\ x^2 +x + 9y - 3y^2&=8 \\ \end{align}

I can find $x=y=1$ by guessing. Please help me solve it without using computer.

Thanks

Edited, sorry, I have to change $x^2$ to $x^3$ to have a nice solution, but still can't done it by hand.

Answer 1

Substracting equation $2$ from $1$ we obtain $y^3 - x - 9y + 3y^2 = -6 \iff y^3 - \sqrt{2-y^3} -9y + 3y^2 = -6$ Now you have an expression that only consists of $y$'s. Try to see if you can manipulate this such that you get your desired result. I am not sure though if you cand find it algebraically. If you can, chances are that long division may be needed.

Answer 2

I think you can use Newton-Raphson or any other approximation methods for multivariate equations (this is an example)