System of Two equations with two unknowns of degree four

Question

I am wondering if there is a direct way to solve exactly a system of two equations of this shape (the A to I are constants):

$Axy + Bxy^2 + Cx^2y + Dx^2y^2 + Ex^2 + Fy^2 + Gx + Hy + I=0$
this problem comes from a geometrical problem

Answer 1

No, it is impossible to solve such a system analytically (which is what I presume you mean by "exactly").

To prove this, suppose that you had an oracle that could solve such a system analytically. You can then use this oracle to solve the quintic equation $Ax^5+Bx^4+Cx^3+Dx^2+Ex+F=0$, by asking the oracle for the solution to the system,

$$ \begin{eqnarray} y&=&x^2 \\ 0&=&Axy^2 + Bx^2y + Cxy + Dx^2 + Ex + F \end{eqnarray} $$

However, the quintic equation is not solvable, so this oracle is a fraud.