How would you go about solving a problem such as:
\begin{matrix} { x }^{ 2 }+3xy-9=0 \quad(1)\\ 2{ y }^{ 2 }-4xy+5=0 \quad(2) \end{matrix}
where $(x,y)\in\mathbb{C}^{2}$.
More generally, how would you solve any set of equations of the form:
\begin{matrix} { ax }^{ 2 }+bxy+c=0 \\ d{ y }^{ 2 }+exy+f=0 \end{matrix}
where $a, b, c, d, e, f \in \mathbb{Q}$ and $(x,y)\in\mathbb{C}^{2}$.
I know that there are four complex solutions to a system of equations in this form, but don't know how one would solve for them.
Best Answer
Multiply first equation by $5$.
Multiply second equation by $9$.
Add both Equations.
Divide this equation by $y^2$.
Let $t={x\over y}$.
You get a quadratic in $t$.
The steps are[Generalization]:
Reduce the two equations to one
$${ ax }^{ 2 }+bxy+{cy}^2=0$$ Then divide by $y^2$ $${ a{x^2 \over y^2} }+b{x\over y}+{c}=0$$ Replace ${x\over y}=t$ $${ a{t} }^{ 2 }+b{t}+{c}=0$$