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.


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 ${x\over y}=t$ $${ a{t} }^{ 2 }+b{t}+{c}=0$$

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    $\begingroup$ It just needed to rhyme :) (And for the second question, you can also generalize: multiply (1) by $-f$, multiply (2) by $c$, add and done) $\endgroup$ – chubakueno Oct 10 '13 at 2:53

It is not necessarily true that a system like this has four complex solutions. In general, two curves of degree 2 which do not have components in common, have four intersections points up to multiplicity, but you may have double, triple, or quadruple points due to tangency, and you may also have intersection points "at infinity": every line through the origin contains a single point on the projective plane which does not lie in $\mathbb{C}^2$, and one or more of your solutions may be out there.

For instance, $xy=1$ and $xy=0$ is a system of the form you gave, but it has two double intersection points, one at "$(\infty,0)$" and one at "$(0,\infty)$".

As the other answer indicates, a system of equations of this form can be manipulated to give a homogeneous quadratic $Ax^2 + Bxy + Cy^2$, which factors into two equations of lines, $\alpha x + \beta y = 0$ and $\gamma x + \delta y=0$. From there, you are looking for the intersection points of lines with quadratic curves, which is a much simpler problem.

There are very general ideas at work here. The general of the solutions of systems of polynomial equations points towards algebraic geometry, one of the deepest, hardest, and most breathtaking fields of mathematics.


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