Roland Manning

2022-06-25

Solving system of multivariable 2nd-degree polynomials
$\begin{array}{c}{x}^{2}+3xy-9=0\phantom{\rule{1em}{0ex}}\left(1\right)\\ 2{y}^{2}-4xy+5=0\phantom{\rule{1em}{0ex}}\left(2\right)\end{array}$
where $\left(x,y\right)\in {\mathbb{C}}^{2}$.
More generally, how would you solve any set of equations of the form:
$\begin{array}{c}{ax}^{2}+bxy+c=0\\ d{y}^{2}+exy+f=0\end{array}$
where $a,b,c,d,e,f\in \mathbb{Q}$ and $\left(x,y\right)\in {\mathbb{C}}^{2}$.

Abigail Palmer

Expert

Multiply first equation by $5$.
Multiply second equation by $9$.
Divide this equation by ${y}^{2}$.
Let $t=\frac{x}{y}$.
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 ${y}^{2}$
$a\frac{{x}^{2}}{{y}^{2}}+b\frac{x}{y}+c=0$
Replace $\frac{x}{y}=t$
${at}^{2}+bt+c=0$

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