<s>[INST] $\begin{array}{|l}x^2-y^2=7 \\ x^2+xy+y^2=37\end{array}$ Any suggestions how to solve: $$\begin{array}{|l}x^2-y^2=7 \\ x^2+xy+y^2=37\end{array}$$
I can simplify the system and get a homogeneous polynomial of degree $2$, but I think it must have an easier way.
 [/INST] Note,
$$\frac17(x^2-y^2)=\frac1{37}( x^2+xy+y^2)$$
or
$$44y^2+7xy-30x^2=0$$
which leads to $x=\frac43y$ and $x=-\frac{11}{10}y$. Plug them into $x^2-y^2=7$ to obtain the solutions
$$(4,3),\>(-4,-3), \>(-\frac{11}{\sqrt3},\frac{10}{\sqrt3}), \>(\frac{11}{\sqrt3},-\frac{10}{\sqrt3})$$
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