$\begin{array}{|l}x^2-y^2=7 \\ x^2+xy+y^2=37\end{array}$ 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.

Add the two to get:
$$y=\frac{44-2x^2}{x}$$
Substitute to the first:
$$x^2-\frac{(44-2x^2)^2}{x^2}=7 \Rightarrow \\
3x^4-169x^2+44^2=0 \Rightarrow \\
x^2=\frac{169\pm \sqrt{169^2-12\cdot 44^2}}{6}=\frac{169\pm 73}{6}=16;\frac{121}{3} \Rightarrow \\
x_{1,2}=\pm 4,x_{3,4}=\pm \frac{11}{\sqrt{3}}\\
y_{1,2}=\pm 3,y_{3,4}=\mp \frac{10}{\sqrt{3}}$$
