\(\left\{{}\begin{matrix}x^2+xy+y^2=3\\x+xy+y=-1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(x+y\right)^2-3xy=3\\x+y=-xy-1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(-xy-1\right)^2-3xy=3\left(1\right)\\x+y=-xy-1\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow x^2y^2+2xy+1-3xy=3\)
\(\Leftrightarrow x^2y^2-xy-2=0\)
\(\Leftrightarrow\left(xy+1\right)\left(xy-2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}xy=-1\\xy=2\end{matrix}\right.\)
TH1: \(xy=-1\Rightarrow x+y=0\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=1\\y=-1\end{matrix}\right.\\\left\{{}\begin{matrix}x=-1\\y=1\end{matrix}\right.\end{matrix}\right.\)
TH2: \(xy=2\Rightarrow x+y=-3\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=-1\\y=-2\end{matrix}\right.\\\left\{{}\begin{matrix}x=-2\\y=-1\end{matrix}\right.\end{matrix}\right.\)
