\(\left\{\begin{matrix}x^2-xy+y^2=7\\x^4+x^2y^2+y^4=21\end{matrix}\right.\)
\(\Leftrightarrow\left\{\begin{matrix}x^2+y^2=7+xy\\x^4+x^2y^2+y^4=21\end{matrix}\right.\)
\(\Leftrightarrow\left\{\begin{matrix}x^4+y^4+2x^2y^2=49+14xy+x^2y^2\\x^4+x^2y^2+y^4=21\end{matrix}\right.\)
\(\Leftrightarrow\left\{\begin{matrix}xy=-2\\x^2-xy+y^2=7\end{matrix}\right.\)
\(\Leftrightarrow\left\{\begin{matrix}x=\frac{-2}{y}\\\frac{4}{y^2}+y^2=5\end{matrix}\right.\)
\(\Leftrightarrow\left[\begin{matrix}\left\{\begin{matrix}x=1\\y=-2\end{matrix}\right.\\\left\{\begin{matrix}x=-1\\y=2\end{matrix}\right.\\\left\{\begin{matrix}x=2\\y=-1\end{matrix}\right.\\\left\{\begin{matrix}x=-2\\y=1\end{matrix}\right.\end{matrix}\right.\)