\(\Leftrightarrow\left\{{}\begin{matrix}x^3-y^3-3\left(x-y\right)=0\\x^6+y^6=27\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(x-y\right)\left(x^2+y^2+xy-3\right)=0\\x^6+y^6=27\end{matrix}\right.\)
TH1: \(\left\{{}\begin{matrix}x-y=0\\x^6+y^6=27\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=y=\sqrt[6]{\frac{27}{2}}\\x=y=-\sqrt[6]{\frac{27}{2}}\end{matrix}\right.\)
TH2: \(\left\{{}\begin{matrix}x^2+y^2+xy=3\\x^6+y^6=27\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+y^2=3-xy\\\left(x^2+y^2\right)^3-3x^2y^2\left(x^2+y^2\right)=27\end{matrix}\right.\)
\(\Rightarrow\left(3-xy\right)^3-3x^2y^2\left(3-xy\right)=27\)
\(\Leftrightarrow2\left(xy\right)^3-27\left(xy\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}xy=0\\xy=\pm\frac{3\sqrt{6}}{2}\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=0\\y=0\\y=\frac{3\sqrt{6}}{2x}\\y=\frac{-3\sqrt{6}}{2x}\end{matrix}\right.\)
Thế vào \(x^6+y^6=27\)
\(\Rightarrow...\)