\(\left\{{}\begin{matrix}x^2-y=y^2-x\\x^2-6y=7\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2-y^2-y+x=0\\x^2-6y=7\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(x-y\right)\left(x+y+1\right)=0\\x^2-6y=7\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}y=x\\y=-1-x\end{matrix}\right.\\x^2-6y=7\end{matrix}\right.\)
TH1: \(\Leftrightarrow\left\{{}\begin{matrix}y=x\\x^2-6y=7\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=x\\x^2-6x-7=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=x\\\left[{}\begin{matrix}x=7\\x=-1\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=y=7\\x=y=-1\end{matrix}\right.\)
TH2: \(\left\{{}\begin{matrix}y=-1-x\\x^2-6\left(-1-x\right)-7=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=-1-x\\x^2+6x-1=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=-1-x\\\left[{}\begin{matrix}x=-3+\sqrt{10}\\x=-3-\sqrt{10}\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=-3+\sqrt{10}\\y=2-\sqrt{10}\end{matrix}\right.\\\left\{{}\begin{matrix}x=-3-\sqrt{10}\\y=2+\sqrt{10}\end{matrix}\right.\end{matrix}\right.\)
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