Question

giải hệ

\(\left\{{}\begin{matrix}x^2+\frac{1}{x^2}+y^2+\frac{1}{y^2}=9\\x+y+\frac{1}{x}+\frac{1}{y}=5\end{matrix}\right.\)

Answer 1

\(DKXD:x,y\ne0\)

\(x+\frac{1}{x}=a;y+\frac{1}{y}=b\Rightarrow x^2+\frac{1}{x^2}=a^2-2;y^2+\frac{1}{y^2}=b^2-2\)

\(\Rightarrow\left\{{}\begin{matrix}a^2-2+b^2-2=9\\a+b=5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(a+b\right)^2-2ab=13\\a+b=5\end{matrix}\right.\)

\(\Leftrightarrow ab=6\)

\(\Rightarrow\left\{{}\begin{matrix}a+b=5\\ab=6\Rightarrow b=\frac{6}{a}\left(a\ne0ko-la-nghiem\right)\end{matrix}\right.\)

\(\Rightarrow a+\frac{6}{a}=5\Leftrightarrow a^2-5a+6=0\Leftrightarrow\left[{}\begin{matrix}a=3\\a=2\end{matrix}\right.\)\(\Rightarrow\left[{}\begin{matrix}b=2\\b=3\end{matrix}\right.\)

Đến đây giải nốt nha :)