Giải hệ phương trình : $\left\{ \begin{array}{l}\left( {x + y} \right)\left( {1 + \dfrac{1}{{xy}}} \right) = 5\\\left( {{x^2} + {y^2}} \right)\left( {1 + \dfrac{1}{{{x^2}{y^2}}}} \right) = 9\end{array} \right.$

$\left\{ \begin{array}{l}\left( {x + y} \right)\left( {1 + \dfrac{1}{{xy}}} \right) = 5\\\left( {{x^2} + {y^2}} \right)\left( {1 + \dfrac{1}{{{x^2}{y^2}}}} \right) = 9\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x + \dfrac{1}{x} + y + \dfrac{1}{y} = 5\\{x^2} + \dfrac{1}{{{x^2}}} + {y^2} + \dfrac{1}{{{y^2}}} = 9\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x + \dfrac{1}{x} + y + \dfrac{1}{y} = 5\\{\left( {x + \dfrac{1}{x}} \right)^2} + {\left( {y + \dfrac{1}{y}} \right)^2} = 13\end{array} \right.$

Đặt $x + \dfrac{1}{x} = a;y + \dfrac{1}{y} = b$ ta có:

$\left\{ \begin{array}{l}a + b = 5\\{a^2} + {b^2} = 13\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}a = 5 - b\\{(5 - b)^2} + {b^2} = 13\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}a + b = 5\\(b - 2)(b - 3) = 0\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}\left\{ \begin{array}{l}a = 2\\b = 3\end{array} \right.\\\left\{ \begin{array}{l}a = 3\\b = 2\end{array} \right.\end{array} \right.$

Giải $\left\{ \begin{array}{l}a = 2\\b = 3\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x + \dfrac{1}{x} = 2\\y + \dfrac{1}{y} = 3\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = 1\\y = \dfrac{{3 \pm \sqrt 5 }}{2}\end{array} \right.$

Giải $\left\{ \begin{array}{l}a = 3\\b = 2\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x + \dfrac{1}{x} = 3\\y + \dfrac{1}{y} = 2\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}y = 1\\x = \dfrac{{3 \pm \sqrt 5 }}{2}\end{array} \right.$

Vậy hệ đã cho có nghiệm$\left( {x;y} \right)$ là : $\left( {1;\dfrac{{3 - \sqrt 5 }}{2}} \right),\left( {1;\dfrac{{3 + \sqrt 5 }}{2}} \right),\left( {\dfrac{{3 - \sqrt 5 }}{2};1} \right),\left( {\dfrac{{3 + \sqrt 5 }}{2};1} \right)$