\(a)\) Điều kiện \(x \ne \pm y.\) Đặt \(u = \dfrac{1}{{x + y}};v = \dfrac{1}{{x - y}}.\) Hệ phương trình trở thành : \(\left\{ \begin{gathered}2u + v = 3 \hfill \\ u - 3v = 1 \hfill \\ \end{gathered} \right.\,\,\,\,\left( * \right)\).
Giải hệ phương trình \(\left( * \right)\) ta được :\(\left\{ \begin{gathered}2u + v = 3 \hfill \\ u - 3v = 1 \hfill \\ \end{gathered} \right.\)\(\Rightarrow\left\{ \begin{gathered}2u + v = 3 \hfill \\ 2u - 6v = 2 \hfill \\ \end{gathered} \right.\)\(\Rightarrow\left\{ \begin{gathered}7v = 1 \hfill \\ u - 3v = 1 \hfill \\ \end{gathered} \right.\)\(\Rightarrow\left\{ \begin{gathered}v = \dfrac{1}{7} \hfill \\ u = 1 +3.\dfrac{1}{7}\hfill \\ \end{gathered} \right.\)
\(\Rightarrow\left\{ \begin{gathered}u = \frac{{10}}{7} \hfill \\ v = \frac{1}{7} \hfill \\\end{gathered} \right. \)\(\Rightarrow \left\{ \begin{gathered} \frac{1}{{x + y}} = \frac{{10}}{7} \hfill \\\frac{1}{{x - y}} = \frac{1}{7} \hfill \\\end{gathered} \right. \)\(\Leftrightarrow \left\{ \begin{gathered} x + y = \frac{7}{{10}} \hfill \\ x - y = 7 \hfill \\\end{gathered} \right. \)\(\Leftrightarrow \left\{ \begin{gathered} y = - \frac{{63}}{{20}} \hfill \\ x = \frac{{77}}{{20}} \hfill \\\end{gathered} \right.\)
Vậy nghiệm của hệ phương trình là : \(\left( {\dfrac{{77}}{{20}}\,\,;\,\, - \dfrac{{63}}{{20}}} \right).\)
\(b)\) Điều kiện \(x \geqslant 0;y \geqslant 0.\) Đặt \(\sqrt x = u\left( {u \geqslant 0} \right),\sqrt y = v\left( {v \geqslant 0} \right).\) Hệ phương trình trở thành :
\(\left\{ \begin{gathered} 3u - 2v = - 2 \hfill \\ 2u - v = 1 \hfill \\\end{gathered} \right. \)\(\Leftrightarrow\left\{ \begin{gathered} 3u - 2v = - 2 \hfill \\ 4u -2 v = 2 \hfill \\\end{gathered} \right. \)\(\Leftrightarrow \left\{ \begin{gathered} u = 0 \hfill \\ v = 1 \hfill \\\end{gathered} \right.\)\( \Rightarrow \left\{ \begin{gathered} \sqrt x = 0 \hfill \\\sqrt y = 1 \hfill \\\end{gathered} \right.\)\( \Rightarrow \left\{ \begin{gathered} x = 0 (tm)\hfill \\ y = 1(tm) \hfill \\\end{gathered} \right.\)
Vậy nghiệm của hệ phương trình là \(\left( {0\,\,;\,\,1} \right).\)
