Giải hệ phương trình sau: \(\left\{\begin{array}{l} x-2 \sqrt{2} y=\sqrt{3} \\ \sqrt{2} x+y=1-\sqrt{6} \end{array}\right.\).

* Hướng dẫn giải

Ta có:

\(\left\{\begin{array}{l} x-2 \sqrt{2} y=\sqrt{3} \\ \sqrt{2} x+y=1-\sqrt{6} \end{array} \Leftrightarrow\left\{\begin{array}{l} x=2 \sqrt{2} y+\sqrt{3} \\ \sqrt{2}(2 \sqrt{2} y+\sqrt{3})+y=1-\sqrt{6} \end{array}\right.\right.\)

\(\begin{array}{l} \Leftrightarrow\left\{\begin{array}{l} x=2 \sqrt{2} y+\sqrt{3} \\ 4 y+\sqrt{6}+y=1-\sqrt{6} \end{array} \Leftrightarrow\left\{\begin{array}{l} x=2 \sqrt{2} y+\sqrt{3} \\ 5 y=1-2 \sqrt{6} \end{array} \Leftrightarrow\left\{\begin{array}{l} x=2 \sqrt{2} \cdot\left(\frac{1-2 \sqrt{6}}{5}\right)+\sqrt{3} \\ y=\frac{1-2 \sqrt{6}}{5} \end{array}\right.\right.\right. \\ \Leftrightarrow\left\{\begin{array}{l} x=2 \sqrt{2} \cdot\left(\frac{1-2 \sqrt{6}}{5}\right)+\sqrt{3} \\ y=\frac{1-2 \sqrt{6}}{5} \end{array}\right. \\ \Leftrightarrow\left\{\begin{array}{l} x=\frac{2 \sqrt{2}-4 \sqrt{12}+5 \sqrt{3}}{5} \\ y=\frac{1-2 \sqrt{6}}{5} \end{array}\right. \end{array}\)

Vậy, nghiệm của hệ phương trình là \(\left(\frac{2 \sqrt{2}-3 \sqrt{3}}{5} ; \frac{1-2 \sqrt{6}}{5}\right)\)