Hệ phương trình \(\left\{\begin{array}{l} x \sqrt{2}-y \sqrt{3}=1 \\ x+y \sqrt{3}=\sqrt{2} \end{array}\right.\) có mấy nghiệm?

* Hướng dẫn giải

Ta có:

\(\begin{aligned} &\text { }\left\{\begin{array}{l} x \sqrt{2}-y \sqrt{3}=1 \\ x+y \sqrt{3}=\sqrt{2} \end{array} \Leftrightarrow\left\{\begin{array}{l} (\sqrt{2}-y \sqrt{3}) \sqrt{2}-y \sqrt{3}=1 \\ x=\sqrt{2}-y \sqrt{3} \end{array}\right.\right.\\ &\Leftrightarrow\left\{\begin{array}{l} 2-y(\sqrt{6}+\sqrt{3})=1 \\ x=\sqrt{2}-y \sqrt{3} \end{array} \Leftrightarrow\left\{\begin{array}{l} y(\sqrt{6}+\sqrt{3})=1 \\ x=\sqrt{2}-y \sqrt{3} \end{array}\right.\right.\\ &\Leftrightarrow\left\{y=\frac{\sqrt{6}-\sqrt{3}}{3} x=\sqrt{2}-y \sqrt{3} \Leftrightarrow\left\{\begin{array}{l} y=\frac{\sqrt{6}-\sqrt{3}}{3} \\ x=1 \end{array}\right.\right.\\ &\text { Vậy hệ phương trình đã cho có nghiệm duy nhất }(x ; y)=\left(1 ; \frac{\sqrt{6}-\sqrt{3}}{3}\right) \text { . } \end{aligned}\)