[tex]$\begin{cases}x^2-y^2=3\\ 2x^2-3xy+2y^2=4\end{cases}\Leftrightarrow \begin{cases}x^2-y^2=3\\ 4\left ( x^2-y^2-3 \right )-3\left ( 2x^2-3xy+2y^2-4 \right )=0\end{cases}\Leftrightarrow \\$[/tex]

[tex]$\begin{cases}x^2-y^2=3\\ -2x^2-10y^2+9xy=0\end{cases}\Leftrightarrow \begin{cases}x^2-y^2=3\\ (x-2y)(2x-5y)=0\end{cases}\\$[/tex]

[tex]$x=2y\Rightarrow 4y^2-y^2=3\Rightarrow y=\pm 1\Rightarrow x=\pm 2\\$[/tex]

[tex]$x=\frac{5}{2}y\Rightarrow \frac{25}{4}y^2-y^2=3\Leftrightarrow \frac{21}{4}y^2=3\Leftrightarrow y^2=\frac{4}{7}\Rightarrow y=\pm \frac{2}{\sqrt{7}}\Rightarrow x=\pm \frac{5}{\sqrt{7}}$[/tex]