Another system of equations!

Here we go. Let's mark the first equation as $(1)$ and the other as $(2)$ . We have:

$(1)$ $\Leftrightarrow$ $x^{2}-xy-2y^{2}-(x+y)=0$

$\Leftrightarrow$ $(x+y)(x-2y)-(x+y)=0$

$\Leftrightarrow$ $(x+y)(x-2y-1)=0$

$\Leftrightarrow$ $x-2y-1=0$ (due to $x+y>0$ )

$\Leftrightarrow$ $x=2y+1$ .

Subtitute $x=2y+1$ back in $(2)$ we got:

$y \sqrt{2y}+ \sqrt{2y}=2y+2$

$\Leftrightarrow$ $(y+1)( \sqrt{2y}-2)=0$

$\Leftrightarrow$ $\sqrt{2y}-2=0$ (due to $y \geq 0 \Rightarrow y+1>0$ )

$\Leftrightarrow$ $2y=4$

$\Leftrightarrow$ $y=2$

$\Rightarrow$ $x=5$ .

Solution ends.