Solve using cross-multiplication :

$\sqrt{2}x - \sqrt{3}y = 0$

$\sqrt{5}x + \sqrt{2}y = 0$

Given, equations :

$\sqrt{2}x - \sqrt{3}y = 0$ ……..(1)

$\sqrt{5}x + \sqrt{2}y = 0$ ……..(2)

Multiplying equation (1) by $\sqrt{2}$, we get :

$\Rightarrow \sqrt{2}(\sqrt{2}x - \sqrt{3}y) = \sqrt{2} \times 0 \\[1em] \Rightarrow 2x - \sqrt{6}y = 0 \text{ …..(1)}$

Multiplying equation (2) by $\sqrt{3}$, we get :

$\Rightarrow \sqrt{3}(\sqrt{5}x + \sqrt{2}y) = \sqrt{3} \times 0 \\[1em] \Rightarrow \sqrt{15}x + \sqrt{6}y = 0 \text{ …..(2)}$

Adding equations (1) and (2), we get :

$\Rightarrow 2x - \sqrt{6}y + \sqrt{15}x + \sqrt{6}y = 0 \\[1em] \Rightarrow 2x + \sqrt{15}x = 0 \\[1em] \Rightarrow x(2 + \sqrt{15}) = 0 \\[1em] \Rightarrow x = 0.$

Substituting value of x in equation (1), we get :

$\Rightarrow 2 \times 0 - \sqrt{6}y = 0 \\[1em] \Rightarrow 0 - \sqrt{6}y = 0 \\[1em] \Rightarrow \sqrt{6}y = 0 \\[1em] \Rightarrow y = 0.$

Hence, x = 0 and y = 0.