An algebra problem by tim oneil

$\begin{aligned} \frac 3{\sqrt y} - \frac 1{\sqrt x} & = \frac 2{\sqrt x + \sqrt y} & \small \color{#3D99F6} \text{Multiply both sides by }\sqrt x \\ 3 \sqrt {\frac xy} - 1 & = \color{#3D99F6} \frac {2\sqrt x}{\sqrt x + \sqrt y} & \small \color{#3D99F6} \text{Divide up and down by }\sqrt y \\ 3 \sqrt {\frac xy} - 1 & = \color{#3D99F6} \frac {2 \sqrt {\frac xy}}{\sqrt {\frac xy}+1} & \small \color{#3D99F6} \text{Multiply both sides by }\sqrt {\frac xy}+1 \\ \left(3 \sqrt {\frac xy} - 1\right)\left(\sqrt {\frac xy}+1\right) & = 2 \sqrt {\frac xy} \\ 3 \times \frac xy + 2\sqrt {\frac xy} - 1 & = 2 \sqrt {\frac xy} \\ 3 \times \frac xy & = 1 \\ \implies \frac xy & = \frac 13 \approx \boxed{0.333} \end{aligned}$