Por substituição:
Primeiro isole o x. Assim:
[tex] \frac{x}{y} = \frac{2}{3} \\\\\boxed{x= \frac{2y}{3}} \\\\\\\\x^2+y^2=52~~(substitua~o~x~nesta~equacao)\\\\(\frac{2y}{3})^2+y^2=52[/tex]
[tex] \frac{4y^2}{9} +y^2=52[/tex]
[tex] \frac{13y^2}{9} =52\\\\13y^2=52.9\\\\y^2= \frac{468}{13}\\\\y= \sqrt{36}\\\\\boxed{y=6} [/tex]
Substituindo o y na proporção:
[tex] \frac{x}{y} = \frac{2}{3} \\\\ \frac{x}{6} = \frac{2}{3}\\\\3x=12\\\\x= \frac{12}{3}\\\\\boxed{x=4 }[/tex]
Beleza..
