Let:
x = 1st unknown number
y = 2nd unknown number
A positive number is 5 larger than another positive number:
[tex]x=5y_{\text{ }}(1)[/tex]the sum of the squares of the two positive number is 53:
[tex]x^2+y^2=53_{\text{ }}(2)[/tex]Replace (1) into (2):
[tex]\begin{gathered} (5y)^2+y^2=53 \\ 25y^2+y^2=53 \\ 26y^2=53 \\ y^2=\frac{53}{26} \\ y=\pm\sqrt[]{\frac{53}{26}} \\ \end{gathered}[/tex]Replace y into (1):
[tex]x=\pm5\sqrt[]{\frac{53}{26}}[/tex]Since the numbers are positive:
[tex]\begin{gathered} x=5\sqrt[]{\frac{53}{26}} \\ y=\sqrt[]{\frac{53}{26}} \end{gathered}[/tex]