Sum of Squares

If you know the Pythagoras triples $8^2 + 6^2 = 10^2$ , then it would be very easy to solve this.

If not, use Euclid's formula, where $m,n$ are positive integers and $m > n$ :

$x=m^2 - n^2 \implies$ Eq.(1)

$y=2mn\implies$ Eq.(2)

$10= m^2+n^2\implies$ Eq.(3)

Since we know that $m$ and $n$ are positive integers, and $4^2 = 16 > 10$ , it means that $m$ and $n$ are a combination of integers $1,2,3$

Try out a few combinations of $m$ and $n$ in Eq.(3), and you would find that $10 = 3^2 + 1^2$

This means that $m=3,n=1$

Substitute into Eq.(1) and Eq. (2):

$x=3^2 - 1^2 = 8\\ y=2(3)(1) = 6$

Therefore, $x+y = 8+6 = \boxed{14}$