## Pythagorean Triples

The expressions $2xy,x^2-y^2$, and $x^2+y^2$, with $x$ and $y$ integers such that $1\leq y, generate Pythagorean triples. Given the triple $(204,253,325)$, find the values for $x$ and $y$ that generate this triple.
Source: NCTM Mathematics Teacher 2008

SOLUTION
$2xy$ is even
$2xy=204\qquad\quad\;\; (1)$
$x^2-y^2=253\qquad (2)$
The hypotenuse has the largest value
$x^2+y^2=325\qquad (3)$

Method 1
Add Eq. $(2)$ to Eq. $(3)$
$x^2-y^2=253$
$x^2+y^2=325$
———————
$2x^2=578$
$x^2=289$
$x=\pm 17$
$x=-17$ is not possible because $x$ is a positive integer
$x=17$
Substitute the value of $x$ into Eq. $(1)$
$2(17)y=204$
$y=6$
$x=17,y=6$

Method 2
$(x+y)^2=x^2+2xy+y^2$
$=325+204$
$=529$
$x+y=\pm 23$
$x+y=-23$ is not possible because both $x$ and $y$ are positive integers
$x+y=23\qquad (4)$
$x^2-y^2=(x+y)(x-y)$
$253=23(x-y)$
$x-y=11\qquad (5)$
Add Eq. $(4)$ to Eq. $(5)$
$x+y=23$
$x-y=11$
—————–
$2x=34$
$x=17$
Substitute the value of $x$ into Eq. $(4)$
$17+y=23$
$y=6$
$x=17,y=6$

Answer: $x=17,y=6$