\[ \begin{cases}
x+\sqrt y = 27 \\ y +\sqrt x = 9 \end{cases} \]
Given that \(x\) and \(y\) are positive integers satisfying the system of equations above, find \(x+y\).
+128407 x + sqrt y = 27
y + sqrt x = 9
Subtract these
x - y + ( sqrt y - sqrt x) = 18 { factor x - y as a difference of roots }
(sqrt x + sqrt y) ( sqrt x - sqrt y) - (sqrt x - sqrt y) = 18 { factor out sqrt x - sqrt y }
(sqrt x - sqrt y) ( sqrt x + sqrt y - 1) = 18
Note that factors of 18 are 1 2 3 6 9 18
And note that
(3) (6) = 18
(sqrt 25 - sqrt 4 ) ( sqrt 25 + sqrt 4 - 1) = 18
(5 - 2) ( 5 + 2 -1) = 18
(3) ( 6) = 18
So
x = 25 y = 4
x + y = 29
