Answer: 529

Solution:

Substituting $y = \frac{3}{4}x$ into $x^y = y^x$ gives $$x^{\frac{3}{4}x} = \left(\tfrac{3}{4}x\right)^{x}.$$ Taking $x$th roots (the quantities here are positive), $x^{3/4} = \frac{3}{4}x,$ so dividing by $x$ yields $x^{-1/4} = \frac{3}{4},$ that is, $x = \left(\frac{4}{3}\right)^4 = \frac{256}{81}.$

Then $y = \frac{3}{4} \cdot \frac{256}{81} = \frac{64}{27},$ and $$x + y = \frac{256}{81} + \frac{192}{81} = \frac{448}{81}.$$ Since $\gcd(448, 81) = 1,$ the answer is $448 + 81 = 529.$