Suppose that $y = \frac34x$ and $x^y = y^x$. The quantity $x + y$ can be expressed as a rational number $\frac {r}{s}$, where $r$ and $s$ are relatively prime positive integers. Find $r + s$.
Substitute $y = \frac34x$ into $x^y = y^x$ and solve.
\[x^{\frac34x} = \left(\frac34x\right)^x\]
\[x^{\frac34x} = \left(\frac34\right)^x \cdot x^x\]
\[x^{-\frac14x} = \left(\frac34\right)^x\]
\[x^{-\frac14} = \frac34\]
\[x = \frac{256}{81}\]
\[y = \frac34x = \frac{192}{81}\]
\[x + y = \frac{448}{81}\]
\[448 + 81 = \boxed{529}\]