The opposite vertices of a rectangle are symmetric with respect to the midpoint of the line segment connecting them.
The midpoint of the line segment connecting $(3,17)$ and $(9,-4)$ is $\left(\frac{3+9}{2},\frac{17+(-4)}{2}\right)=(6,6.5)$.
The other two vertices are symmetric to $(3,17)$ and $(9,-4)$ with respect to $(6,6.5)$.
The $y$-coordinate of the first vertex is $2(6.5)-17=13-17=-4$.
The $y$-coordinate of the second vertex is $2(6.5)-(-4)=13+4=17$.
The sum of the $y$-coordinates of the other two vertices is $(-4)+17=\boxed{13}$.
The answer is: 13
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
If $(3,17)$ and $(9,-4)$ are the coordinates of two opposite vertices of a rectangle, what is the sum of the $y$-coordinates of the other two vertices?

### Answer:
