jmc

If we have two points $(x_1,y_1)$ and $(x_2,y_2)$ on a line, we can find the slope of the line by using the formula $\frac{y_1-y_2}{x_1-x_2}.$ So, for the line we are given, the slope is $\frac{(-5)-(-14)}{2-p}=\frac{9}{2-p},$ and the slope is also $\frac{(-14)-(-17)}{p-(p+2)}=\frac{3}{-2}.$ Setting these values equal, we get $$\frac{9}{2-p}=-\frac{3}{2}.$$ Multiplying both sides by the product of the denominators and simplifying gives $$\begin{array}{rl}(2-p)(3)& =(-2)(9)\\ 6-3p& =-18\\ p& =8.\end{array}$$ Now we need to find $q.$ Using the same strategy as above, we find that $$\begin{array}{rl}\frac{q-(-5)}{13-2}& =\frac{3}{-2}\\ (11)(3)& =(-2)(q+5)\\ 33& =-2q-10\\ q& =-\mathrm{21.5.}\end{array}$$ Therefore, $p+q=8+(-21.5)=\boxed{-13.5}.$