1. **State the problem:** Find the slope of the line passing through points $P(5, -5)$ and $Q(4, -5)$. 2. **Recall the slope formula:** The slope $m$ of a line through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 3. **Substitute the coordinates:** $$m = \frac{-5 - (-5)}{4 - 5} = \frac{-5 + 5}{4 - 5} = \frac{0}{-1}$$ 4. **Simplify the fraction:** $$m = 0$$ 5. **Interpretation:** Since the slope is zero, the line through points $P$ and $Q$ is horizontal. **Final answer:** $$\boxed{0}$$