To find the slope of the line passing through the points $(10, -5)$ and $(10, -9)$, we use the slope formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Here, $(x_1, y_1) = (10, -5)$ and $(x_2, y_2) = (10, -9)$. Plugging these values into the formula:

$m = \frac{-9 - (-5)}{10 - 10} = \frac{-9 + 5}{0} = \frac{-4}{0}$

Since division by zero is undefined, the slope does not exist. This means the line is vertical.

Would you like further details or have any questions?

Here are five related questions you might find useful:

1. What is the equation of a vertical line through a given point?
2. How do you find the slope of a horizontal line?
3. Can a line have both undefined and zero slopes?
4. How does the slope affect the steepness and direction of a line?
5. What is the difference between vertical and horizontal lines in terms of slope?

Tip: If the $x$-coordinates of two points are the same, the line is vertical, and its slope is undefined.