Slope

The formula

Between two points $(x_1, y_1)$ and $(x_2, y_2)$:

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

Reading the sign

Slope	The line
positive	rises left to right
negative	falls left to right
zero	is horizontal
undefined	is vertical

Zero and undefined are opposites, not variations of each other. A horizontal line has slope $0$ because the rise is zero; a vertical line has no slope because the run is zero, and dividing by zero is undefined.

Why it is the same everywhere on a line

Pick any two points and drop a right triangle beneath them. Any other pair gives a triangle with the same angles, so the two are similar and their sides are in the same ratio. Scaling a triangle changes the rise and the run together and leaves the fraction alone. See what slope is.

Slope as a rate

When the axes carry units, slope carries the units of $y$ per unit of $x$ — dollars per gigabyte, litres per minute, miles per hour. That is why it is also called a rate of change.