Slope

Pick any two points on a line. The slope is how much it climbs for how much it moves across:

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

The number carries two pieces of meaning at once:

• Sign — the direction. A line rising left-to-right has positive slope; one falling left-to-right has negative slope.
• Size — the steepness. A slope of $3$ is steeper than $\tfrac{1}{2}$.

Two special lines: a horizontal line never rises, so $\Delta y = 0$ and the slope is $0$. A vertical line never runs, so $\Delta x = 0$ — and since you can't divide by zero, its slope is undefined.

Take $(0, 0)$ and $(4, 2)$. Going from the first point to the second, $y$ rises by $2$ and $x$ runs by $4$:

$m = \frac{\Delta y}{\Delta x} = \frac{2 - 0}{4 - 0} = \frac{2}{4} = \frac{1}{2}$

The order of the points doesn't matter as long as you stay consistent: from $B$ to $A$ the rise is $-2$ and the run is $-4$, and $\tfrac{-2}{-4}$ is still $\tfrac{1}{2}$.

When the line falls, the slope comes out negative. From $(0, 5)$ to $(2, 1)$ the rise is $1 - 5 = -4$ over a run of $2$, so $m = \tfrac{-4}{2} = -2$ — a steep downhill line.

Find $\Delta y$ and $\Delta x$, divide, and watch the sign. Predict before you check.