Two-point formula for slope

Property The slope of the line joining points $P 1(x 1, y 1)$ and $P 2(x 2, y 2)$ is $$m = \frac{y 2 y 1}{x 2 x 1} \quad \text{if} \quad x 2 \neq x 1$$.

Examples To find the slope of the line through $(1, 2)$ and $(4, 8)$, we let $(x 1, y 1) = (1, 2)$ and $(x 2, y 2) = (4, 8)$. The slope is $m = \frac{8 2}{4 1} = \frac{6}{3} = 2$. The slope of the line containing points $( 2, 5)$ and $(3, 5)$ is calculated as $m = \frac{ 5 5}{3 ( 2)} = \frac{ 10}{5} = 2$. For the points $(5, 3)$ and $( 1, 1)$, the slope is $m = \frac{ 1 ( 3)}{ 1 5} = \frac{2}{ 6} = \frac{1}{3}$. It doesn't matter which point you choose as first or second.

Explanation This formula is a precise way to calculate 'rise over run.' It finds the vertical change (the 'rise,' $y 2 y 1$) and divides it by the horizontal change (the 'run,' $x 2 x 1$) between any two points on a line.