Section 4.2: Slope of a Line

Property

The Greek letter $\Delta$ ('delta') is used in mathematics to indicate change. The slope of a line is defined by the ratio

as we move from one point to another on the line. In symbols, the slope, denoted by $m$, is:

Examples

• For a line passing through points $(2, 5)$ and $(4, 9)$, the change in $y$ is $\Delta y = 9 - 5 = 4$ and the change in $x$ is $\Delta x = 4 - 2 = 2$. The slope is $m = \frac{4}{2} = 2$.

• If we move from point $A(1, 3)$ to point $B(5, 6)$, the slope is $m = \frac{\Delta y}{\Delta x} = \frac{6-3}{5-1} = \frac{3}{4}$.

Property

The slope of the line joining points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is

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.