Lesson 44: Finding Slope Using the Slope Formula

New Concept

A rate of change is a ratio that compares the change in one quantity with the change in another. The slope of a line represents this rate:

What’s next

This lesson builds the foundation. Next, you'll apply the slope formula in worked examples, analyze data from tables, and solve real-world rate-of-change problems.

Property

The slope $m$ of a line containing points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Examples

For points $(2, 4)$ and $(6, 6)$, the slope is $m = \frac{6 - 4}{6 - 2} = \frac{2}{4} = \frac{1}{2}$.
For points $(-4, 4)$ and $(4, -2)$, the slope is $m = \frac{-2 - 4}{4 - (-4)} = \frac{-6}{8} = -\frac{3}{4}$.
For points $(0, -5)$ and $(5, 10)$, the slope is $m = \frac{10 - (-5)}{5 - 0} = \frac{15}{5} = 3$.

Explanation

Think of slope as the ultimate measure of a line's steepness, or "rise over run." This formula calculates exactly that: for every step you take horizontally (the run), it tells you how many steps you go up or down (the rise). It’s a simple way to put a number to a slant.

Property

A horizontal line has a slope of 0. The slope of a vertical line is undefined.

Examples

A horizontal line through $(-6, 4)$ and $(6, 4)$ has a slope of $m = \frac{4 - 4}{6 - (-6)} = \frac{0}{12} = 0$.
A vertical line through $(7, 1)$ and $(7, -5)$ has a slope of $m = \frac{-5 - 1}{7 - 7} = \frac{-6}{0}$, which is undefined.

Explanation

A horizontal line is perfectly flat, so it has zero rise for its run. A vertical line is like a cliff—you can't run sideways at all! Since the formula would make you divide by zero (an impossible task in math), we just call its super-steep slope "undefined."

Property

The slope of a line represents a rate of change. It's a ratio that compares the change in one quantity with the change in another.

Examples

A train travels 1320 feet in 26 seconds. The rate (speed) is $\frac{1320}{26} \approx 50.8$ feet per second.
A boa grows from 22 to 42 inches in 10 months. The rate is $\frac{42-22}{10} = 2$ inches per month.

Explanation

Slope isn't just for graphs; it shows how fast something changes in real life! Think of speed in feet per second or growth in inches per month. The slope formula is the perfect tool for calculating these everyday rates when you have two data points, like a start and end measurement.