The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).

What is slope?

Slope is a measure of the steepness of a line.
Slope=riserun=ΔyΔx\text{Slope} = \dfrac{\text{rise}}{\text{run}}=\dfrac{\Delta y}{\Delta x}
Want an in-depth introduction to slope? Check out this video.

Example: Slope from graph

We're given the graph of a line and asked to find its slope.
The line appears to go through the points (0,5)(0,5) and (4,2)(4,2).
Slope=ΔyΔx=2540=34\text{Slope}=\dfrac{\Delta y}{\Delta x}=\dfrac{2-5}{4-0}=\dfrac{-3}{4}
In other words, for every three units we move vertically down the line, we move four units horizontally to the right.
Want to learn more about finding slope from graphs? Check out this video.

Example: Slope from two points

We're told that a certain linear equation has the following two solutions:
Solution: x=11.4   y=11.5x=11.4 ~~~ y=11.5
Solution: x=12.7   y=15.4x=12.7 ~~~ y=15.4
And we're asked to find the slope of the graph of that equation.
The first thing to realize is that each solution is a point on the line. So, all we need to do is find the slope of the line through the points (11.4,11.5)(11.4,11.5) and (12.7,15.4)(12.7,15.4).
Slope=ΔyΔx=15.411.512.711.4=3.91.3=3913=3\begin{aligned} \text{Slope}=\dfrac{\Delta y}{\Delta x}&=\dfrac{15.4-11.5}{12.7-11.4}\\\\ &=\dfrac{3.9}{1.3}\\\\ &=\dfrac{39}{13}\\\\ &=3\end{aligned}
The slope of the line is 33.
Want to learn more about finding slope from two points? Check out this video.

Practice

Problem 1
What is the slope of the line below?
Give an exact number.
Want more practice? Check out this Slope from graphs exercise and this Slope from points exercise.
Loading