Slope of a Line
Simple Explanation
Slope measures how steep a line is β how much it rises or falls vertically for each step it takes horizontally. It is the change in y divided by the change in x between any two points on the line.
Why Do We Need It?
Slope tells you the rate at which one quantity changes compared to another β steepness, speed, or incline β anywhere two related quantities change together.
See It
A coordinate plane with two draggable points, the line through them, and the live slope value
Formula
Slope Formula
m = (yβ β yβ) / (xβ β xβ)
How steep a line is β the change in y for each unit change in x.
- m
- β the slope (gradient) of the line
- xβ, yβ
- β the coordinates of the first point on the line
- xβ, yβ
- β the coordinates of the second point on the line
When to use it: Whenever you need to measure or compare how steep a line is, or check whether two lines are parallel or perpendicular.
Worked Example
Find the slope through two points
Find the slope of the line through A(1, 2) and B(5, 14).
Why Does This Work?
Slope is a ratio, so it stays the same no matter which two points on the line you pick β it measures the line's constant steepness, not the specific points chosen.
Real-Life Example
Wheelchair ramp incline
A ramp rises 1 metre in height over a 12 metre horizontal run.
Its slope is 1/12 β the same rise-over-run idea used to check whether a ramp meets accessibility standards.
Practice
What is the slope of the line through (0, 0) and (4, 8)?
EasyWhat is the slope of the line through (3, 10) and (7, 2)?
MediumCommon mistake
Swapping the order of subtraction between numerator and denominator (using yββyβ but xββxβ), which flips the sign of the slope.
Quick Review
- m = (yββyβ)/(xββxβ) β 'rise over run'.
- A positive slope rises left to right; a negative slope falls.
- Slope is the same wherever you measure it along a straight line.