Slope is the change in y divided by the change in x, between any two points on the line.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Keep the order consistent
If you subtract the second point’s y first, subtract the second point’s x first too. Mixing them flips the sign.
What a negative slope means
The line falls left to right. The y decreases while the x increases.
Flat and vertical
A horizontal line has slope 0. A vertical line has no slope at all, because the run is nought.
Rise over run, from coordinates
For points (x₁, y₁) and (x₂, y₂), the slope is (y₂ − y₁) ÷ (x₂ − x₁). The top is the vertical change and the bottom the horizontal one — rise over run, computed rather than counted off a graph.
Keep the order consistent
Whichever point you call first, use it first in both subtractions. Subtracting y in one order and x in the other flips the sign and gives a slope pointing the wrong way. Consistency is the only rule.
Either point can go first
Swapping which point is first negates both the top and the bottom, and the two negatives cancel. So both orders give the same slope, provided you are consistent. Checking both ways is a quick way to confirm you were.
The vertical case
If the two points share an x-coordinate, the denominator is zero and the slope is undefined. That corresponds to a vertical line, which cannot be written as y = mx + b at all. It is the one case the formula cannot handle.
Step 2: Try It Yourself
Two slope triangles on one line, deliberately different sizes.
Both triangles give the same slope, 2, even though one is bigger. They are the same shape, so their sides are in the same ratio.
Step 3: Watch an Example
One step at a time.
Watch Ana Find a Negative Slope
A line passes through (1, 8) and (5, 0).
- Step 1
She takes the change in y first: 0 − 8 = −8.
Step 4: Your Turn
Practice makes it stick.
The Rise
Problem 1 of 2
Through (0, 0) and (3, 12). What is the slope?
The Fall
Problem 2 of 2
Through (2, 10) and (6, 2). What is the slope?
Rise over Run
1 of 4
Through (0, 1) and (2, 7). Slope?
2 of 4
Through (1, 5) and (4, 5). Slope?
3 of 4
Through (0, 9) and (3, 0). Slope?
4 of 4
Through (2, 1) and (5, 10). Slope?
Step 5: Quick Check
Show what you know.
Question 1 of 1
Through (1, 2) and (4, 11). Slope?
What You Learned
- Slope is the change in y divided by the change in x.
- Subtract the two points in the same order, or the sign flips.
- A falling line has a negative slope; a flat line has slope nought.