What is slope?
Slope measures how steep a line is: how far it goes up (the rise) for each step across (the run). A slope of 2 rises 2 for every 1 across. A negative slope goes down from left to right, a horizontal line has a slope of 0, and a vertical line has no slope at all: its run is 0, and you can’t divide by 0.
The equation of a line is usually written y = mx + b: m is the slope and b is where the line crosses the y-axis. To find it from two points, work out m, then put one point into y − y₁ = m(x − x₁) and solve for y.
Source OpenStax · Intermediate Algebra 2e, 3.2 Slope of a Line ↗OpenStax · Intermediate Algebra 2e, 3.3 Find the Equation of a Line ↗Common Core State Standards · Grade 8, slope and y = mx + b (8.EE.B.6) ↗
How to calculate slope
Here it is with your numbers:
- m = (7 − 1) ÷ (6 − 2)
- m = 6 ÷ 4
- m = 1.5
- b = 1 − 1.5 × 2
- b = −2
It means the line is y = 1.5x − 2: it rises 3 for every 2 it goes across and crosses the y-axis at −2.
Slope as a percent, angle and ratio
Builders, road signs and the ADA describe slope as a percent grade (100 × rise ÷ run), an angle, or a ratio such as 1 : 12 (1 up for every 12 across). The angle is the inverse tangent of the rise over the run.
| Grade | Angle | Ratio |
|---|---|---|
| 1% | 0.5729° | 1 : 100 |
| 2% | 1.1458° | 1 : 50 |
| 5% | 2.8624° | 1 : 20 |
| 8.3333% | 4.7636° | 1 : 12 |
| 10% | 5.7106° | 1 : 10 |
| 25% | 14.0362° | 1 : 4 |
| 50% | 26.5651° | 1 : 2 |
| 100% | 45° | 1 : 1 |
Source U.S. Access Board · Guide to the ADA Standards, Chapter 4: Ramps and Curb Ramps ↗OpenStax · Precalculus 2e, 6.3 Inverse Trigonometric Functions ↗
FAQs
What is the formula for the slope of a line?
The slope is the rise over the run: m = (y₂ − y₁) ÷ (x₂ − x₁), the change in y divided by the change in x. Through (−2, −3) and (−7, 4) it is 7 ÷ (−5), which is −7/5. A horizontal line has slope 0; a vertical line has no slope (it is not defined, because the run is 0).
Source OpenStax · Intermediate Algebra 2e, 3.2 Slope of a Line ↗
How to find slope with y = mx + b?
In y = mx + b, m is the slope and b is where the line crosses the y-axis. So y = −4/7x − 2 has a slope of −4/7 and crosses the y-axis at −2.
How steep is a 5% slope?
A 5% slope rises 5 for every 100 across: a ratio of 1 : 20, or 2.86°. Under the ADA, a path steeper than 5% counts as a ramp, and a ramp can be no steeper than 1 : 12 (8.33%).
Source U.S. Access Board · Guide to the ADA Standards, Chapter 4: Ramps and Curb Ramps ↗
What’s a 2% slope in degrees?
A 2% slope is 1.15°: the angle is the inverse tangent of the rise over the run, tan⁻¹(2 ÷ 100), which is 1.1458°. It rises 2 for every 100 across, a ratio of 1 : 50.
Source OpenStax · Precalculus 2e, 6.3 Inverse Trigonometric Functions ↗
Sources
- OpenStax: Intermediate Algebra 2e, 3.2 Slope of a Line
- OpenStax: Intermediate Algebra 2e, 3.3 Find the Equation of a Line
- Common Core State Standards: Grade 8, slope and y = mx + b (8.EE.B.6)
- U.S. Access Board: Guide to the ADA Standards, Chapter 4: Ramps and Curb Ramps
- OpenStax: Precalculus 2e, 6.3 Inverse Trigonometric Functions