How to Find Slope? (+FREE Worksheet!)
Finding slope tells you how steep a line is and which direction it goes. Slope is the ratio of vertical change (rise) to horizontal change (run) between any two points on a line, and it appears in almost every linear equation topic in Algebra 1. Once you can find slope from a graph, from two points, or from an equation, the rest of linear algebra becomes much easier.
What Is Slope?
Slope (\(\color{blue}{m}\)) measures how much a line rises or falls for every unit it moves to the right. A positive slope rises left to right; a negative slope falls left to right. A zero slope means a flat horizontal line; an undefined slope means a vertical line.
\(\color{blue}{m = \frac{\text{ rise }}{\text{ run }} = \frac{(y<\text{ sub }>2 – y<\text{ sub }>1)}{(x<\text{ sub }>2 – x<\text{ sub }>1)}}\)
Methods for Finding Slope
Method 1 — Using Two Points
Label the points \(\color{blue}{(x_{1}, y_{1})}\) and \(\color{blue}{(x_{2}, y_{2})}\), then apply the formula.
Example: points \(\color{blue}{(2, 3)}\) and \(\color{blue}{(6, 7)}\):
\(\color{blue}{m = \frac{(7 – 3)}{(6 – 2)} = \frac{4}{4} = 1}\)
Method 2 — From a Graph
Pick two clear lattice points on the line. Count the rise (vertical distance) and the run (horizontal distance). Write \(\color{blue}{m = \frac{\text{ rise }}{\text{ run }}}\) and assign the correct sign.
Example: a line that goes up 3 units for every 2 units to the right has slope \(\color{blue}{m = \frac{3}{2}}\).
Method 3 — From Slope-Intercept Form
If the equation is \(\color{blue}{y = \text{ mx } + b}\), the slope is the coefficient of \(\color{blue}{x}\).
Example: \(\color{blue}{y = -4x + 7}\) has slope \(\color{blue}{m = -4}\).
Method 4 — Horizontal and Vertical Lines
Any horizontal line has slope 0. Any vertical line has undefined slope (the run is zero, and division by zero is undefined).
Step-by-Step Summary
- Identify or read two points on the line: \(\color{blue}{(x_{1}, y_{1})}\) and \(\color{blue}{(x_{2}, y_{2})}\).
- Subtract the y-values: \(\color{blue}{y_{2} – y_{1}}\) (rise).
- Subtract the x-values in the same order: \(\color{blue}{x_{2} – x_{1}}\) (run).
- Divide: \(\color{blue}{m = \frac{\text{ rise }}{\text{ run }}}\).
- Simplify the fraction and assign the sign.
Watch: How to Find Slope (Video Lesson)
Math with Mr. J walks through multiple methods for finding slope with clear visuals:
Finding Slope – Worked Examples
Example 1: Find the slope through \(\color{blue}{(2, 3)}\) and \(\color{blue}{(6, 7)}\).
\(\color{blue}{m = \frac{(7 – 3)}{(6 – 2)} = \frac{4}{4} = 1}\)
Example 2: Find the slope through \(\color{blue}{(-1, 4)}\) and \(\color{blue}{(3, -2)}\).
\(\color{blue}{m = \frac{(-2 – 4)}{(3 – (-1))} = -\frac{6}{4} = -\frac{3}{2}}\)
Example 3: Find the slope through \(\color{blue}{(0, 5)}\) and \(\color{blue}{(4, 5)}\).
\(\color{blue}{m = \frac{(5 – 5)}{(4 – 0)} = \frac{0}{4} = 0}\) (horizontal line)
Example 4: Find the slope through \(\color{blue}{(3, -2)}\) and \(\color{blue}{(3, 6)}\).
\(\color{blue}{m = \frac{(6 – (-2))}{(3 – 3)} = \frac{8}{0}}\) ⇒ undefined (vertical line)
More Practice: Finding Slope from a Graph (Video)
Khan Academy shows how to read rise and run directly from a graph to determine slope:
Exercises for Finding Slope
Find the slope of the line through each pair of points.
- \(\color{blue}{(1, 2)}\) and \(\color{blue}{(5, 10)}\)
- \(\color{blue}{(0, -3)}\) and \(\color{blue}{(4, 5)}\)
- \(\color{blue}{(-2, 6)}\) and \(\color{blue}{(4, -3)}\)
- \(\color{blue}{(5, 7)}\) and \(\color{blue}{(5, -1)}\)
- \(\color{blue}{(-3, -4)}\) and \(\color{blue}{(3, -4)}\)
- \(\color{blue}{(2, -1)}\) and \(\color{blue}{(-4, 5)}\)
Answers
- \(\color{blue}{m = 2}\)
- \(\color{blue}{m = 2}\)
- \(\color{blue}{m = -\frac{3}{2}}\)
- Undefined
- \(\color{blue}{m = 0}\)
- \(\color{blue}{m = -1}\)
Frequently Asked Questions
Does it matter which point I label (x₁, y₁)?
No. As long as you subtract the coordinates in the same order (both second minus first), you will get the same slope. Switching the order of both top and bottom still gives the same result.
What does a slope \(\color{blue}{\text{ of } -2}\) mean?
A slope \(\color{blue}{\text{ of } -2}\) means the line falls 2 units for every 1 unit it moves to the right, so it has a steep negative direction.
How is slope related to the slope-intercept equation?
In the equation \(\color{blue}{y = \text{ mx } + b}\), the slope \(\color{blue}{m}\) is the coefficient of \(\color{blue}{x}\). You can read the slope directly without needing two points.
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