Tutorial
How to Find the Slope of a Line: Formula, Methods, and Examples
Published 14 March 2026 · 9 min read
Slope measures the steepness and direction of a line. It tells you how much a line rises or falls for each unit of horizontal movement. Slope is one of the most fundamental concepts in algebra, geometry, and calculus, and it has practical applications in fields like engineering, economics, and data science.
In this tutorial, you will learn the slope formula, the rise-over-run method, how to read slope from an equation, and the special relationships between parallel and perpendicular lines. Each concept is illustrated with worked examples.
Key Concepts
The Slope Formula
Given two points and , the slope is:
The numerator is the vertical change (rise) and the denominator is the horizontal change (run).
Positive Slope
The line rises from left to right. As increases, increases. Example: means the line goes up 2 units for every 1 unit to the right.
Negative Slope
The line falls from left to right. As increases, decreases. Example: means the line goes down 3 units for every 1 unit to the right.
Zero Slope
A horizontal line has slope . It does not rise or fall. Example: .
Undefined Slope
A vertical line has undefined slope because the run is zero, and you cannot divide by zero. Example: .
Slope-Intercept Form
The slope-intercept form of a line makes the slope immediately visible:
where is the slope and is the y-intercept (the point where the line crosses the y-axis).
If you are given an equation like , rearrange it to slope-intercept form to read off the slope. Subtract from both sides, then divide by 2: . The slope is .
Step-by-Step Method: Finding Slope from Two Points
- Label the two points as and .
- Subtract the y-coordinates: (the rise).
- Subtract the x-coordinates in the same order: (the run).
- Divide: .
- Simplify the fraction if possible.
Worked Examples
Example 1: Basic Slope Calculation
Find the slope of the line through and .
The slope is 2, meaning the line rises 2 units for every 1 unit to the right.
Example 2: Negative Slope
Find the slope of the line through and .
The negative slope tells us the line falls from left to right.
Example 3: Fractional Slope
Find the slope of the line through and .
The line rises 3 units for every 4 units to the right, a gentle upward slope.
Example 4: Slope from an Equation
Find the slope of .
Rearrange to slope-intercept form:
The slope is and the y-intercept is .
Example 5: Horizontal and Vertical Lines
Find the slope of the line through and .
This is a horizontal line with slope 0.
Now find the slope through and .
This is a vertical line. Division by zero means the slope is undefined.
Parallel and Perpendicular Lines
Parallel Lines
Parallel lines have the same slope but different y-intercepts.
Example: and are parallel because both have slope 3.
Perpendicular Lines
Perpendicular lines have slopes that are negative reciprocals of each other.
Example: if one line has slope , a perpendicular line has slope .
Example 6: Are These Lines Perpendicular?
Line A passes through and . Line B passes through and .
Yes, the lines are perpendicular because the product of their slopes is .
Common Mistakes
Subtracting coordinates in different orders
If you compute in the numerator, you must use in the denominator (same order). Mixing them up gives the wrong sign.
Confusing rise and run
Slope is rise over run, not run over rise. The vertical change goes on top; the horizontal change goes on the bottom.
Saying vertical lines have slope 0
Vertical lines have undefined slope (division by zero). Horizontal lines have slope 0. These are often confused.
Forgetting to simplify
A slope of should be simplified to . While both are technically correct, simplified form is standard and easier to work with.
Try It Yourself
Ready to practise finding slopes? Use our free slope calculator to check your answers instantly. Enter two points or an equation and see the slope, y-intercept, and graph.
Open Slope CalculatorRelated Articles
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