Slope Calculator
Calculate the slope (rise over run) of a line between two coordinate points, plus the distance between points and y-intercept of the line.
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Formula
m = (y₂ − y₁) / (x₂ − x₁)
Slope m is the ratio of vertical change (rise) to horizontal change (run). The y-intercept b is found by substituting the slope and one point into y = mx + b and solving for b. The full line equation is y = mx + b.
How to use the Slope Calculator
- 1
Enter your x₁
- 2
Enter your y₁
- 3
Enter your x₂
- 4
Enter your y₂
- 5
Read your results instantly
Results update in real time as you type.
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What slope means and how to interpret it
Slope quantifies how steep a line is and its direction. A slope of 2 means for every 1 unit you move right, you move up 2 units. A slope of −0.5 means for every 2 units right, you drop 1 unit.
Special cases: a horizontal line has slope 0 (no rise). A vertical line has undefined slope (division by zero — infinite steepness). Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals of each other (m₁ × m₂ = −1).
The slope-intercept form of a line
Every non-vertical line can be written as y = mx + b, where m is the slope and b is the y-intercept (where the line crosses the y-axis). Given any two points, you can determine the unique line through them.
This form is the foundation of linear regression in statistics, where the 'line of best fit' through data points minimizes the sum of squared vertical distances from each point to the line.
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Real-world slope applications
Slope is everywhere. In civil engineering, road grade (slope) determines how steep a highway is — a 6% grade rises 6 feet for every 100 horizontal feet, which is steep for a road. ADA ramp requirements specify a maximum slope of 1:12 (8.33%) for wheelchair accessibility.
In economics, the slope of a supply or demand curve tells you how sensitive quantity is to price changes. In physics, the slope of a velocity-time graph is acceleration.
Tips & Insights
Point order doesn't matter
Whether you use (x₁,y₁) → (x₂,y₂) or reverse them, slope is the same. (y₂−y₁)/(x₂−x₁) = (y₁−y₂)/(x₁−x₂). Order matters for direction when graphing, but not for the slope value.
Road grades are expressed as percentages
A slope of 0.08 = 8% grade = rises 8 feet per 100 horizontal feet. This is the standard way civil engineers and GPS systems express grade.
Worked Examples
Linear function
Slope: 2. Y-intercept: 2. Line equation: y = 2x + 2. Distance between points: 8.25 units.
Negative slope (descending line)
Slope: −2. Y-intercept: 7. Line equation: y = −2x + 7. For every unit right, the line drops 2 units.
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Frequently Asked Questions
What does slope mean?
Slope measures the steepness of a line — how much the y-value changes for every unit change in x. A positive slope goes up left to right; negative goes down.
What is an undefined slope?
A vertical line has undefined slope because the denominator (x₂−x₁) equals zero. Division by zero is undefined in mathematics.
What is the slope of a horizontal line?
A horizontal line has a slope of 0. There is no vertical change (rise) regardless of horizontal change (run), so rise/run = 0/run = 0.
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