Slope Calculator
Calculate slope and line equation from two points
Slope Calculator – Find the Slope of a Line Online
Calculate the slope or gradient of any line using two points or a linear equation with our free online slope calculator. Find slope, intercepts, and line equations easily.
Point 1 (x₁, y₁)
Point 2 (x₂, y₂)
Understanding Slope
Slope measures how steep a line is. It tells you how much the line rises (or falls) for every unit it runs horizontally. A positive slope climbs from left to right; a negative slope descends. Zero slope means a horizontal line; undefined slope means vertical.
This slope calculator finds the slope from two points, calculates the angle of inclination, and gives you the complete line equation in slope-intercept form (y = mx + b). Enter your coordinates and see each step of the calculation.
Slope Formula and Line Equation
Slope Formula
Slope equals rise over run – the change in y divided by the change in x between any two points on the line.
Slope-Intercept Form
Where m is the slope and b is the y-intercept (where the line crosses the y-axis).
Positive Slope
Line rises from left to right. As x increases, y increases.
Negative Slope
Line falls from left to right. As x increases, y decreases.
Zero Slope
Horizontal line. y stays constant regardless of x.
Undefined Slope
Vertical line. x stays constant; slope calculation divides by zero.
Step-by-Step Calculation
Find Rise (Δy)
Subtract y-coordinates: Δy = y₂ - y₁
Find Run (Δx)
Subtract x-coordinates: Δx = x₂ - x₁
Calculate Slope
Divide rise by run: m = Δy / Δx
Find Y-Intercept
Use b = y₁ - m × x₁ to find where line crosses y-axis
Worked Examples
Example 1: Positive Slope
Find the slope between (1, 2) and (4, 8)
Example 2: Negative Slope
Find the slope between (-2, 5) and (3, -1)
Example 3: Zero Slope
Find the slope between (2, 4) and (7, 4)
Example 4: Undefined Slope
Find the slope between (3, 1) and (3, 6)
Real-World Applications
Road Grade and Construction
Road signs showing "6% grade" describe slope. A 6% grade means the road rises 6 feet for every 100 feet horizontally. Engineers calculate slope to ensure roads are safe – too steep and vehicles struggle; too flat and water doesn't drain.
Roof Pitch
Roofers express slope as "rise over run" – a 6/12 roof rises 6 inches for every 12 inches horizontally. Steeper roofs shed water and snow better but require more materials and are harder to work on.
Economics and Trends
In graphs showing price over time, slope represents the rate of change. A positive slope means prices are rising; negative means falling. The steeper the slope, the faster the change. Analysts use slope to identify trends.
Physics and Motion
On a position-time graph, slope equals velocity. On a velocity-time graph, slope equals acceleration. Understanding slope is fundamental to analyzing motion in physics.
Accessibility Ramps
The ADA requires wheelchair ramps to have a maximum slope of 1:12 – one inch of rise for every 12 inches of run. This ensures ramps are usable by people with limited strength or mobility.
Frequently Asked Questions
What is the formula for slope?
Slope = (y₂ - y₁) / (x₂ - x₁), also written as rise over run. It measures the steepness of a line by comparing vertical change to horizontal change between two points.
What does a positive vs negative slope mean?
Positive slope means the line goes up from left to right (increasing). Negative slope means the line goes down from left to right (decreasing). Zero slope is a horizontal line.
What is an undefined slope?
Undefined slope occurs when x₂ = x₁ (vertical line). Division by zero makes the slope undefined. A vertical line has equation x = constant with no y-intercept.
How do you find the y-intercept?
Use the formula b = y₁ - m × x₁, where m is the slope and (x₁, y₁) is any point on the line. The y-intercept is where the line crosses the y-axis (x = 0).
What is slope-intercept form?
Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. This form makes it easy to graph a line and understand its behavior.
Can slope be a fraction or decimal?
Yes. Slope can be any real number – whole numbers, fractions, decimals, positive, negative, or zero. A slope of 2/3 means the line rises 2 units for every 3 units it runs.
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