Linear Function Calculator
Find the slope a and intercept b of y = ax + b from two points, with the substituted rate-of-change formula. Useful for middle-school linear functions and trend analysis.
(x₁, y₁)
(x₂, y₂)
Try an example
Enter two points to find the slope a and intercept b of y = ax + b.
How to use
- 1
Enter two points
Fill in (x₁, y₁) and (x₂, y₂) that the line passes through.
- 2
Read a and b
Slope a = (y₂−y₁)÷(x₂−x₁) and intercept b = y₁ − a×x₁ are shown with substituted working.
- 3
Read the equation
The result appears as y = ax + b, ready to copy.
Features
- Slope and intercept from two points with substituted working
- Formatted y = ax + b output (y = ax when b = 0, − |b| when negative)
- x₁ = x₂ (vertical line) reported as an error
- Inputs are never transmitted; all computation happens locally in your browser
Use cases
Check linear homework
Verify "find the line through two points" problems from school math.
Analyze data trends
Compute the rate of change between two data points.
Sketch graphs
Confirm the slope sign to see whether the line rises or falls.
Details
A linear function y = ax + b is determined by its slope a (the change in y per unit change in x) and intercept b (the value of y at x = 0). Since two points determine a unique line, a and b follow from the two coordinates: a = (y₂−y₁)÷(x₂−x₁) and b = y₁ − a×x₁.
The slope a is also called the rate of change — how much y increases when x increases by 1. A positive slope rises to the right, a negative slope falls, and zero slope gives the horizontal line y = b. The intercept b is where the line crosses the y-axis at (0, b).
Conditions: x₁ must not equal x₂ (that would be the vertical line x = x₁, which cannot be written as y = ax + b). Coordinates accept absolute values up to one million. Inputs are never transmitted; all computation happens locally in your browser.
FAQ
What is the slope (rate of change)?
It is how much y changes when x increases by 1, computed as a = (y₂−y₁)÷(x₂−x₁). Positive slopes rise to the right, negative slopes fall, and zero gives a horizontal line.
What is the intercept?
The value of y at x = 0 — where the line crosses the y-axis. In y = ax + b, the intercept b means the line passes through (0, b).
Why can't I compute when x₁ equals x₂?
Two points with the same x-coordinate define the vertical line x = x₁. Since x is fixed regardless of y, it cannot be written in the form y = ax + b.
When is the slope zero?
When both points share the same y-coordinate. The line is horizontal, taking the form y = b (a constant), and y never changes no matter how x varies.
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Verified: Known values ((2,3),(5,9)→y=2x−1, (-1,4),(3,-8)→y=−3x+1, horizontal→y=5), the x₁=x₂ error and the English page are covered by browser tests
Did you know?
A line with slope a = 2 gains 2 in y for every 1 in x. On y = 2x − 1 through (2, 3), x = 5 gives y = 9 — the "rate of change × change" pattern predicts y, handy for pricing, distance and growth estimates.