Case 1
- x of the first point (x₁)
- 1
- y of the first point (y₁)
- 2
- x of the second point (x₂)
- 3
- y of the second point (y₂)
- 6
2.0000
Open with these valuesSlope Calculator
2.0000
Result: 2.0000Slope is the change in y divided by the change in x: from (1, 2) to (3, 6) the line climbs 4 for every 2 across, so the slope is 2. It is a bare ratio with no unit, the same number in metres as in miles. A vertical line has no slope, because the change in x is zero.
The numbers at a glance
Held fixed: x of the first point (x₁) 1.0000, y of the first point (y₁) 2.0000, x of the second point (x₂) 3.0000.
| y of the second point (y₂) | Result |
|---|---|
| 0.0000 | -1.0000 |
| 2.0000 | 0.0000 |
| 4.0000 | 1.0000 |
| 6.0000Your value | 2.0000 |
| 8.0000 | 3.0000 |
| 10.0000 | 4.0000 |
| 12.0000 | 5.0000 |
Worked examples
Case 2
- x of the first point (x₁)
- 0
- y of the first point (y₁)
- 1
- x of the second point (x₂)
- 4
- y of the second point (y₂)
- 3
0.5000
Open with these valuesCase 3
- x of the first point (x₁)
- 2
- y of the first point (y₁)
- 5
- x of the second point (x₂)
- 4
- y of the second point (y₂)
- 1
-2.0000
Open with these valuesHow it's calculated
m = (y₂ − y₁) ÷ (x₂ − x₁)
- StepEnter both points; the order is free as long as you keep it the same above and below.
- StepSubtract the y values, then the x values, in that same order.
- ResultDivide the change in y by the change in x — that ratio is the slope.
What this number means
A slope measures how steeply a straight line climbs or falls between two points. It is the rise divided by the run — the change in y divided by the change in x, or (y₂ − y₁) ÷ (x₂ − x₁). Subtract the coordinates in the same order top and bottom, second point minus first, or the sign flips. For the points (1, 2) and (3, 6) the rise is 4 over a run of 2, so the slope is 2. Read the sign for direction and the size for steepness. A positive slope rises from left to right, a negative one falls, and a slope of zero is a flat horizontal line where y never changes. A slope of 2 climbs twice as fast as a slope of 1; a slope of 0.5 is a gentle incline. It is the same quantity as a real-world grade: a five percent wheelchair ramp, a roof pitch, the rate at which one thing changes against another on a graph. The caveat that matters most is the vertical line. If the two x-values are equal the run is zero, the line stands straight up, and the slope is undefined because nothing can be divided by zero — the calculator returns no result rather than a misleading number. And a slope describes a straight line: on curved data it gives only the average rate of change between the two points you picked.
Sign for direction, size for steepness
A positive slope rises from left to right, a negative one falls, and zero is a flat line. A slope of 2 climbs twice as fast as a slope of 1.
A vertical line has no slope
If both x-values are equal the run is zero and nothing can be divided by it. The calculator returns no result rather than a misleading number.
The same quantity as a grade
A ramp gradient, a roof pitch and the rate one thing changes against another on a graph are all this number. A five percent ramp is a slope of 0.05.
Commonly misread
It does not matter which point I call the first.
It does not — as long as you subtract in the same order on top and bottom. Mixing the order flips the sign.
The slope describes my curved data.
A slope describes a straight line. Between two points on a curve it gives the average rate of change, not the rate at either point.
Reference table
| x₁, y₁, x₂, y₂ | Line through the points | Slope |
|---|---|---|
| 0, 0, 1, 1 | y = x | 1 |
| 1, 2, 3, 6 | y = 2x | 2 |
| 0, 1, 4, 3 | y = 0.5x + 1 | 0.5 |
| 0, 3, 2, 3 | y = 3, horizontal | 0 |
| 2, 5, 4, 1 | y = -2x + 9, falling | -2 |
| -1, -1, 1, 1 | y = x, through negative ground | 1 |
Questions
How do I calculate the slope between two points?
Divide the change in y by the change in x — slope = (y₂ − y₁) ÷ (x₂ − x₁), often called rise over run. For the points (1, 2) and (3, 6) that is (6 − 2) ÷ (3 − 1) = 2. Subtract the coordinates in the same order top and bottom and you get the gradient.
Where does the line cross the y axis?
Once you have the slope m, the y-intercept is b = y₁ − m × x₁ and the line reads y = mx + b. For (1, 2) and (3, 6) the slope is 2, so b = 2 − 2 × 1 = 0 and the line is y = 2x. This page reports the slope itself; the intercept follows from it in one step.
Why is the slope of a vertical line undefined?
A vertical line has the same x value at both points, so the run x₂ − x₁ is zero, and dividing by zero is undefined. The line still exists — x = 4, for instance — it simply has no finite slope. This calculator shows no result in that case rather than a misleading number.
What do positive, negative and zero slopes mean?
A positive slope rises from left to right, a negative slope falls, and a zero slope is a flat horizontal line. The larger the absolute value, the steeper the line: a slope of 2 climbs twice as fast as a slope of 1.
Can the coordinates be negative or zero?
Yes. Any of x₁, y₁, x₂ and y₂ can be negative, zero or positive — points live anywhere on the plane. The only restriction is that the two x values must differ; if x₂ equals x₁ the line is vertical and the slope is undefined.
Sources and last check
Information, not professional advice.