OneTaskBy Tim King · Updated
Slope Intercept Form Calculator
Two-Point or Equation Input, Vertical and Horizontal Lines, Worked Steps Shown
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How the Slope-Intercept Form Calculator works
This slope intercept form calculator takes two points, such as (2, 3) and (6, 11), or a line equation such as 3x + 4y = 12, and gives the slope, both intercepts and the line in slope-intercept, point-slope and standard form. Vertical and horizontal lines are handled too.
Slope-intercept form writes a straight line as y = mx + b. The slope m is how far the line rises for each step to the right, and b is where it crosses the y-axis. Once a line is in this form you can read its steepness and starting point at a glance, or plot it in two moves.
How the slope intercept form calculator works
From two points the formula for slope is m = (y₂ − y₁) ÷ (x₂ − x₁). Substituting either point into y = mx + b gives b = y₁ − m × x₁. From an equation the calculator collects every term on one side and solves for y.
First example. For (2, 3) and (6, 11), m = (11 − 3) ÷ (6 − 2) = 2, and b = 3 − 2 × 2 = −1, so y = 2x − 1.
Second example. For 3x + 4y = 12, subtract 3x and divide by 4: y = −(3/4)x + 3. The slope is −3/4 and the line crosses the axes at (0, 3) and (4, 0).
How many points do you need to fix a line? Two distinct points, or one point and the slope.
Slope and slope-intercept form for 22 pairs of points
| Points | Slope | Equation |
|---|---|---|
| (0, 0) and (1, 1) | 1 | y = x |
| (2, 3) and (6, 11) | 2 | y = 2x − 1 |
| (1, 2) and (3, 8) | 3 | y = 3x − 1 |
| (0, 5) and (2, 1) | -2 | y = −2x + 5 |
| (-1, -1) and (3, 7) | 2 | y = 2x + 1 |
| (2, 4) and (4, 5) | 1/2 | y = (1/2)x + 3 |
| (0, 0) and (4, 2) | 1/2 | y = (1/2)x |
| (1, 1) and (5, 1) | 0 | y = 1 |
| (3, 0) and (0, 6) | -2 | y = −2x + 6 |
| (-2, 1) and (2, 9) | 2 | y = 2x + 5 |
| (1, 4) and (2, 7) | 3 | y = 3x + 1 |
| (0, -3) and (6, 1) | 2/3 | y = (2/3)x − 3 |
| (5, 2) and (1, -6) | 2 | y = 2x − 8 |
| (-3, 4) and (3, -2) | -1 | y = −x + 1 |
| (2, 2) and (8, 5) | 1/2 | y = (1/2)x + 1 |
| (1, -1) and (4, 8) | 3 | y = 3x − 4 |
| (0, 10) and (5, 0) | -2 | y = −2x + 10 |
| (-4, -2) and (2, 1) | 1/2 | y = (1/2)x |
| (6, 3) and (2, 3) | 0 | y = 3 |
| (1, 0) and (3, 10) | 5 | y = 5x − 5 |
| (-1, 5) and (1, -1) | -3 | y = −3x + 2 |
| (0, 1) and (3, 3) | 2/3 | y = (2/3)x + 1 |
Tips
- Same x, no slope. Two points with equal x-values make a vertical line, x = constant.
- Perpendicular slopes multiply to −1. A slope of 2 is perpendicular to −1/2.
- Order does not matter. Swap the two points and the slope comes out the same.
FAQ
How do you find the equation of a line from two points?
Work out the slope by dividing the change in y by the change in x. Then put one point and the slope into y = mx + b and solve for b. For (1, 2) and (3, 8) the slope is 6 ÷ 2 = 3 and b = 2 − 3 = −1, so the line is y = 3x − 1.
What is the difference between slope-intercept and standard form?
Slope-intercept form, y = mx + b, shows the slope and y-intercept directly. Standard form, Ax + By = C with whole numbers and A positive, is tidier for intercepts and for solving systems. Both describe the same line, and the calculator gives each one for every input.
What is point-slope form used for?
Point-slope form, y − y₁ = m(x − x₁), lets you write a line straight from one known point and a slope without solving for b first. It is the quickest route when a question gives a point and a gradient, and it expands into slope-intercept form in one step.
Why is the slope of a vertical line undefined?
Every point on a vertical line has the same x, so the change in x is zero and the slope formula divides by zero. The line is written x = c instead. A horizontal line is the opposite case, with a slope of exactly 0 and an equation of the form y = c.
See also: all Algebra and Geometry tools, Arithmetic, Number Theory, Statistics, Trigonometry and Calculus.
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