# What is my line in point-slope form?

Finds the point-slope form y − y₁ = m(x − x₁) of a line through two points, or through one point with a given slope, in exact fractions, with the slope-intercept and standard forms.

- Page: https://www.acalculator.org/math/point-slope-form-calculator
- JSON spec: https://www.acalculator.org/math/point-slope-form-calculator.json
- Version: 79a42702a5be

## Default answer

Example with the default inputs (I know Two points, x₁ 1, y₁ 3, x₂ 3, y₂ 7): In point-slope form the line is y − 3 = 2(x − 1), which is y = 2x + 1.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| given | I know | What you know about the line: two points on it, or its slope and one point. |
| x1 | x₁ | The x coordinate of a point on the line. |
| y1 | y₁ | The y coordinate of the same point. |
| x2 | x₂ | The x coordinate of a second point on the line. |
| y2 | y₂ | The y coordinate of the second point. |
| m | Slope (m) | How much y changes when x goes up by 1. |
| x | x | Optional: an x value at which to read y on the line. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| equation | Slope-intercept form | The line as y = mx + b, with the slope m and the y-intercept b as exact fractions. |
| slope | Slope m | The rise over the run: (y₂ − y₁) ÷ (x₂ − x₁). |
| intercept | y-intercept b | Where the line crosses the y axis: the value of y at x = 0. |
| xIntercept | x-intercept | Where the line crosses the x axis: the value of x at y = 0, which is −b ÷ m. None for a flat line. |
| pointSlope | Point-slope form | The line as y − y₁ = m(x − x₁), through the first point, with the slope as an exact fraction. |
| standard | Standard form | The same line as Ax + By = C, with whole numbers A, B, C that share no factor. |
| y | y on the line | The y value on the line at the x you typed: mx + b. |

## Method

m = (y₂ − y₁) ÷ (x₂ − x₁) (or the slope as typed), b = y₁ − m × x₁, and the line is y = mx + b; all in exact fractions.

## Assumptions

- Every number is read exactly as typed: 0.1 is 1/10, so the slope and intercept are exact fractions.
- The two points must have different x values. A vertical line, x = c, has no slope-intercept form.
- The point-slope form uses the first point, (x₁, y₁).
- The standard form Ax + By = C uses whole numbers with no common factor, with A ≥ 0.
- y at x is shown when it is at most 10³⁰⁰ in size.

## Worked examples

1. given = points, x1 = -2, y1 = -4, x2 = 0, y2 = 2 gives pointSlope = y + 4 = 3(x + 2), equation = y = 3x + 2, slope = 3, intercept = 2. Source: OpenStax, Algebra and Trigonometry 2e, §4.1 Linear Functions (point-slope form y − y₁ = m(x − x₁)). https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-1-linear-functions, retrieved 2026-10-02, Example 5: through (0, 2) and (−2, −4), y + 4 = 3(x + 2), so y = 3x + 2.
2. given = points, x1 = 3, y1 = -2, x2 = 8, y2 = 1 gives pointSlope = y + 2 = (3/5)(x − 3), equation = y = (3/5)x − 19/5, slope = 0.6, intercept = -3.8. Source: OpenStax, Algebra and Trigonometry 2e, §4.1 Linear Functions (point-slope form y − y₁ = m(x − x₁)). https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-1-linear-functions, retrieved 2026-10-02, Example 7: f(3) = −2 and f(8) = 1 give y + 2 = (3/5)(x − 3), so y = (3/5)x − 19/5.
3. given = slope, m = -0.5, x1 = 4, y1 = 1, x = 10 gives pointSlope = y − 1 = −(1/2)(x − 4), equation = y = −(1/2)x + 3, standard = x + 2y = 6, y = -2. Source: OpenStax, Algebra and Trigonometry 2e, §4.1 Linear Functions (point-slope form y − y₁ = m(x − x₁)). https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-1-linear-functions, retrieved 2026-10-02.
4. given = points, x1 = 0, y1 = 5, x2 = 4, y2 = 5 gives pointSlope = y − 5 = 0, equation = y = 5, slope = 0. Source: OpenStax, Algebra and Trigonometry 2e, §4.1 Linear Functions (point-slope form y − y₁ = m(x − x₁)). https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-1-linear-functions, retrieved 2026-10-02.
5. given = points, x1 = 0.1, y1 = 0.2, x2 = 0.3, y2 = 0.5 gives pointSlope = y − 1/5 = (3/2)(x − 1/10), slope = 1.5, intercept = 0.05. Source: OpenStax, Algebra and Trigonometry 2e, §4.1 Linear Functions (point-slope form y − y₁ = m(x − x₁)). https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-1-linear-functions, retrieved 2026-10-02.

## FAQ

### What is point-slope form?

A way to write a straight line from one point (x₁, y₁) on it and its slope m: y − y₁ = m(x − x₁). The line through (−2, −4) with slope 3 is y + 4 = 3(x + 2).

### How do I write point-slope form from two points?

First find the slope, m = (y₂ − y₁) ÷ (x₂ − x₁). Then put m and one of the points into y − y₁ = m(x − x₁). For (3, −2) and (8, 1): m = 3 ÷ 5 = 3/5, so y + 2 = (3/5)(x − 3).

### Why does the equation say y + 4 when the point is (−2, −4)?

Because y − (−4) is y + 4, and x − (−2) is x + 2. Subtracting a negative coordinate gives a plus sign.

### How do I turn point-slope form into y = mx + b?

Multiply out the bracket and move y₁ to the right side. y + 4 = 3(x + 2) gives y + 4 = 3x + 6, so y = 3x + 2.

### Does it matter which point I use?

No, both give the same line, but the equations look different. Through (−2, −4) the line is y + 4 = 3(x + 2); through (0, 2) it is y − 2 = 3x. This calculator uses the first point you type.

### What if both points have the same x value?

Then the line is vertical, x = c. Its slope is undefined because the run x₂ − x₁ is 0, so it has no point-slope form. The calculator says so instead of giving a number.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §4.1 Linear Functions (point-slope form y − y₁ = m(x − x₁); Example 5: through (0, 2) and (−2, −4) the line is y + 4 = 3(x + 2), or y = 3x + 2; Example 7: f(3) = −2 and f(8) = 1 give y + 2 = (3/5)(x − 3), or y = (3/5)x − 19/5). https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-1-linear-functions (retrieved 2026-10-02)