What is my line in point-slope form?
Type two points on a line, or its slope and one point. The calculator writes the line in point-slope form with exact fractions, and also as y = mx + b and in standard form.
- Point-slope form
- y − 3 = 2(x − 1)
In point-slope form the line is y − 3 = 2(x − 1), which is y = 2x + 1.
- Slope-intercept form
- y = 2x + 1
- Slope m
- 2
- y-intercept b
- 1
- x-intercept
- -0.5
- Standard form
- 2x − y = −1
Point-slope form: y − 3 = 2(x − 1). In point-slope form the line is y − 3 = 2(x − 1), which is y = 2x + 1.
y on the line by x
How to calculate
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.
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.
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.
- 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
Each example is checked against the calculator on every build.
- I know Two points, x₁ -2, y₁ -4, x₂ 0, y₂ 2 gives Point-slope form y + 4 = 3(x + 2), Slope-intercept form y = 3x + 2, Slope m 3, y-intercept b 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
- I know Two points, x₁ 3, y₁ -2, x₂ 8, y₂ 1 gives Point-slope form y + 2 = (3/5)(x − 3), Slope-intercept form y = (3/5)x − 19/5, Slope m 0.6, y-intercept b -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
- I know Slope and a point, Slope (m) -0.5, x₁ 4, y₁ 1, x 10 gives Point-slope form y − 1 = −(1/2)(x − 4), Slope-intercept form y = −(1/2)x + 3, Standard form x + 2y = 6, y on the line -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
- I know Two points, x₁ 0, y₁ 5, x₂ 4, y₂ 5 gives Point-slope form y − 5 = 0, Slope-intercept form y = 5, Slope m 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
- I know Two points, x₁ 0.1, y₁ 0.2, x₂ 0.3, y₂ 0.5 gives Point-slope form y − 1/5 = (3/2)(x − 1/10), Slope m 1.5, y-intercept b 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
How it works
A non-vertical line through a point (x₁, y₁) with slope m is y − y₁ = m(x − x₁).
From two points (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁) ÷ (x₂ − x₁). The point-slope form uses the first point.
From a slope and one point: the slope as typed, with the point (x₁, y₁).
The calculator also shows, from the same m and the first point:
- slope-intercept form y = mx + b, with b = y₁ − m × x₁
- standard form Ax + By = C: multiply y = mx + b by the least common multiple L of the denominators of m and b to get (mL)x − Ly = −bL, make A 0 or more (and B positive when A is 0), and divide out any common factor of A, B and C
- the x-intercept −b ÷ m when m is not 0, and y = mx + b at an x you type
This page uses the same inputs and the same arithmetic as the slope-intercept form calculator; it leads with the point-slope form.
Rules
- Every number is read exactly as typed, as a fraction: 0.1 is 1/10. The slope and the forms are exact fractions in lowest terms.
- Two points with the same x give no answer (a vertical line has no slope). Two equal points give no answer (they do not fix one line).
- The decimal slope, intercepts and y are the exact fractions rounded to double precision; y at x is left out when it is larger than 10³⁰⁰ in size.
How the equations are written
- Fractions are in lowest terms; a whole number has no denominator. A fraction coefficient is in brackets: (3/5)(x − 3).
- A coefficient of 1 is left out: y − 1 = x − 4, and −1 shows as a minus sign.
- y − (−4) is written y + 4 and x − (−2) is written x + 2; a coordinate of 0 leaves just x or y. A slope of 0 is written y − y₁ = 0.
- Signs between terms use the minus sign −.
Worked examples by hand
Through (−2, −4) and (0, 2). m = (2 − (−4)) ÷ (0 − (−2)) = 6 ÷ 2 = 3. From the first point: y + 4 = 3(x + 2). Multiplying out: y = 3x + 6 − 4 = 3x + 2.
Through (3, −2) and (8, 1). m = (1 − (−2)) ÷ (8 − 3) = 3/5. y + 2 = (3/5)(x − 3). Then y = (3/5)x − 9/5 − 2 = (3/5)x − 19/5 (slope 0.6, intercept −3.8).
Slope −1/2 through (4, 1), read at x = 10. y − 1 = −(1/2)(x − 4). b = 1 − (−1/2) × 4 = 3, so y = −(1/2)x + 3, or x + 2y = 6. At x = 10: y = −5 + 3 = −2.
Through (0, 5) and (4, 5). m = 0 ÷ 4 = 0, so y − 5 = 0, which is y = 5.
Through (0.1, 0.2) and (0.3, 0.5). m = 0.3 ÷ 0.2 = 3/2. The first point is (1/10, 1/5), so y − 1/5 = (3/2)(x − 1/10). b = 1/5 − 3/20 = 1/20 = 0.05.
Other questions people ask
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.