Quick answer
Point-slope form is y − y₁ = m(x − x₁). With slope 2 through (1, 5): y − 5 = 2(x − 1), which is y = 2x + 3 in slope-intercept form.
Enter the slope and a point. The calculator returns the point-slope and slope-intercept forms.
Related tools: Slope-Intercept Form Calculator: Line equation y = mx + b from a slope and a point. Standard Form to Slope-Intercept Form Calculator: Convert Ax + By = C to y = mx + b. Slope Calculator: Slope, angle, distance and line equation through two points.
How to use the Point-Slope Form Calculator
- Enter slope (m).
- Enter point x₁.
- Enter point y₁.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Slope (m) | A number | — |
| Point x₁ | A number | — |
| Point y₁ | A number | — |
Formula and method
Formula
y − y₁ = m(x − x₁)
IEEE-754 double precision; shown to at most 6 decimal places.
Point-slope form with the equivalent slope-intercept form.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Slope (m) = 2; Point x₁ = 1; Point y₁ = 5.
| Result | Value |
|---|---|
| Point-slope form | y − 5 = 2(x − 1) |
| Slope-intercept | y = 2x + 3 |
The first line is the headline result (point-slope form). Below it the point-Slope Form Calculator lists slope-intercept, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Converting
Expand: y − 5 = 2x − 2, so y = 2x + 3. The y-intercept is 3. Check: at x = 1, y = 5 ✓.
When to use it
Point-slope form is the quickest way to write a line when you know one point and the slope. Vertical lines, which have no slope, can't be written this way and are x = c.
How the result changes with slope
Holding every other input at the first worked example, this table changes only slope. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Slope | Point-slope form | Slope-intercept |
|---|---|---|
| 1 | y − 5 = (x − 1) | y = x + 4 |
| 2 | y − 5 = 2(x − 1) | y = 2x + 3 |
| 4 | y − 5 = 4(x − 1) | y = 4x + 1 |
| 8 | y − 5 = 8(x − 1) | y = 8x − 3 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Slope (m): entering "a" shows "Slope (m) must be a number." beside that field and no result.
Choosing between the Point-Slope Form Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Slope-Intercept Form Calculator](/math/slope-intercept-form/) | slope, point x and point y | Equation |
| [Standard Form to Slope-Intercept Form Calculator](/math/standard-form-to-slope-intercept-form/) | a, b and c | Slope-intercept form |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [Vertex Form Calculator](/math/vertex-form/) | a, b and c | Vertex (h, k) |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Non-vertical lines
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for slope (m) to confirm this.
Frequently asked questions
What is the point-slope form for slope −1 through (2, 3)?
y − 3 = −1(x − 2), which simplifies to y = −x + 5.
How do I get the slope from two points?
The slope is rise over run: (y₂ − y₁) ÷ (x₂ − x₁).
What is the y-intercept here?
The y-intercept for slope 2 through (1, 5) is 3, from y = 2x + 3.
What does the Point-Slope Form Calculator calculate?
Point-slope and slope-intercept equations of a line. It returns Point-slope form and Slope-intercept.
What formula does the Point-Slope Form Calculator use?
y − y₁ = m(x − x₁). Point-slope form with the equivalent slope-intercept form.
What do I need to enter in the point-Slope Form Calculator?
You enter Slope (m), Point x₁ and Point y₁.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Non-vertical lines.
What is an example of a point-Slope Form result?
With slope (m) 2, point x₁ 1 and point y₁ 5, the point-slope form is y − 5 = 2(x − 1).
Related calculators
- Slope-Intercept Form Calculator — Line equation y = mx + b from a slope and a point.
- Standard Form to Slope-Intercept Form Calculator — Convert Ax + By = C to y = mx + b.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- Vertex Form Calculator — Convert y = ax² + bx + c to vertex form.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Row Echelon Form Calculator — Reduced row echelon form and rank of a matrix.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.