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What Is Point-Slope Form?
Point-slope form is the line equation y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is a point the line goes through. You use it when you already know a point and the slope, and you haven't bothered to find the y-intercept yet.
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What each letter is
- m — slope (rise/run).
- (x₁, y₁) — any known point on the line. The subscripts just mean "this specific point," not a second variable.
- x and y — the coordinates of every other point on the line.
Worked example: slope 2, point (3, 1).
- y − 1 = 2(x − 3)
- That's already the line. Distribute if you need slope-intercept: y − 1 = 2x − 6 → y = 2x − 5.
Point-slope vs. slope-intercept vs. standard
| Form | Equation | When it's handy |
|---|---|---|
| Point-slope | y − y₁ = m(x − x₁) | You have one point + slope |
| Slope-intercept | y = mx + b | You want graphing from the y-intercept |
| Standard | Ax + By = C | Some textbooks / integer-coefficient questions |
If you have two points and no slope yet: first compute m = (y₂ − y₁) / (x₂ − x₁), then plug either point into point-slope. Both points give equivalent equations.
Vertical lines don't have a defined slope, so they don't get a point-slope equation of this type; they're x = a constant.
The closest sibling pages are What Is Second Person Point of View? and What Is Slope Intercept Form?.
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Frequently Asked Questions
What does point-slope form look like?
y − y₁ = m(x − x₁), where m is slope and (x₁, y₁) is a known point on the line.
When do you use point-slope instead of y = mx + b?
When you already have a point and a slope and don't need the intercept yet. You can always rearrange into slope-intercept after.
Can you use either point if you have two points?
Yes. Compute the slope from both points, then plug either point into point-slope; the lines match.