Point Slope Form Calculator
Build a line's equation from one point and a slope. Enter (x₁, y₁) and m — decimals and negatives are fine — and get point-slope form with the signs handled correctly, plus the slope-intercept and standard-form versions of the same line.
Example: with x₁ (known point) 1 · y₁ (known point) 3 · Slope (m) 2 → Point-slope form: y - 3 = 2(x - 1).
- Slope-intercept formy = 2x + 1
- Standard form2x - y = -1
- y-intercept (b)1
Computed by the calculator below using its default values. Change any input to see your own numbers.
y − y₁ = m(x − x₁): plug the point and slope straight in. Distribute and solve for y to get y = mx + b, where b = y₁ − m·x₁.
Why point-slope form exists
Slope-intercept form (y = mx + b) is great for graphing, but it demands the one point you often do not have: the y-intercept. Point-slope form works from any point. It restates the definition of slope — between your known point (x₁, y₁) and a generic point (x, y), the rise over run must equal m — and cross-multiplies it into y − y₁ = m(x − x₁). Nothing to solve; just substitute.
The only trap is the built-in subtraction. The template subtracts the point's coordinates, so the point (1, 3) with slope 2 gives y − 3 = 2(x − 1); a negative coordinate flips the sign, so the point (−2, 5) gives y − 5 = m(x + 2). Distribute the m and move y₁ across, and you land on slope-intercept form with b = y₁ − m·x₁ — this calculator shows every version at once.
How it’s calculated
Point-slope: y − y₁ = m(x − x₁), with signs simplified when coordinates are negative. Slope-intercept: y = mx + b with b = y₁ − m·x₁. Standard form: mx − y = −b, scaled so the x-coefficient is positive. Values are shown to 4 decimal places.
Requires a defined slope — a vertical line has no point-slope or slope-intercept form and is written x = x₁ instead.
Three forms of the same line (point (1, 3), slope 2)
| Form | Equation | Best when you know / want |
|---|---|---|
| Point-slope | y − 3 = 2(x − 1) | Any point plus the slope |
| Slope-intercept | y = 2x + 1 | Graphing; reading m and b at a glance |
| Standard | 2x − y = −1 | Integer-coefficient work; finding both intercepts |
Standard algebra conventions; each line converts to the others by distributing and rearranging.
Common mistakes
- Flipping a sign inside the parentheses — the point (1, 3) gives (x − 1), but the point (−1, 3) gives (x + 1), because subtracting −1 adds.
- Plugging the slope into a coordinate slot, or the x-value into the y slot — label the pieces before substituting.
- Distributing m to x only and forgetting the x₁ term when converting to slope-intercept form.
- Trying to write a vertical line this way — undefined slope means no point-slope form; use x = x₁.
Frequently asked questions
What is point-slope form?
y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is any known point on the line. With point (1, 3) and slope 2: y − 3 = 2(x − 1).
How do I convert point-slope form to slope-intercept form?
Distribute the slope, then add y₁ to both sides. y − 3 = 2(x − 1) becomes y = 2x − 2 + 3 = 2x + 1. In general b = y₁ − m·x₁.
Does it matter which point on the line I use?
No — any point on the line produces an equation for the same line. The point-slope strings look different, but they all simplify to the identical y = mx + b.
What if the slope is negative or a fraction?
Substitute it as-is, parentheses and all: point (4, −2) with m = 0.5 gives y + 2 = 0.5(x − 4), which simplifies to y = 0.5x − 4. The form handles any real slope except an undefined (vertical) one.
Why is there no point-slope form for vertical lines?
A vertical line's slope is undefined — the run is zero, so rise over run divides by zero. Vertical lines are written x = x₁, a form with no y in it at all.