Write the equation of a line in point-slope, slope-intercept, standard and general form. Start from a point and a slope, two points, a slope given as an angle or as rise over run. Each result lists the intercepts, the worked steps and a graph of the line.
Quick Summary: What Is Point-Slope Form?
Point-slope form writes the equation of a straight line from one point on the line and its slope: y − y₁ = m(x − x₁). Here (x₁, y₁) is the known point and m is the slope. Expanding it gives slope-intercept form, y = mx + b.
- Formula
- y − y₁ = m(x − x₁)
- Slope from two points
- m = (y₂ − y₁) / (x₂ − x₁), which is rise over run, or tan θ for an angle θ.
- Convert to slope-intercept form
- y = mx + (y₁ − m·x₁). The y-intercept is b = y₁ − m·x₁.
- x-intercept
- x = x₁ − y₁ / m, when m ≠ 0.
- Parallel and perpendicular lines
- Parallel lines share the slope m. Perpendicular lines have slope −1/m.
- Vertical lines
- The slope is undefined, so the equation is x = x₁ and point-slope form does not apply.
Last updated: October 28, 2026
What Is Point-Slope Form?
Any point (x, y) on a line has the same slope to a fixed point (x₁, y₁) on that line. Setting that slope equal to m and clearing the fraction gives y − y₁ = m(x − x₁). The slope measures steepness: positive when the line rises from left to right, negative when it falls, zero when it is horizontal. To measure the slope between two points on its own, use the slope calculator.
The slope between any point (x, y) and the known point (x₁, y₁) is constant.
Which Form of a Line Equation Should You Use?
| Form | Equation | Needs | Best for |
|---|---|---|---|
| Point-slope | y − y₁ = m(x − x₁) | One point and the slope | Writing an equation quickly from given data |
| Slope-intercept | y = mx + b | Slope and y-intercept | Graphing and reading a rate and a starting value |
| Standard | Ax + By = C | Integer coefficients | Systems of equations and intercepts |
| General | Ax + By + C = 0 | Integer coefficients | Distance and geometry formulas |
| Two-point | y − y₁ = [(y₂ − y₁)/(x₂ − x₁)](x − x₁) | Two points | Lines through data points |
| Vertical line | x = a | One x-value | Lines with undefined slope |
Slope-intercept form is point-slope form for the special point (0, b): y − b = m(x − 0) simplifies to y = mx + b.
Point Slope Form Formulas
Point-slope form: y − y₁ = m(x − x₁)
Slope from two points: m = (y₂ − y₁) / (x₂ − x₁)
Slope from an angle: m = tan θ
Slope-intercept form: y = mx + b, with b = y₁ − m·x₁
x-intercept: x = x₁ − y₁ / m, m ≠ 0
Standard form from m = p/q: −p·x + q·y = q·y₁ − p·x₁
Parallel line: same m
Perpendicular line: m₂ = −1 / m
Slope from two points: m = (y₂ − y₁) / (x₂ − x₁)
Slope from an angle: m = tan θ
Slope-intercept form: y = mx + b, with b = y₁ − m·x₁
x-intercept: x = x₁ − y₁ / m, m ≠ 0
Standard form from m = p/q: −p·x + q·y = q·y₁ − p·x₁
Parallel line: same m
Perpendicular line: m₂ = −1 / m
How Do You Use the Point Slope Form Calculator?
- Point and Slope: enter x₁ and y₁, then give the slope directly, as an angle in degrees, or as rise and run.
- Two Points: enter both points. The calculator finds the slope first, then writes the line through the first point.
- Find a Coordinate: enter a point and slope, choose whether you know x or y, and read the matching coordinate on the line.
- Parallel and Perpendicular: enter a given line as y = mx + b and a new point. The result is the parallel or perpendicular line through that point.
- Read the equation forms, intercepts and steps, and check the graph.
How to Write the Equation of a Line from a Point and a Slope
- Identify the point (x₁, y₁) and the slope m.
- Substitute them into y − y₁ = m(x − x₁), keeping negative numbers in brackets.
- Simplify the double signs. For example, y − (−3) becomes y + 3.
- Distribute m across the brackets.
- Add y₁ to both sides to isolate y and get y = mx + b.
- Check by substituting (x₁, y₁) into the final equation. Both sides must match.
Point Slope Form Examples with Answers
Example 1: slope 2 through (2, −3)
Point-slope: y + 3 = 2(x − 2). Expand: y + 3 = 2x − 4. Slope-intercept: y = 2x − 7. Standard form: 2x − y = 7. The y-intercept is −7 and the x-intercept is 3.5.Example 2: taxi fare
A ride of 5 km costs $14 and each kilometre costs $2. With (5, 14) and m = 2: y − 14 = 2(x − 5), so y = 2x + 4. The flat starting fee is $4.Example 3: through (1, 2) and (5, 10)
m = (10 − 2) / (5 − 1) = 2. Using (1, 2): y − 2 = 2(x − 1), so y = 2x. Using (5, 10) gives the same line.Example 4: perpendicular to y = 2x + 3 through (4, 1)
The new slope is −1/2. Point-slope: y − 1 = −½(x − 4). Slope-intercept: y = −½x + 3. Standard form: x + 2y = 6.Example 5: a line at 45° through (0, 1)
m = tan 45° = 1, so y − 1 = 1(x − 0) and y = x + 1.Straight-line equations also describe real ground and structures. Engineers apply the same slope idea to terrain, as in slope stability monitoring.
Special Cases of Slope
| Slope | Line | Equation through (x₁, y₁) |
|---|---|---|
| Positive | Rises left to right | y − y₁ = m(x − x₁) |
| Negative | Falls left to right | y − y₁ = −|m|(x − x₁) |
| Zero | Horizontal | y = y₁ |
| Undefined | Vertical | x = x₁ |
| Fraction p/q | Rise p for every run q | q(y − y₁) = p(x − x₁) |
Roads, roofs and ramps usually quote slope as a grade in percent. This guide to slope percentage and angle shows how to convert a rise and run into either one.
Common Point Slope Form Mistakes
- Sign errors on the point. A point with x₁ = −4 gives (x + 4), not (x − 4).
- Swapping x and y. The y-values belong on the left side with y₁.
- Skipping the distribution. Multiply m by both terms inside the brackets.
- Using point-slope form for a vertical line. Its slope is undefined. Write x = x₁.
- Rounding the slope early. Keep fractions such as 1/3 exact until the last step.
- Flipping the perpendicular slope without the sign. The slope is the negative reciprocal.
Point Slope Form FAQs
What is point-slope form?
Point-slope form is y − y₁ = m(x − x₁). It describes the line with slope m that passes through the point (x₁, y₁).
How do you convert point-slope form to slope-intercept form?
Expand the right side to get y − y₁ = mx − m·x₁, then add y₁ to both sides. The result is y = mx + (y₁ − m·x₁), so the y-intercept is y₁ − m·x₁.
How do you find the y-intercept from point-slope form?
Use b = y₁ − m·x₁. For y − 1 = 2(x − 3), the intercept is 1 − 2 × 3 = −5.
What is point-slope form when the slope is zero?
The equation reduces to y − y₁ = 0, which is y = y₁. It is a horizontal line that crosses the y-axis at y₁.
Can a vertical line be written in point-slope form?
No. A vertical line has an undefined slope. Its equation is x = x₁, where x₁ is the x-value of every point on the line.
How do you write point-slope form from two points?
Find the slope m = (y₂ − y₁) / (x₂ − x₁), then substitute m and either point into y − y₁ = m(x − x₁). Both points give the same line.
How do you find a perpendicular line in point-slope form?
Use the negative reciprocal of the original slope, −1/m, and the new point. A line with slope 2 has a perpendicular line with slope −1/2.
How do you find the slope from an angle?
The slope equals the tangent of the angle the line makes with the positive x-axis: m = tan θ. An angle of 45° gives m = 1, and 90° gives an undefined slope.
References
- Larson, R. (2021). Algebra and Trigonometry, 10th ed. Cengage Learning.
- Stewart, J., Redlin, L. and Watson, S. (2016). Precalculus: Mathematics for Calculus, 7th ed. Cengage Learning.
- National Governors Association and Council of Chief State School Officers (2010). Common Core State Standards for Mathematics: Algebra and Functions.
- Weisstein, E. W. Point-Slope Form. MathWorld, A Wolfram Web Resource.
- National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics.