Point-slope form writes a straight line as y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is any point on the line. Use it when you know a slope and a point, or two points, but not the y-intercept: plug the numbers in, then distribute and solve for y to get slope-intercept form, or clear fractions to get standard form.
This guide shows where the formula comes from, works examples from a point and slope, from two points and from a word problem, converts one line into all three forms side by side, and covers the cases that trip people up: negative coordinates, fractional slopes, parallel and perpendicular lines, and vertical lines.
What Is Point-Slope Form?
y − y₁ = m(x − x₁), where m is the slope of the line and (x₁, y₁) is one known point on it. The letters x and y stay as variables; only m, x₁ and y₁ get numbers.It is one of three standard ways to write a line, alongside slope-intercept form y = mx + b and standard form Ax + By = C (OpenStax Precalculus lists point-slope form among its key equations for linear functions). All three describe the same kind of object. They differ in which facts about the line you can read off at a glance: point-slope form shows a point and the slope, slope-intercept form shows the slope and where the line crosses the y-axis, and standard form makes both intercepts quick to find.
Where the Formula Comes From
You don't need to memorize point-slope form as a separate fact. It is the slope formula rearranged. Take the known point (x₁, y₁) and any other point (x, y) on the line. The slope between them must equal m:
(y − y₁) / (x − x₁) = m
Multiply both sides by (x − x₁) and you get y − y₁ = m(x − x₁). That's the whole derivation. It also tells you what the equation means: a point (x, y) is on the line exactly when the rise from (x₁, y₁) is m times the run. The multiplied version has one bonus over the fraction: it is still true at x = x₁ itself, where the fraction would divide by zero.
Add y₁ to both sides: y = m(x − x₁) + y₁. This is the line y = mx shifted right by x₁ and up by y₁, the same shifts you meet in function transformations. The point (0, 0) on y = mx moves to (x₁, y₁).
Writing the Equation from a Point and a Slope
- Label the numbers
Write down
m,x₁andy₁before substituting. This is where sign errors are caught. - Substitute into the template
Put each number in brackets inside
y − y₁ = m(x − x₁), then tidy the signs: subtracting a negative becomes adding. - Convert if the question asks
Distribute
mand isolateyfor slope-intercept form, or move everything to one side for standard form.
Write the equation of the line with slope 3 that passes through (2, −5), in point-slope and slope-intercept form.
Label: m = 3, x₁ = 2, y₁ = −5.
Substitute: y − (−5) = 3(x − 2), which tidies to y + 5 = 3(x − 2).
Distribute: y + 5 = 3x − 6. Subtract 5 from both sides: y = 3x − 11.
Check with the given point: 3(2) − 11 = 6 − 11 = −5 ✓.
y + 5 = 3(x − 2), or y = 3x − 11For slope 4 through (−3, 2), students often write y − 2 = 4(x − 3). The template already contains a minus, so x − (−3) becomes x + 3: the correct equation is y − 2 = 4(x + 3). The second classic slip is distributing the slope to x only, giving y − 2 = 4x + 3. The 4 multiplies the whole bracket: y − 2 = 4x + 12, so y = 4x + 14. A quick test catches both: substitute the given point. With x = −3, 4(−3) + 14 = 2 ✓.
Writing Point-Slope Form from Two Points
With two points, find the slope first with m = (y₂ − y₁) / (x₂ − x₁), then use either point in the template. Students often ask which point is the "right" one. Both are. Each choice gives an equation that looks different in point-slope form and turns into the same equation once simplified, because both points sit on the same line with the same slope.
Find the equation of the line through (−1, 4) and (3, −4).
Slope: m = (−4 − 4) / (3 − (−1)) = −8 / 4 = −2.
Using (−1, 4): y − 4 = −2(x + 1). Distribute: y − 4 = −2x − 2, so y = −2x + 2.
Using (3, −4): y + 4 = −2(x − 3). Distribute: y + 4 = −2x + 6, so y = −2x + 2. Same line.
Standard form: add 2x to both sides of y = −2x + 2 to get 2x + y = 2.
Check both points in 2x + y = 2: 2(−1) + 4 = 2 ✓ and 2(3) + (−4) = 2 ✓.
y − 4 = −2(x + 1), equivalently y = −2x + 2 or 2x + y = 2If a teacher's answer key shows y + 4 = −2(x − 3) and yours shows y − 4 = −2(x + 1), you're both right. Simplify both to slope-intercept form to compare.
Converting Between Point-Slope, Slope-Intercept and Standard Form
Here is the line from Example 2 in all three forms, with the move that takes you from one to the next.
| Form | The line from Example 2 | What you read off | How to get here |
|---|---|---|---|
| Point-slope | y − 4 = −2(x + 1) | Slope −2, point (−1, 4) | From standard or slope-intercept form: find the slope and any one point, then substitute |
| Slope-intercept | y = −2x + 2 | Slope −2, y-intercept (0, 2) | From point-slope: distribute m, then add y₁ to both sides |
| Standard | 2x + y = 2 | x-intercept (1, 0), y-intercept (0, 2) | Move the x term left, clear any fractions, make A positive |
Standard form has a convention most courses follow: A, B and C are integers with no common factor, and A is not negative. When the slope is a fraction, multiply by its denominator early so the fractions vanish before you rearrange.
Write the line through (4, 1) and (−2, 5) in point-slope, standard and slope-intercept form.
Slope: m = (5 − 1) / (−2 − 4) = 4 / (−6) = −2/3.
Point-slope with (4, 1): y − 1 = −2/3 (x − 4).
Multiply every term by 3 to clear the fraction: 3y − 3 = −2(x − 4) = −2x + 8.
Add 2x and 3 to both sides: 2x + 3y = 11. Here A = 2 is positive and 2, 3, 11 share no factor.
Slope-intercept: 3y = −2x + 11, so y = −2/3 x + 11/3.
Check both points in 2x + 3y = 11: 2(4) + 3(1) = 8 + 3 = 11 ✓ and 2(−2) + 3(5) = −4 + 15 = 11 ✓.
y − 1 = −2/3 (x − 4), 2x + 3y = 11, y = −2/3 x + 11/3Notice the y-intercept is 11/3. Starting from slope-intercept form would have meant hunting for that awkward number first. Point-slope form lets you write a correct equation straight from the data and leave the fraction work to the end. The slope-intercept post covers converting standard form to y = mx + b in more detail.
Graphing a line from point-slope form
Point-slope form is already a set of graphing instructions. For y − 4 = −2(x + 1), read the point as (−1, 4) (flip the sign of each number next to x and y) and plot it. The slope −2 means down 2 for every 1 to the right, so step to (0, 2) and (1, 0), then draw the line through the points. Both of those points match the intercepts in the table above, which is a free check. For more on plotting, see graphing linear equations.
Parallel and Perpendicular Lines Through a Point
This is where point-slope form earns its place. "Find the line through this point, parallel (or perpendicular) to that line" hands you a point and, indirectly, a slope.
Parallel lines share a slope. Perpendicular lines have slopes whose product is −1, so you flip the fraction and change the sign (the negative reciprocal). Neither rule works for vertical lines, which have no numerical slope; those get their own section below.
Find the lines through (6, −1) that are parallel and perpendicular to 3x − 2y = 8.
Find the given slope: −2y = −3x + 8, so y = 3/2 x − 4 and the slope is 3/2.
Parallel (slope 3/2): y + 1 = 3/2 (x − 6). Distribute: y + 1 = 3/2 x − 9, so y = 3/2 x − 10, or 3x − 2y = 20.
Perpendicular (slope −2/3, since 3/2 × (−2/3) = −1): y + 1 = −2/3 (x − 6). Distribute: y + 1 = −2/3 x + 4, so y = −2/3 x + 3, or 2x + 3y = 9.
Check (6, −1): 3(6) − 2(−1) = 18 + 2 = 20 ✓ and 2(6) + 3(−1) = 12 − 3 = 9 ✓.
y = 3/2 x − 10. Perpendicular: y = −2/3 x + 3A Word Problem Where Point-Slope Form Fits Best
Real data rarely includes the moment x = 0. You measure something partway through, at two times, and want the rule. That's two points and no intercept, which is exactly the situation point-slope form was built for.
A candle burns at a steady rate. It is 18 cm tall after 2 hours and 13.5 cm tall after 5 hours. Write its height h (cm) in terms of time t (hours), then find its starting height and when it burns out.
The points are (2, 18) and (5, 13.5). Slope: m = (13.5 − 18) / (5 − 2) = −4.5 / 3 = −1.5 cm per hour. Negative, because the candle gets shorter.
Point-slope with (2, 18): h − 18 = −1.5(t − 2).
Distribute: h − 18 = −1.5t + 3, so h = −1.5t + 21. The intercept 21 is the height at t = 0.
Burns out when h = 0: 1.5t = 21, so t = 14 hours.
Check the second data point: −1.5(5) + 21 = −7.5 + 21 = 13.5 ✓.
h = −1.5t + 21; it started at 21 cm and burns out after 14 hoursHorizontal and Vertical Lines
A horizontal line has slope 0, so point-slope form still works: through (5, −2) it gives y + 2 = 0(x − 5), which is y = −2.
A vertical line is the one line point-slope form can't write. Through (5, −2) and (5, 3), the slope formula gives 5 / 0, which is undefined, so there's no m to substitute. Write it directly as x = 5. If you see a zero denominator while finding the slope, that's your signal the line is vertical, not that you've made a mistake.
How to Check Your Answer
- Substitute every given point into your final equation. A point-and-slope question has one point to test; a two-point question has two, and the second one is the real test, because the first was built in.
- Match the slope. In slope-intercept form, the coefficient of
xmust equal themyou calculated. In standard form, the slope is−A/B; for2x + 3y = 11that's−2/3✓. - Sanity-check the sign. If
ygoes up asxgoes up, the slope must be positive. A candle that grows taller over time means a sign slipped.
Once you've done a problem by hand, Solver AI on iPhone and iPad can scan a printed or handwritten line problem and show its steps, so you can compare each line with your own working and spot where a sign changed.
In calculus, the tangent line to a curve is written in point-slope form: the point is on the curve and the slope is the derivative there. For y = x² at x = 3, the point is (3, 9) and the slope is 2(3) = 6, so the tangent is y − 9 = 6(x − 3), or y = 6x − 9. The implicit differentiation guide ends its examples the same way.
Practice Problems
Write each answer in point-slope form first, then convert as asked. Answers follow each problem.
- Slope
−4through(−2, 7), in slope-intercept form.
Answer:y − 7 = −4(x + 2), soy = −4x − 1. - Through
(1, −3)and(5, 9), in standard form.
Answer:m = 12/4 = 3;y + 3 = 3(x − 1), soy = 3x − 6, or3x − y = 6. - Through
(−4, 0), parallel toy = 1/2 x + 7, in both slope-intercept and standard form.
Answer:y − 0 = 1/2 (x + 4), soy = 1/2 x + 2, orx − 2y = −4. - A seedling is 12 cm tall on day 4 and 19.5 cm tall on day 9, growing at a steady rate. Write its height
hin terms of daydand predict day 20.
Answer:m = 7.5/5 = 1.5cm per day;h − 12 = 1.5(d − 4), soh = 1.5d + 6. On day 20,h = 36cm. - Find the line through
(2, −3)perpendicular to the line through(−1, 4)and(5, 1), then find where the two lines cross.
Answer: the second line has slope−3/6 = −1/2, so the perpendicular slope is2:y + 3 = 2(x − 2), soy = 2x − 7. The other line isy − 4 = −1/2 (x + 1), soy = −1/2 x + 7/2. Setting2x − 7 = −1/2 x + 7/2gives5/2 x = 21/2, sox = 21/5 = 4.2andy = 1.4. They cross at(4.2, 1.4). For more on that last step, see solving systems of equations.