Point Slope Form Calculator
This point-slope form calculator writes the equation y − y₁ = m(x − x₁) from a point and a slope — or from two points, in which case it shows you both valid answers and proves they are the same line.
Give it (3, 5) and a slope of 2 and you get y − 5 = 2(x − 3). It also shows the substitution before it is tidied, which is where the marks are usually lost.
What Point-Slope Form Is
y − y₁ = m(x − x₁)
Here (x₁, y₁) is any point the line passes through and m is its slope. That is all the form needs: one point and a steepness. No y-intercept, no rearranging, no solving for anything.
It comes straight out of the definition of slope. Take any other point (x, y) on the line. The slope between it and (x₁, y₁) is (y − y₁) ÷ (x − x₁), and that must equal m. Multiply both sides by (x − x₁) and you have the form. So it is not a separate fact to memorise — it is the slope formula with the fraction cleared away.
That is also why it is the form questions ask for when they hand you a point and a slope. Of the several ways of writing the equation of a line, this is the one that needs the least work when those are your two starting facts — any other makes you rearrange to answer a question that was already answered.
How to Write the Equation
Two steps if the numbers are friendly, three if they are not.
1. Substitute. The slope goes where m is; the coordinates go where x₁ and y₁ are. Write it with the brackets in place, exactly as the formula has them.
2. Resolve the signs. Only if a coordinate is negative — see the next section.
3. Stop. Point-slope form is a finished answer. If the question says "leave it in point-slope form", it is asking you not to tidy it away into something else.
With (2, 3) and slope 4: y − 3 = 4(x − 2).
With (11, 4) and slope 12: y − 4 = 12(x − 11).
With (0, 7) and slope 3: y − 7 = 3x — when x₁ is 0 the bracket holds a bare x, so the brackets come off.
The Sign Trap With Negative Coordinates
This is where most of the marks in this topic go, and it is worth slowing down for.
The formula already contains a minus. If the coordinate is also negative, the two meet and make a plus. Take (−3, −5) with slope 2. Substituting exactly as written:
y − (−5) = 2(x − (−3))
and only then tidying:
y + 5 = 2(x + 3)
The trap is skipping the first line. Written straight from the point, it is very easy to produce "y − 5 = 2(x − 3)" by copying the numbers without their signs — which is a completely different line. Write the brackets in first, resolve them second. This calculator shows both lines for exactly that reason, and only when a coordinate is negative, so the step does not become noise when there is nothing to resolve.
A fractional slope changes nothing about it: (−13, −12) with slope 7/2 gives y + 12 = 7/2(x + 13).
From Two Points — and Why Two Answers Can Both Be Right
Given two points instead of a slope, work the slope out first, then substitute. For (3, 5) and (5, 9): the slope is (9 − 5) ÷ (5 − 3) = 2.
Now the part that causes real anxiety. Either point may be used as (x₁, y₁):
Using (3, 5): y − 5 = 2(x − 3)
Using (5, 9): y − 9 = 2(x − 5)
These look nothing like each other, and both are correct. They are the same line, written from different starting points. If your answer disagrees with the book's, check whether the book simply started from the other point before assuming you have made a mistake.
You can prove it in one step. Expand each one and see where it lands:
y − 5 = 2x − 6, so y = 2x − 1
y − 9 = 2x − 10, so y = 2x − 1
Identical. That expansion is the real test of whether two equations describe the same line, and this calculator does it for you rather than asking you to take it on trust. It also catches a case nothing else does: if you type a slope and two points that disagree with it, it stops and tells you both values instead of quietly preferring one.
When It Does Not Work: Vertical Lines
A vertical line has no point-slope form at all. Its slope is undefined, so there is nothing to put in for m — the formula has no way to start.
It is written x = c instead, where c is the x-coordinate that every point on it shares. The points (4, 1) and (4, 9) give x = 4. This is not a shortcoming of point-slope form so much as of the whole y = … family: no equation of that shape can describe a line that never moves sideways.
A horizontal line is fine, by contrast. Its slope is 0, which is an ordinary number, so (1, 3) and (6, 3) give y − 3 = 0 — which simplifies to y = 3, as you would expect.
How to Use This Calculator
The point is always needed. After that you give either the slope or a second point — whichever the question handed you.
| x₁, y₁ | A point on the line. Always needed |
| m | The slope, if you were given it |
| x₂, y₂ | Or a second point instead, and the slope is worked out |
| ← → ↑ ↓ | Move between the boxes |
| ± | Flips the sign, for a negative coordinate |
| ⁄ and . | Fraction bar and decimal point |
| ⌫ / AC | Delete one character, or clear everything |
| = | Works it out |
Whichever route you are not using fades back as soon as you start on the other, so you never have to wonder how many boxes to fill. Fractions like 7/2 and decimals both work, and everything stays exact.
Open the working and you get the substitution with its brackets, the signs resolved when there are any, both answers when you gave two points, the expansion proving they match, and the same line written the other two common ways for reference.
Point-Slope Form FAQ
What is point-slope form?
How do you write an equation in point-slope form?
What happens with negative coordinates?
Can two different point-slope answers both be right?
How do you find point-slope form from two points?
Can a vertical line be written in point-slope form?
One point, one slope, brackets in place before the signs are resolved — and two different-looking answers that are the same line.