The point-slope formula
y − y₁ = m(x − x₁)
Where m is the slope and (x₁, y₁) is
any point the line passes through. Any point on the line works, and each
gives a different-looking but equivalent equation — so two correct
answers can look different while describing exactly the same line.
Watch the signs
This is where point-slope questions are usually lost. The formula subtracts, so a negative coordinate produces a double negative that turns into a plus.
For the point (−3, 5) with slope −2/3, substituting gives
y − 5 = −2/3(x − (−3)), which tidies
to:
y − 5 = −2/3(x + 3)
The + 3 catches people out constantly. This calculator shows
the raw substitution and the tidied version as separate steps, so you can
see exactly where the sign flips.
The three forms, and when each is useful
- Point-slope —
y − y₁ = m(x − x₁). The quickest to write when you know a point and the slope, and the form most homework asks for. - Slope-intercept —
y = mx + b. The easiest to graph, becausebtells you where to start. - Standard —
Ax + By = Cusually written with integer coefficients and no common factor. It is useful for systems of equations and, unlike point-slope and slope-intercept form, can represent a vertical line.
All three describe the same line. Converting between them is rearranging, not recalculating.
If you have two points rather than a slope, the calculator first finds the
slope from the change in y divided by the change in
x. See the Slope Calculator
for a closer look at that calculation.
Why fractions rather than decimals
A slope of 2/3 is exact; 0.6666667 is a rounded approximation. Using the rounded value to calculate the intercept can introduce further rounding. This calculator accepts slopes as fractions, keeps them exact throughout, and gives an intercept like 5/3 rather than 1.67.
Vertical lines have no point-slope form
Point-slope form is built around m. A vertical line has no
slope — the run between any two of its points is zero, and dividing
by zero has no answer — so there is nothing to substitute.
Such a line can only be written x = c. If you enter two
points with the same x-value, the calculator says so rather than
producing an equation that looks right and isn’t.
Related: the equation of the line
To go from two points to the equation of the line through them, the y = mx + b from the same two points does it in one step, with the slope and intercept worked out separately.
Frequently asked questions
What is point-slope form?
y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is a point on the line. It is the most direct way to write a line's equation when those two things are what you know.
Why does my answer look different from the book's?
Probably because you used the other point. Any point on the line gives a valid point-slope equation, and two such equations look different while describing the same line. Convert both to slope-intercept form to check: if they match there, they are the same line.
How do I convert point-slope to slope-intercept form?
Multiply out the bracket, then add y₁ to both sides so y stands alone. The calculator shows this as a separate step so you can follow the rearrangement rather than just take the result.
Can I enter the slope as a fraction?
Yes — type it as 2/3 or -3/4. Whole numbers and decimals work too, and all of them are converted to exact fractions before anything is calculated.
Can I find point-slope form from two points?
Yes. The calculator first finds the slope using
m = (y₂ − y₁) / (x₂ − x₁),
then uses one of the points in the point-slope formula. If the two
points have the same x-value, they form a vertical line, which has
no point-slope form.
What if the two points have the same x-coordinate?
The points define a vertical line. Its slope is undefined because
the change in x is zero, so it cannot be written in point-slope or
slope-intercept form. Its equation is x = c, where
c is the shared x-coordinate.