Point-slope form is the quickest way to write the equation of a line when you know its slope and one point on it:
y − y₁ = m(x − x₁)
Here m is the slope and (x₁, y₁) is the known point. The x and y without numbers stay as letters.
Where it comes from
Take any point (x, y) on the line. The slope between it and the known point (x₁, y₁) must equal m, so (y − y₁) ÷ (x − x₁) = m. Multiply both sides by (x − x₁) and you get point-slope form.
Worked examples
Example 1
Write the equation of the line with slope 4 through (1, −2).
y − (−2) = 4(x − 1), which simplifies to y + 2 = 4(x − 1).
To change it to slope-intercept form, expand and rearrange: y + 2 = 4x − 4, so y = 4x − 6.
Example 2: from two points
Find the equation through (3, 1) and (5, 7).
Slope: m = (7 − 1) ÷ (5 − 3) = 6/2 = 3.
Using (3, 1): y − 1 = 3(x − 3), which expands to y = 3x − 8.
Using (5, 7) instead gives y − 7 = 3(x − 5). It looks different, but it also expands to y = 3x − 8. Both are correct.
See Example 2 in Slope Finder →
Watch the signs
The form has minus signs built in, so a negative coordinate turns into a plus:
- Point (−4, 3), slope 2: y − 3 = 2(x + 4)
- Point (5, −6), slope −1: y + 6 = −(x − 5)
- Point (0, 2), slope ½: y − 2 = ½x
Going the other way, you can read the point off an equation by flipping the signs. y + 1 = 5(x − 2) passes through (2, −1) and has slope 5.
When to use point-slope form
- A question gives you a point and a slope.
- You have two points and have already worked out the slope.
- You need a line parallel or perpendicular to another line through a given point.
If the question asks for the answer in slope-intercept form or standard form, start with point-slope form and rearrange.