The formula
y − y₁ = m(x − x₁)
It is called point-slope form because the equation is built directly from one point and the slope. If you have any two points on a line, you can find the slope first and then write the equation in seconds — no graph paper required.
What it looks like on a graph
The picture below shows the line from Worked Example 1. The slope triangle makes the formula visible: from (1, 3), move 3 to the right (the run) and 6 up (the rise). Slope = rise ÷ run = 6 ÷ 3 = 2 — computed exactly, not eyeballed.
Worked examples
Two worked examples, step by step
Worked example 1: Find the equation through two points
A line passes through (1, 3) and (4, 9). Find its equation in point-slope form.
Find the slope. m = (y₂ − y₁) / (x₂ − x₁) = (9 − 3) / (4 − 1) = 9 − 3 = 6 / 4 − 1 = 3 = 2
Pick a point. Either point works. Using (1, 3): x₁ = 1, y₁ = 3.
Plug into the formula. y − 3 = 2(x − 1) — that is the equation in point-slope form.
Check with the other point. Does (4, 9) satisfy it? 9 − 3 = 6 and 2(4 − 1) = 6. ✓ It checks out.
Worked example 2: Use a slope and a point to get slope-intercept form
A line has slope −1/2 and passes through (2, 5). Write the equation in slope-intercept form.
Start from point-slope form. y − 5 = −1/2 · (x − 2)
Distribute the slope. y − 5 = −1/2 · x + 1 (because −1/2 · (−2) = 1)
Isolate y. y = −1/2 · x + 1 + 5 = −1/2 · x + 6
Sanity check. At x = 2: y = −1 + 6 = 5. ✓ The point (2, 5) is on the line.
Common mistakes
The four mistakes that cost the most points
These come up over and over when students first meet point-slope form. Each one is a small slip, not a big gap — but each one breaks the equation.
Mixing up the two points
In m = (y₂ − y₁)/(x₂ − x₁), the subscripts must match: both from the same point. Using (y₂ − y₁) over (x₁ − x₂) flips the sign of the slope and the whole line tilts the wrong way.
Sign slips with negative coordinates
With the point (−2, 5), the equation is y − 5 = m(x − (−2)), which simplifies to y − 5 = m(x + 2). Subtracting a negative means adding — write out the double-negative step.
Forgetting the parentheses
y − 3 = 2(x − 1) is not y − 3 = 2x − 1. The slope multiplies the whole quantity (x − 1). Distribute before you simplify anything else.
Stopping one step early
Point-slope form is an answer only when the question asks for it. If the question asks for slope-intercept form or standard form, keep going: distribute, then isolate y.
FAQ
Quick answers
What is the point-slope formula?
y − y₁ = m(x − x₁). It is the equation of a line with slope m passing through the point (x₁, y₁).
How is point-slope form different from slope-intercept form?
Slope-intercept form (y = mx + b) shows the slope and the y-intercept. Point-slope form shows the slope and any one point on the line. You can convert between them with one or two algebra steps.
Can I use either point when two points are given?
Yes. The slope is the same no matter which point you call (x₁, y₁), and the equation you get will be equivalent even if it looks different at first.
Drill the algebra behind it
Fraction fluency is what makes slope arithmetic fast. Print 20 fraction problems with an answer key.