Part 9 of 36 · LinesSee the whole course
Point-slope form, explained
Point-slope form writes the equation of a line when you know one point (x₁, y₁) on it and its slope m: y − y₁ = m(x − x₁). Put the point and the slope in, and the equation is done. There is no need to find b first.
Why a second form of the equation?
With slope-intercept form, y = mx + b, you must know b, the height where the line crosses the y-axis. Often you do not. You are told a point somewhere in the middle of the graph, and a slope. Point-slope form lets you start from exactly that.
It says one thing in symbols: from the point, the rise is m times the run. y − y₁ is how far up you are from the point. x − x₁ is how far across you are from the point. For a line, up = slope × across.
The recipe
- Write the form: y − y₁ = m(x − x₁).
- Replace m with the slope.
- Replace x₁ and y₁ with the point. Leave plain x and y alone.
- If you want y = mx + b, open the bracket and move the number to the right.
Try it. Move the point and the slope and watch all three forms of the same line.
- y − 2 = 3(x − 1)
- y − 2 = 3x − 3
- y = 3x − 1
Example 1: slope 2 through (3, 5)
Write the form: y − y₁ = m(x − x₁)
Put in m = 2, x₁ = 3, y₁ = 5: y − 5 = 2(x − 3)
Check the point: 5 − 5 = 2(3 − 3), and 0 = 0 ✓
As y = mx + b: y − 5 = 2x − 6, so y = 2x − 1
The point (4, 7) should be on this line too: 7 − 5 = 2(4 − 3) gives 2 = 2 ✓.
Example 2: with negatives, slope −3 through (−2, 4)
Put in m = −3, x₁ = −2, y₁ = 4: y − 4 = −3(x − (−2))
Subtracting a negative adds: y − 4 = −3(x + 2)
Open the bracket: y − 4 = −3x − 6
Add 4: y = −3x − 2
Check (−2, 4): −3(−2) − 2 = 6 − 2 = 4 ✓
This is where most slips happen. A point with x₁ = −2 turns x − x₁ into x + 2. The sign in the bracket is the opposite of the sign of the point.
Example 3: from two points, (1, 2) and (3, 8)
You do not have the slope yet, so find it first: (8 − 2) ÷ (3 − 1) = 3. Now pick either point.
With (1, 2): y − 2 = 3(x − 1), which is y = 3x − 1
With (3, 8): y − 8 = 3(x − 3), which is y = 3x − 1
Both give the same line, because the two point-slope equations are the same line written from two different starting points. This is the same answer as in finding the equation from two points, with one step less: you never have to solve for b.
Where the formula comes from
Take the point (x₁, y₁) and any other point (x, y) on the line. The slope between them is rise ÷ run:
(y − y₁) ÷ (x − x₁) = m
Multiply both sides by (x − x₁): y − y₁ = m(x − x₁)
So point-slope form is just the slope formula, rearranged so that it has no fraction in it. If the slope formula is new to you, start with how to find the slope from two points.
Watch it explained in under a minute
Read the video transcript
You know one point and the slope. The point is (3, 5). The slope is 2.
Point-slope form looks like this. y − y_1 = m(x − x_1). The point goes in x_1 and y_1. The slope goes in m.
Put them in. y − 5, = 2 times, (x − 3). That is the line.
Check it with another point. When x = 4, y − 5 = 2 × 1, so y = 7.
Want y = mx + b? Open the bracket, and add 5. y = 2x − 1.
Want practice questions and more examples? Full guide: pipmath.app/blog
The five mistakes to avoid
- Getting the signs of the point wrong. For the point (−2, 4) the equation starts y − 4 = m(x + 2), not (x − 2).
- Swapping the coordinates: using the x-value in the y slot. The y of the point goes next to y.
- Replacing x and y as well. The plain x and y stay as letters. Only x₁, y₁ and m are replaced.
- Forgetting to multiply the whole bracket when converting. −3(x + 2) is −3x − 6, not −3x + 2.
- Using the form for a vertical line. A vertical line has no slope, so it is just x = a number.
Try it yourself
Write each line in point-slope form, then as y = mx + b. Tap a question to see the answer.
1. Slope 4 through (1, 3)
y − 3 = 4(x − 1). Open the bracket: y − 3 = 4x − 4, so y = 4x − 1. Check (1, 3): 4 − 1 = 3 ✓.
2. Slope −½ through (4, 1)
y − 1 = −½(x − 4). Open the bracket: y − 1 = −½x + 2, so y = −½x + 3. Check (4, 1): −2 + 3 = 1 ✓.
3. Through (−1, 2) and (1, 6)
Slope: (6 − 2) ÷ (1 − (−1)) = 4 ÷ 2 = 2. Using (−1, 2): y − 2 = 2(x + 1). Open it: y − 2 = 2x + 2, so y = 2x + 4. Check (1, 6): 2 + 4 = 6 ✓.
Where this fits in
Now you have two ways to write a line: slope-intercept form when you know where it starts, and point-slope form when you know one point on it. Next in this course: the two special cases that do not fit y = mx + b, horizontal and vertical lines.