One point and a slope pin down a line
Any line on the coordinate plane is fixed by exactly two facts: how steep it is (the slope, m) and one place it passes through (a point, (x₁, y₁)). Point-slope form is the equation template that takes those two facts and hands back the whole line:
y - y₁ = m(x - x₁)
The interactive above is that template made draggable. Move the anchor point, slide Slope m, and the Point-slope form box writes the equation from exactly those two ingredients, while the Same line, slope-intercept box shows the identical line rewritten as y = mx + b. A dashed triangle at the point shows the slope as a step of +1 across and m up.
The point-slope formula
The formula is not something to memorize cold. It falls out of the definition of slope in one step, and seeing that step is the “aha” of this topic. The slope between the known point (x₁, y₁) and any other point (x, y) on the line is
m = (y - y₁) / (x - x₁)
Multiply both sides by (x - x₁) to clear the fraction:
y - y₁ = m(x - x₁)
That is the entire derivation. Point-slope form is the slope formula with the division undone. It says: pick any point (x, y) on this line, measure the slope back to (x₁, y₁), and you must get m. Every point that satisfies the equation is on the line, and every point on the line satisfies it.
| Symbol | Meaning |
|---|---|
| m | the slope of the line (rise over run) |
| (x₁, y₁) | one known point on the line; any point of the line works |
| (x, y) | stands for every point on the line; stays as letters |
The subscript 1 just means “the specific point we know.” x₁ and y₁ get replaced by numbers; x and y do not. In y - 2 = 2(x - 1), the known point is (1, 2), the slope is 2, and that is everything the equation says.
How to use point-slope form
Given a slope and a point, there are three steps, and none of them is algebra:
- Identify m, the slope.
- Identify the known point and label it (x₁, y₁).
- Substitute them into y - y₁ = m(x - x₁). Leave x and y as letters.
Worked example. Write the equation of the line with slope 3 through (4, 7).
Here m = 3 and (x₁, y₁) = (4, 7), so:
y - 7 = 3(x - 4)
That is a complete, correct equation of the line. Unless a question specifically asks for slope-intercept form, you can stop here; converting is a separate, easy move covered below.
The sign trap: subtracting a negative
The formula subtracts the coordinates of the known point. So when a coordinate is negative, the substitution collides with the minus sign and flips it to a plus, and this flip is the single most common point-slope error.
Worked example. Write the equation of the line through (-1, 3) with slope -2.
Substituting x₁ = -1 literally gives x - (-1):
y - 3 = -2(x - (-1))
and subtracting a negative is adding:
y - 3 = -2(x + 1)
The trap runs in both directions. Writing the equation, students often produce y - 3 = -2(x - 1) instead, which passes through (1, 3), a different line entirely. Reading an equation, a plus sign inside the parentheses always signals a negative x₁: in y + 5 = 2(x - 4) the known point is (4, -5), because y + 5 is really y - (-5).
Converting to slope-intercept form
Two moves turn point-slope form into y = mx + b: distribute the slope, then add y₁ to both sides.
Worked example. Convert y - 3 = -2(x + 1) to slope-intercept form.
Distribute the -2:
y - 3 = -2x - 2
Add 3 to both sides:
y = -2x + 1
The slope is still -2 (converting never changes it) and the y-intercept turns out to be
- The Same line, slope-intercept box in the interactive performs exactly this conversion live every time you move the point or the slider. See slope-intercept form for everything the y = mx + b version is good for.
Two points? Find the slope first
Point-slope form is also the natural tool when a problem gives you two points and no slope. Compute m from the slope formula, then plug in either point; both give the same line.
Worked example. Write the equation of the line through (1, 5) and (3, 9).
First the slope:
m = (9 - 5) / (3 - 1) = 4 / 2 = 2
Using the first point, (1, 5): y - 5 = 2(x - 1), which distributes to y - 5 = 2x - 2, so y = 2x + 3.
Using the second point, (3, 9): y - 9 = 2(x - 3), which distributes to y - 9 = 2x - 6, so y = 2x + 3.
Identical, exactly as the last Investigate showed: the point you pick changes how the point-slope version looks, never which line it is.
When to prefer point-slope form
| You are given | Best starting form |
|---|---|
| the slope and the y-intercept | slope-intercept, y = mx + b |
| the slope and any other point | point-slope |
| two points | point-slope, after computing m |
| an equation you want to graph quickly | slope-intercept |
Slope-intercept form is really a special case of point-slope form: anchor the line at its y-intercept (0, b), and y - b = m(x - 0) simplifies straight to y = mx + b. Point-slope is the general tool; slope-intercept is that tool pre-aimed at one particular point. That is also why point-slope keeps showing up later, from writing equations of parallel and perpendicular lines through a given point to tangent lines in calculus: it is the form that never needs the y-intercept handed to it.
The interactive above also works as a point-slope form calculator: drag the anchor to your point, set m, and both equation boxes fill themselves in.
It’s free to embed on your own site or LMS. Next, see where m itself comes from in slope, or work with the converted form directly in slope-intercept form.