Formula

Point-slope form

Also written: y minus y1 equals m times x minus x1 · point slope equation

y - y_1 = m(x - x_1)

Point-slope form writes the equation of a line from one point on it and its slope, with no need to find the y-intercept first.

What each part means

$m$: the slope
$(x_1, y_1)$: any known point on the line
$(x, y)$: a general point on the line — these stay as letters

When to use it

You know a point and a slope and want the equation immediately. It is the fastest route when the given point is not the y-intercept.

It is the slope formula rearranged

Not a new fact — the slope formula with the denominator cleared.

m = \frac{y - y_1}{x - x_1}

Multiply both sides by $x - x_1$ :

y - y_1 = m(x - x_1)

That is the whole derivation. Anything true of one is true of the other.

Why the subscripts matter

$x_1$ and $y_1$ are numbers you substitute. Plain $x$ and $y$ are variables that stay as letters, because they stand for every point on the line at once.

Substituting for all four is the most common error here, and it collapses the equation to a statement about one point rather than a line.

Worked examples

Write the line through $(2, 5)$ with slope $3$ .

y - 5 = 3(x - 2)

That is a complete answer. Expanding gives slope-intercept form:

y = 3x - 1

Write the line through $(-4, 1)$ with slope $-\tfrac{1}{2}$ .

Subtracting a negative becomes addition:

y - 1 = -\tfrac{1}{2}(x + 4)

The form subtracts both coordinates. A point at $(-4, 1)$ gives $x - (-4)$ , which is $x + 4$ . Negative coordinates flip the sign inside the bracket, and that is where most slips happen — see point-slope form.

Lessons that teach this

Point-Slope Form: Writing a Line from One Point
Learn point-slope form, why it is the slope formula rearranged, and how to use it to write a line's equation from a single point and a slope.
How to Write the Equation of a Line from Two Points
Find a line's equation from any two points: compute the slope, substitute into point-slope form, then simplify — with worked examples and full solutions.
Parallel and Perpendicular Lines: The Slope Rules
Parallel lines have equal slopes and perpendicular lines have negative reciprocal slopes — why both rules are true, and how to write equations using them.