Formula

Slope-intercept form

Also written: y = mx + b · y equals mx plus b · slope intercept equation

y = mx + b

Slope-intercept form writes a line so its slope and its y-intercept can be read straight off the equation.

What each part means

$m$: the slope — how much y changes per step of 1 in x
$b$: the y-intercept — the y-value where the line crosses the y-axis

When to use it

You want to graph a line quickly, or compare two lines, or read a rate and a starting value out of a real-world equation.

Why this form is the useful one

Every other arrangement hides the two facts you most often want. Here they are both visible without any work:

y = 3x - 4 \quad\Rightarrow\quad m = 3,\ b = -4

To graph it, plot $(0, b)$ , then step across $1$ and up $m$ , and repeat. Two points fix a line.

Why b is the y-intercept

The $y$ -axis is the set of points where $x = 0$ . Substituting:

y = m(0) + b = b

So the line passes through $(0, b)$ , whatever $m$ is. The intercept is not a separate fact bolted on — it is what the equation says at $x = 0$ .

In a real situation

C = 15h + 40

The $15$ is a rate: dollars per hour. The $40$ is a fixed starting amount charged before any hours are worked. Slope is the per-unit cost and the intercept is the flat fee, in almost every problem of this shape.

Worked examples

Write the line with slope $-2$ through $(0, 7)$ .

The point is on the $y$ -axis, so $b = 7$ :

y = -2x + 7

Put $6x + 2y = 10$ into slope-intercept form.

2y = -6x + 10 \quad\Rightarrow\quad y = -3x + 5

Slope $-3$ , intercept $5$ . Converting from standard form means solving for $y$ .

The line it cannot write

A vertical line has undefined slope, so no value of $m$ describes it. Vertical lines are written $x = k$ instead.

Lessons that teach this

Slope-Intercept Form: y = mx + b
Understand y = mx + b, what m and b each control, how to graph a line from it in two moves, and how to rewrite any linear equation into this form.
How to Graph Linear Equations
Graph a line three ways — from slope-intercept form, from intercepts, or from a table — and understand why every point on the line is a solution.
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.
Standard Form of a Linear Equation: Ax + By = C
What standard form is, when it beats slope-intercept form, how to convert between the two, and how to find the slope without rearranging.