Math Core

A line is completely determined by two facts: how steep it is and where it crosses the $y$ -axis. Slope-intercept form puts both facts right in the equation, so you can read a line's graph straight off its formula, and write a formula straight off its graph.

Where the form comes from

Think about a gym that charges a $20 sign-up fee plus $15 per month. After $x$ months you've paid

y = 15x + 20.

The two numbers each have a job:

$15$ is the rate of change: each extra month adds $15. On the graph, that's the slope.
$20$ is the starting value: at $x = 0$ months, you've paid $20. On the graph, that's where the line crosses the $y$ -axis.

Every non-vertical line has an equation of this shape.

Definition

Slope-intercept form

The slope-intercept form of a line is

y = mx + b,

where $m$ is the slope and $b$ is the $y$ -intercept. The line crosses the $y$ -axis at the point $(0, b)$ .

Why is $b$ the $y$ -intercept? Every point on the $y$ -axis has $x = 0$ . Substituting $x = 0$ gives $y = m(0) + b = b$ .

Since a linear function is just a line written with function notation, you'll also see $f(x) = mx + b$ . It means exactly the same thing.

Reading $m$ and $b$

To read the slope and intercept, the equation must be in the exact shape $y = mx + b$ , with $y$ alone on one side. Take each sign with its number.

equation	slope $m$	$y$ -intercept $b$
$y = 4x - 7$	$4$	$-7$
$y = -x + 2$	$-1$	$2$
$y = \dfrac{2}{3}x$	$\dfrac{2}{3}$	$0$
$y = 5$	$0$	$5$
$y = 6 - 3x$	$-3$	$6$

The last row is a trap: the terms are in a different order, but the slope is still the coefficient of $x$ , which is $-3$ .

Common mistake

In $y = 4x - 7$ the $y$ -intercept is $-7$ , not $7$ . The form is $y = mx + b$ , so subtracting $7$ means $b = -7$ . The line crosses the $y$ -axis below the origin.

Graphing from the equation

Slope-intercept form gives you a two-step recipe for graphing.

Graphing y = mx + b

Plot the $y$ -intercept $(0, b)$ .
From that point, use the slope $\dfrac{\text{rise}}{\text{run}}$ to find a second point. Then draw the line through both.

Worked example: Graphing with a fractional slope

Graph $y = -\dfrac{2}{3}x + 4$ .

The $y$ -intercept is $4$ , so plot $(0, 4)$ . The slope is $-\dfrac{2}{3}$ : from $(0, 4)$ go down $2$ and right $3$ to reach $(3, 2)$ . Repeat to reach $(6, 0)$ . Draw the line.

Start at the y-intercept (0, 4), then go down 2 and right 3, again and again.Open in grapher →

Check a point: at $x = 3$ , $y = -\dfrac{2}{3}(3) + 4 = -2 + 4 = 2$ , which matches.

A negative slope can be thought of as $\dfrac{-2}{3}$ (down $2$ , right $3$ ) or $\dfrac{2}{-3}$ (up $2$ , left $3$ ). Both land on the same line.

Writing the equation of a line

To write $y = mx + b$ , you need two numbers: $m$ and $b$ . How you find them depends on what you're given.

Worked example: From a graph

Write the equation of the line.

y = 3x/2 - 2(0, -2)(2, 1)Open in grapher →

The line crosses the $y$ -axis at $(0, -2)$ , so $b = -2$ . From $(0, -2)$ to $(2, 1)$ the line rises $3$ and runs $2$ , so $m = \dfrac{3}{2}$ .

y = \frac{3}{2}x - 2.

When the $y$ -intercept isn't given, find the slope first, then substitute one point to solve for $b$ .

Worked example: From two points

Write the equation of the line through $(2, 1)$ and $(6, 9)$ .

Slope. $m = \dfrac{9 - 1}{6 - 2} = \dfrac{8}{4} = 2$ , so the equation looks like $y = 2x + b$ .

Intercept. The point $(2, 1)$ is on the line, so it must make the equation true:

\begin{aligned} 1 &= 2(2) + b \\ 1 &= 4 + b \\ -3 &= b \end{aligned}

The equation is $y = 2x - 3$ . Check the other point: $2(6) - 3 = 9$ , as it should.

Tip

Always test your equation with a point you didn't use to find $b$ . If it satisfies the equation, you almost certainly have the right line.

Rewriting into slope-intercept form

Sometimes an equation arrives in a different shape, like $3x + 2y = 10$ . Solve for $y$ (you practiced this with literal equations):

\begin{aligned} 3x + 2y &= 10 \\ 2y &= -3x + 10 && \text{subtract } 3x \\ y &= -\frac{3}{2}x + 5 && \text{divide every term by } 2 \end{aligned}

Now you can read it: slope $-\dfrac{3}{2}$ , $y$ -intercept $5$ .

Worked example: A model

A candle is $30$ cm tall and burns down $2.5$ cm per hour. Write an equation for its height $y$ after $x$ hours. When does it burn out?

The starting value is $30$ and the rate is $-2.5$ (the height goes down), so $y = -2.5x + 30$ .

It burns out when $y = 0$ : $0 = -2.5x + 30$ , so $2.5x = 30$ and $x = 12$ hours.

Practice

Practice 1

What is the slope of the line $y = -4x + 7$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

What is the $y$ -intercept of the line $y = 5 - 3x$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Write the equation of the line with slope $3$ and $y$ -intercept $-2$ . Enter it in the form $y = mx + b$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 4

Write the equation of the line shown, in the form $y = mx + b$ .

y = -2x + 3(0, 3)(2, -1)Open in grapher →

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 5

Which is $3x + 2y = 10$ written in slope-intercept form?

Practice 6

A line has slope $\dfrac{1}{2}$ and passes through $(4, 10)$ . Write its equation in the form $y = mx + b$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 7

Write the equation of the line through $(-1, 4)$ and $(3, -4)$ , in the form $y = mx + b$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 8

A plumber charges a $60 visit fee plus $45 per hour. Write an equation for the cost $y$ of a job lasting $x$ hours. How much, in dollars, does a $3.5$ -hour job cost?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Where the form comes from

Reading mmm and bbb

Graphing from the equation

Writing the equation of a line

Rewriting into slope-intercept form

Practice

Reading $m$ and $b$