Math Core

If you know where a line crosses the $y$ -axis and how steep it is, you know the whole line. Slope-intercept form packs both facts into one short equation, so you can read a line's graph straight from its equation, and the other way around.

The y-intercept

The y-intercept is where a line crosses the $y$ -axis. Every point on the $y$ -axis has $x = 0$ , so the $y$ -intercept is the output when the input is 0.

Look at the line $y = 2x + 3$ . When $x = 0$ , $y = 2(0) + 3 = 3$ . So the line crosses the $y$ -axis at $(0, 3)$ . And each time $x$ goes up by 1, $y$ goes up by 2, so the slope is 2.

y = 2x + 3 crosses the y-axis at 3 and rises 2 for every 1 step right.Open in grapher →

Both numbers are sitting right there in the equation.

Slope-intercept form

y = mx + b

$m$ is the slope.
$b$ is the y-intercept: the line crosses the $y$ -axis at $(0, b)$ .

Worked example: Reading m and b

Find the slope and $y$ -intercept of each line.

$y = -3x + 5$
$y = \dfrac{1}{2}x - 4$
$y = 7 - 2x$

Solutions.

$m = -3$ , $b = 5$ .
$m = \dfrac{1}{2}$ , $b = -4$ . (Subtracting 4 is adding $-4$ .)
Rearrange first: $y = -2x + 7$ . So $m = -2$ and $b = 7$ .

Common mistake

The slope is the number multiplying $x$ , not the first number you see. In $y = 7 - 2x$ the slope is $-2$ , not 7. Rewrite the equation in the order $mx + b$ if that helps.

Graphing from the equation

To graph $y = mx + b$ :

Plot the $y$ -intercept $(0, b)$ .
Use the slope as rise over run to step to a second point.
Repeat once more to check, then draw the line.

Worked example: Graph y = (2/3)x − 1

The $y$ -intercept is $-1$ , so start at $(0, -1)$ . The slope is $\dfrac{2}{3}$ : go right 3 and up 2 to reach $(3, 1)$ . Do it again to reach $(6, 3)$ .

y = 2x/3 - 1(0, -1)(3, 1)(6, 3)Open in grapher →

Writing the equation of a line

To write $y = mx + b$ , you need two things: $m$ and $b$ .

Worked example: From a graph

Write the equation of this line.

y = -2x + 4(0, 4)(2, 0)Open in grapher →

The line crosses the $y$ -axis at 4, so $b = 4$ . From $(0, 4)$ to $(2, 0)$ the rise is $-4$ and the run is 2, so $m = \dfrac{-4}{2} = -2$ .

The equation is $y = -2x + 4$ .

Worked example: From two points

Write the equation of the line through $(2, 7)$ and $(5, 13)$ .

Step 1: slope. $m = \dfrac{13 - 7}{5 - 2} = \dfrac{6}{3} = 2$ .

Step 2: intercept. So far, $y = 2x + b$ . The point $(2, 7)$ is on the line, so put in $x = 2$ and $y = 7$ :

7 = 2(2) + b \quad\Rightarrow\quad 7 = 4 + b \quad\Rightarrow\quad b = 3

The equation is $y = 2x + 3$ .

Check with the other point: $2(5) + 3 = 13$ . ✓

Tip

After you write an equation, test it on a point you were given. If a given point doesn't fit, something went wrong.

Practice

Practice 1

What is the slope of the line $y = -4x + 9$ ?

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

Practice 2

What is the $y$ -intercept of the line $y = 5x - 8$ ?

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 $-2$ .