Math Core

You've seen that a line has a slope and a $y$ -intercept. When a linear equation is written as $y = mx + b$ , both numbers are sitting right there in the equation. This form is the one you'll use most in Algebra 1, because it lets you read a line's graph straight from its equation.

Reading m and b

Look at $y = 2x + 3$ . Make a quick table:

$x$	0	1	2	3
$y$	3	5	7	9

When $x = 0$ , $y = 3$ , so the line crosses the $y$ -axis at $(0, 3)$ . Each time $x$ goes up by 1, $y$ goes up by 2, so the slope is 2. The two numbers in the equation are exactly the slope and the $y$ -intercept.

Slope-intercept form

A linear equation in slope-intercept form looks like

y = mx + b

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

Worked example: Finding m and b

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

$y = -3x + 4$
$y = \dfrac{1}{2}x - 6$
$y = 5x$
$y = 8 - x$

Solutions.

$m = -3$ , $b = 4$ .
$m = \dfrac{1}{2}$ , $b = -6$ . Subtracting 6 is adding $-6$ .
$m = 5$ , $b = 0$ . The line goes through the origin.
Rewrite as $y = -x + 8$ . Then $m = -1$ and $b = 8$ .

Common mistake

The slope is the number multiplied by $x$ , not simply the first number you see. In $y = 8 - x$ , the slope is $-1$ and the $y$ -intercept is 8. Also keep the sign: in $y = 4x - 7$ , $b$ is $-7$ , not 7.

Graphing from the equation

You don't need a table to graph $y = mx + b$ .

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

Worked example: Graphing y = (2/3)x − 1

Graph $y = \dfrac{2}{3}x - 1$ .

Start at the $y$ -intercept, $(0, -1)$ . The slope is $\dfrac{2}{3}$ , so go right 3 and up 2 to reach $(3, 1)$ . Do it again to reach $(6, 3)$ . Draw the line through the three points.

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

For a negative slope such as $-\dfrac{3}{4}$ , go right 4 and down 3.

Writing the equation of a line

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

Worked example: From two points

Write the equation of the line through $(2, 5)$ and $(4, 9)$ .

Find the slope: $m = \dfrac{9 - 5}{4 - 2} = \dfrac{4}{2} = 2$ .

Find $b$ : the equation so far is $y = 2x + b$ . Substitute one of the points, say $(2, 5)$ :

\begin{aligned} 5 &= 2(2) + b \\ 5 &= 4 + b \\ 1 &= b \end{aligned}

The equation is $y = 2x + 1$ . Check with the other point: $2(4) + 1 = 9$ . ✓

Rewriting into slope-intercept form

Some equations, like $3x + y = 5$ , have $x$ and $y$ on the same side. Solve for $y$ to put them in slope-intercept form, the same way you solved equations earlier in this course.

Worked example: Solving for y

Write $6x + 2y = 8$ in slope-intercept form.

\begin{aligned} 6x + 2y &= 8 \\ 2y &= -6x + 8 && \text{subtract } 6x \\ y &= -3x + 4 && \text{divide every term by 2} \end{aligned}

The slope is $-3$ and the $y$ -intercept is 4.

Starting value and rate

In a real situation, $b$ is often a starting amount and $m$ is a rate.

A gym charges a $25 sign-up fee plus $15 per month. The cost after $x$ months is

y = 15x + 25.

The slope 15 is the cost per month, and the $y$ -intercept 25 is what you pay at month 0, before any monthly fees.

Tip

After you write an equation, test it with a point you know. If the line should pass through $(4, 9)$ , plug in $x = 4$ and make sure you get 9.

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 $y = 7 - 2x$ ?

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

Practice 3

Write the equation of the line with slope 5 and $y$ -intercept $-3$ .

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

Practice 4

Write the equation of this line.

y = -3x/4 + 2(0, 2)(4, -1)Open in grapher →

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

Practice 5

Write the equation of the line through $(1, 4)$ and $(3, 10)$ .

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

Practice 6

Which equation is $4x + 2y = 10$ written in slope-intercept form?

Practice 7

A plumber charges $40 to come to your house plus $65 for each hour of work. Write an equation for the total cost $y$ , in dollars, for $x$ hours of work.

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

Practice 8

A line has slope $-2$ and passes through $(3, 1)$ . What is its $y$ -intercept?

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