Find the Slope and y-intercept: Mini-Lesson Review

Mini Lesson Question

Question #2: Find the Slope and $y$ -intercept

What is the slope and

y

-intercepts of the line

3x - y= 12

$m=\frac{3}{12}$ ; y-intercept $= (3, - 12)$
$m=\frac{12}{3}$ ; y-intercept $= (3, - 1)$
$m=\frac13$ ; y-intercept $= (- 12, 0)$
$m=\frac31$ ; y-intercept $= (0, - 12)$

Graph a Line Using Its Slope and $y$ -intercepts

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using one point and the slope of the line. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.

Let’s look at the graph of the equation $y = \frac{1}{2} x + 3$ and find its slope and $y$ -intercept.

The figure shows the graph of a straight line on the coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, 3), (2, 4), and (4, 5). A right triangle is drawn by connecting the three points (2, 4), (2, 5), and (4, 5). The vertical side of the triangle is labeled “Rise equals 1”. The horizontal side of the triangle is labeled “Run equals 2”. The line is labeled y equals 1 divided by 2 times x plus 3.

The red lines in the graph show us the rise is 1 and the run is 2. Substituting into the slope formula:

$m = \frac{r i s e}{r u n}$

$m = \frac{1}{2}$

The $y$ -intercept is 3.

Look at the equation of this line.

Look at the slope and $y$ -intercepts.

Slope $m$ equals 1 divided by 2 and $y$ -intercept (0, 3). The 1 divided by 2 is emphasized in red. The 3 is emphasized in blue.

When a linear equation is solved for $y$ , the coefficient of the $x$ term is the slope and the constant term is the $y$ -coordinate of the $y$ -intercept. We say that the equation $y = \frac{1}{2} x + 3$ is in slope–intercept form. Sometimes the slope–intercept form is called the “y-form.”

$m equals 1 divided by 2; $y$-intercepts is (0, 3). y equals 1 divided by 2 x plus 3. y equals m x plus b. The m and 1 divided by 2 are emphasized in red. The b and 3 are emphasized in blue.$

The slope–intercept form of an equation of a line with slope $m$ and $y$ -intercept, (0,b) is $y = m x + b$ .

Let’s practice finding the values of the slope and $y$ -intercept from the equation of a line.

Example

Identify the slope and $y$ -intercept of the line from the equation:

$y = - \frac{4}{7} x - 2$

Write the slope-intercept form of the equation of the line.	$y = m x + b$
Write the equation of the line.	$y = - \frac{4}{7} x - 2$
Identify the slope.	$m = - \frac{4}{7}$
Identify the $y$ -intercept.	$y$ -intercept is (0, −2)

This table illustrates the process of writing and analyzing the slope-intercept form of a linear equation, showing how to identify its slope and y-intercept.

Try it

Identifying the Slope and $y$ -intercepts of a Line

Identify the slope and $y$ -intercept of the line from the equation:

$y = - \frac{1}{3} x + 3$

Here is how to find the slope and $y$ -intercept of the equation $y$ = − $\frac{1}{3} x + 3$ :

Write the slope-intercept form of the equation of the line.	$y = m x + b$
Write the equation of the line.	$y$ = − $\frac{1}{3} x + 3$
Identify the slope.	$m$ = − $\frac{1}{3}$
Identify the $y$ -intercept.	$y$ -intercept is (0, 3)

Illustrates how to identify the slope, y-intercept, and the general slope-intercept form of a linear equation.

Check Your Understanding

What is the slope and $y$ -intercepts of the line $6 x + 2 y = 18$ ?

$m = - 3$ ; $y$ -intercepts = $(9, 0)$
$m = 6$ ; $y$ -intercepts = $(0, 18)$
$m = - 6$ ; $y$ -intercepts = $(0, 18)$
$m = - 3$ ; $y$ -intercepts = $(0, 9)$

$m = - 3$ ; $y$ -intercepts $= (0, 9)$ .

Check yourself: Write the equation in slope-intercept form, $y = m x + b$ , where $m$ is the slope and $b$ is the $y$ -intercepts: $6 x + 2 y = 18$ ; $y = - 3 x + 9$ .