7.6.1 Slope-Intercept Form of a Linear Equation

Learning Objectives

Write a linear equation in slope-intercept form.
Identify the slope and $y$ -intercept given a linear equation.
Write the linear equation of a graphed line.
Graph a linear equation in slope-intercept form.

Key words

Slope-intercept form: a linear equation in the form $y=mx+b$

Slope-Intercept Form

When an equation is written in the form $y = mx+b$ it is said to be in slope-intercept form. Let’s see why it has that name. Consider the equation $y=2x+3$ . If we set $x=0$ , then $y=2(0)+3=3$ . This means that the point $(0, 3)$ is the $y$ -intercept of the line formed by the equation.

Figure 1. Graph of $y=2x+3$

The graph in figure 1, also shows that the point $(-2 -1)$ lies on the graph. Consequently, the slope of the line is $m=\frac{y_2-y_1}{x_2-x_1}=\frac{3-(-1)}{0-(-2)}=\frac{4}{2}=2$ . This is the coefficient of $x$ in the equation $y=2x+3$ . So with slope = $m$ and the $y$ -intercept $= (0, b)$ , the equation written in the form $y=mx+b$ immediately tells us the slope, $m$ , and the $y$ -intercept, $(0, b)$ .

Slope-Intercept Form of a Linear Equation

In the equation $y=mx+b$ ,

$m$ is the slope of the graph.
$b$ is the $y$ -coordinate of the $y$ -intercept $(0, b)$ of the graph.

Examples

Find the slope and $y$ -intercept of the line with the given equation:

1. $y=-3x+7$

2. $y=\frac{2}{3}x-\frac{1}{5}$

3. $2x-3y=6$

Solution

1. $m=-3\text{ and }b=7\text{ so the slope is }-3\text{ and the }y\text{-intercept is the point } (0, 7)$

2. $m=\frac{2}{3}\text{ and }b=-\frac{1}{5}\text{ so the slope is }\frac{2}{3}\text{ and the }y\text{-intercept is the point } \left (0,-\frac{1}{5}\right )$

3. We first have to rearrange the equation to solve for $y$ :

$\begin{equation}\begin{aligned}2x-3y & = 6 \\ -3y & = -2x+6 \\ y & = \frac{2}{3}x-2\end{aligned}\end{equation}$

$\text{So, }m=\frac{2}{3}\text{ and }b=-2\text{ so the slope is }\frac{2}{3}\text{ and the }y\text{-intercept is the point } \left (0,2\right )$

Try It

Find the slope and $y$ -intercept of the line with the given equation:

1. $y=6x-1$

2. $y=\frac{4}{7}x-\frac{2}{3}$

3. $5x-y=10$

Show Answer

1. $m=6\text{ and }b=-1\text{ so the slope is }6\text{ and the }y\text{-intercept is the point } (0, -1)$

2. $m=\frac{4}{7}\text{ and }b=-\frac{2}{3}\text{ so the slope is }\frac{4}{7}\text{ and the }y\text{-intercept is the point } \left (0,-\frac{2}{3}\right )$

3. We first have to rearrange the equation to solve for $y$ :

$\begin{equation}\begin{aligned}5x-y & = 10 \\ y & = 5x-10\end{aligned}\end{equation}$

$\text{So, }m=-5\text{ and }b=10\text{ so the slope is }-5\text{ and the }y\text{-intercept is the point } \left (0,10\right )$

We can also find the linear equation that represents te graph of a line If we know the slope and the $y$ -intercept.

Example

Write the equation of the line that has a slope of $\displaystyle \frac{1}{2}$ and a y-intercept of $(0, −5)$ .

Solution

Substitute the slope, $m$ , into $y=mx+b$ .

$\displaystyle y=\frac{1}{2}x+b$

Substitute $b$ , into the equation.

$\displaystyle y=\frac{1}{2}x-5$

Answer

$y=\frac{1}{2}x-5$

Example

Write the equation of the line in the graph by identifying the slope and $y$ -intercept.

Solution

Identify the point where the graph crosses the $y$ -axis $(0,3)$ . This means that $b=3$ . Identify one other point and draw a slope triangle to find the slope.

Slope: $m=\frac{-2}{3}$ .

Substitute $m$ and $b$ into the slope-intercept equation.

$y=mx+b\\y=\frac{-2}{3}x+b\\y=\frac{-2}{3}x+3$

Answer

$y=\frac{-2}{3}x+3$

When we are given an equation in the slope-intercept form of $y=mx+b$ , we can identify the slope and y-intercept and use these to graph the equation. When we have an equation in slope-intercept form we can graph it by first plotting the $y$ -intercept, then using the slope to find a second point, and connecting the dots.

Example

Graph $y=\frac{1}{2}x-4$ using the slope-intercept equation.

Solution

First, plot the $y$ -intercept.

The y-intercept plotted at negative 4 on the y axis.

Now use the slope to count up or down and over left or right to the next point. This slope is $\frac{1}{2}$ , so we can count up one and right two—both positive because both parts of the slope are positive.

Connect the dots.

A line crosses through negative 4 on the y-axis and has a slope of 1/2.

NOTE: it is important for the equation to first be in slope-intercept form. If it is not, we have to solve it for $y$ so we can identify the slope and the $y$ -intercept.

Try It

Watch the video below for another example of how to write the equation of the line, when given a graph, by identifying the slope and y-intercept.

NEW CHAPTER 7 Linear Equations in Two Variables