Slope-Intercept Form (of a Linear Equation)

You can graph any line with its slope and y-intercept, but visualizing what the line looks like could be difficult depending on how that information is presented. One of the formats that makes it as easy to work with as possible is called the slope-intercept form of a line:

$y = m x + b$

The m represents the slope of the line while b is the y-intercept value. Put another way, you know this line intersects the y-axis at $(0, b)$ with a defined slope m. That means you have all of the information you need to graph the line easily!

The slope-intercept form of a linear equation can also be seen as a special case of the point-slope form of the equation where $(x_{1}, y_{1})$ is the point $(0, b)$ . Let's take a closer look at how you can use the slope-intercept form to your advantage!

Working with slope-intercept form

The easiest application of slope-intercept form is writing an equation given the slope and y-intercept of a line. For example, let's say that a line has a slope of 3 and crosses the y-axis at $(0, -2)$ . We would write the equation as:

$y = 3 x + (- 2)$

$y = 3 x - 2$

The $y = m x + b$ format tells us to plug in the 3 for m and the -2 for b. Adding -2 is the same as subtracting 2, giving us the equation above.

Finding the slope before writing in slope-intercept form

Of course, you may not always be directly presented with both the intercept and the slope, and a few extra steps may be required. For instance, you might not be given the slope and have to figure it out yourself before writing the $y = m x + b$ form of the equation. Let's look at a line with a y-intercept of $(0, 4)$ that also passes through $(2, 9)$ as an example.

We don't know what m is, but we can figure it out based on the information we have. Remember that the formula for determining slope is $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . We have two points to work with, giving us $\frac{9 - 4}{2 - 0}$ . That works out to $\frac{5}{2}$ , which is our value for m!

From here, the problem is the same as in the example above. Using $y = m x + b$ format, we get:

$y = \frac{5}{2} x + 4$

Getting slope-intercept form through the distributive property

You might also be given an equation that's presented in other formats, such as point-slope, allowing you to simplify it into slope-intercept form with the distributive property. Here is an example:

$y - 1 = - 3 (x + 2)$

First, let's get rid of those parentheses by distributing the -3:

$y - 1 = - 3 x - 6$

Next, we need to get y by itself by adding one to each side:

$y = - 3 x - 5$

The resulting equation is already in slope-intercept form, so we can stop here. You may need to combine like terms to finish some problems though.

Graphing slope-intercept form

The last common usage involving slope-intercept form is graphing an equation on graph paper. For example, let's say that you've been asked to graph $y = - 2 x + 3$ . The equation is already in the right format, so you know that your line intersects the y-axis at $\frac{0}{3}$ and has a slope of -2. Since the slope is telling us that "for every change of 1 unit in the x direction, changes y by the value of the slope", we can easily find other points to plot on our graph. Starting with the point $\frac{0}{3}$ and going right one unit and down 2 (down because the slope is negative) gives us $(1, 3 - 2) = (1, 1)$ .

Remember to extend your line in both directions to indicate that it's infinitely long and not limited to the two specific points you have.

Slope-intercept form practice questions

a. Write an equation in slope-intercept form for a line with a slope of 5 and a y-intercept of $(0, -1)$ .

$y = 5 x - 1$

b. If a line has a y-intercept of. $(0, 4)$ and also passes through $(1, 7)$ , write an appropriate equation for it in slope-intercept form.

First, find the slope using the two points $(0, 4)$ and $(1, 7)$ :

$m = \frac{y2 - y1}{x2 - x1} = \frac{7 - 4}{1 - 0} = 3$

Now we have the slope $m = 3$ and the y-intercept $b = 4$ . Plug them into the slope-intercept form:

$y = 3 x + 4$

c. Rewrite the following equation in slope-intercept form: $y + 5 = 4 (x + 2)$

First, distribute the 4 on the right side:

$y + 5 = 4 x + 8$

Now, subtract 5 from both sides to get the equation in slope-intercept form:

$y = 4 x + 3$

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Help with slope-intercept form is available with Varsity Tutors

Students are expected to have a strong understanding of slope-intercept form after completing Algebra I, with subsequent math courses often asking students to apply it within other contexts. If your student is a little fuzzy on how it works, hoping the problem resolves itself is not the solution. Instead, a private tutor could create personalized lesson plans to help the student in your life develop a deeper understanding of slope-intercept form and related concepts such as one-step linear equations. Reach out to an Educational Director at Varsity Tutors today to learn more about the potential benefits of private instruction.