## Changing an Equation into Slope-Intercept Form

## Introduction

Before doing this lesson, you should have a grasp of the concept of slope as well as a good idea of how to use a table to draw lines on a coordinate plane. See the menu of algebra links for lessons on these topics.

The slope-intercept form of an equation is y = mx + b, where m is the slope and b is the y-intercept. However, not all equations are given in this form.

Equations that are not in this form may be more difficult to graph. Before looking at the lesson, consider the equation 8y = 24 – 4x. Can you find any coordinates that work for this equation? Can you determine the slope of this line or the x or y intercept?

Drawing the line of the equation 8y = 24 – 4x can be done, but this line can be graphed more easily if the equation is rewritten in slope-intercept form. In this lesson, you will learn how to change equations into slope-intercept form to allow you to analyze them and draw their graph more easily.

## Lesson

In the introduction, you were asked to take a closer look at the equation 8y = 24 – 4x. Finding coordinates for this equation can be done by “plugging in” values of x.

If x = 0, then 8y = 24 and y = 3. This is the coordinate (0, 3)

If x = 1, then 8y = 24 – (4)1, 8y = 20, and y = 2.5. Coordinate (1, 2.5)

If x = 2, then 8y = 24 – (4)(2), 8y = 16 and y = 2. Coordinate (2, 2)

The graph of the equation 8y = 24 – 4x is shown to the left. The graph makes a straight line and this line appears to have a negative slope and a y-intercept of 3. One can look at the graph and determine the slope and the y-intercept visually, but it is also possible to find these two characteristics of the line using algebra.

The slope-intercept form of an equation is y = mx + b, where m is the slope and b is the y-intercept. To change our original equation into slope-intercept form, simply solve the equation for y.

In the equation above, the y-term has been isolated on the left side of the equation and the right side has been rearranged into slope-intercept form (mx + b). So the equation has been 8y = 24 – 4x can be changed into y = -½x + 3. The slope is -½ and the y-intercept is 3.

Finding the slope and y-intercept of an equation can often be done without drawing a graph. See if you can find the slope and y-intercept of the equation without drawing a graph.

**Example 1:** Find the slope and y-intercept of the line 5x + 5y = 10.

Solution:

**Example 2:** Find the slope and y-intercept of the line 2y = 6(x + 3)

Solution:

Examples 1 and 2 result in equations whose slopes and y-intercepts are integers. When simplifying many equations, however, you will often run into fractions for the slope, y-intercept, or both. Example 3 demonstrates fractional results for the slope and y-intercept.

**Example 3:** Find the slope and y-intercept of the line 5y = 24 + 8x

Solution:

You can use the rules of algebra to change any 2-variable equation into slope-intercept form. Remember that the simplified (slope-intercept) form can be useful to quickly identify the slope and y-intercept of the line.

Even though graphing is not covered in this lesson, the purpose of changing an equation into slope-intercept form is often to draw the graph. Drawing the graph of a line is easiest when the equation is in slope-intercept form.

**Related Links:**** **

For more information on graphing lines and related topics, try one of the links below.

**Resource Pages**

- Graphing Linear Equations (Intro) Resource Page
- Graphing Linear Equations (Advanced) Resource Page

**Related Lessons**

- The Slope-Intercept Form of an Equation
- Find the equation (given 1 point and a slope)
- Find the equation (given 2 points)

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