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## 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.

## Try It

Find the slope and y-intercept for each equation:

1)  3y = 3x + 9

2)  5(x + y) = 25

3)   2x = 4y + 8

4)  10x + 2y = 20

5)  4y = 13x - 20

Solutions:

1)  Solution:

2)  Solution:

3)  Solution:

4)  Solution:

5)  Solution: