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4.5 Use the slope–intercept form of an equation of a line

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By the end of this section, you will be able to:
  • Recognize the relation between the graph and the slope–intercept form of an equation of a line
  • Identify the slope and y-intercept form of an equation of a line
  • Graph a line using its slope and intercept
  • Choose the most convenient method to graph a line
  • Graph and interpret applications of slope–intercept
  • Use slopes to identify parallel lines
  • Use slopes to identify perpendicular lines

Before you get started, take this readiness quiz.

  1. Add: x 4 + 1 4 .
    If you missed this problem, review [link] .
  2. Find the reciprocal of 3 7 .
    If you missed this problem, review [link] .
  3. Solve 2 x − 3 y = 12 for y .
    If you missed this problem, review [link] .

Recognize the relation between the graph and the slope–intercept form of an equation of a line

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. 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.

In Graph Linear Equations in Two Variables , we graphed the line of the equation y = 1 2 x + 3 by plotting points. See [link] . Let’s find the slope of this line.

This figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line is labeled with the equation y equals one half x, plus 3. The points (0, 3), (2, 4) and (4, 5) are labeled also. A red vertical line begins at the point (2, 4) and ends one unit above the point. It is labeled “Rise equals 1”. A red horizontal line begins at the end of the vertical line and ends at the point (4, 5). It is labeled “Run equals 2. The red lines create a right triangle with the line y equals one half x, plus 3 as the hypotenuse.

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

m = rise run m = 1 2

What is the y -intercept of the line? The y -intercept is where the line crosses the y -axis, so y -intercept is ( 0 , 3 ) . The equation of this line is:

The figure shows the equation y equals one half x, plus 3. The fraction one half is colored red and the number 3 is colored blue.

Notice, the line has:

The figure shows the statement “slope m equals one half and y-intercept (0, 3). The slope, one half, is colored red and the number 3 in the y-intercept is colored 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 = 1 2 x + 3 is in slope–intercept form.

The figure shows the statement “m equals one half; y-intercept is (0, 3). The slope, one half, is colored red and the number 3 in the y-intercept is colored blue. Below that statement is the equation y equals one half x, plus 3. The fraction one half is colored red and the number 3 is colored blue. Below the equation is another equation y equals m x, plus b. The variable m is colored red and the variable b is colored blue.

Slope-intercept form of an equation of a line

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

y = m x + b

Sometimes the slope–intercept form is called the “ y -form.”

Use the graph to find the slope and y -intercept of the line, y = 2 x + 1 .

Compare these values to the equation y = m x + b .

Solution

To find the slope of the line, we need to choose two points on the line. We’ll use the points ( 0 , 1 ) and ( 1 , 3 ) .

.
Find the rise and run. .
.
.
Find the y -intercept of the line. The y -intercept is the point (0, 1).
. .

The slope is the same as the coefficient of x and the y -coordinate of the y -intercept is the same as the constant term.

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Use the graph to find the slope and y -intercept of the line y = 2 3 x − 1 . Compare these values to the equation y = m x + b .

The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line goes through the points (0, negative 1) and (6, 3).

slope m = 2 3 and y -intercept ( 0 , −1 )

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Use the graph to find the slope and y -intercept of the line y = 1 2 x + 3 . Compare these values to the equation y = m x + b .

The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line goes through the points (0, 3) and (negative 6, 0).

slope m = 1 2 and y -intercept ( 0 , 3 )

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Identify the slope and y -intercept from an equation of a line

In Understand Slope of a Line , we graphed a line using the slope and a point. When we are given an equation in slope–intercept form, we can use the y -intercept as the point, and then count out the slope from there. Let’s practice finding the values of the slope and y -intercept from the equation of a line.

Identify the slope and y -intercept of the line with equation y = −3 x + 5 .

Solution

We compare our equation to the slope–intercept form of the equation.

.
Write the equation of the line. .
Identify the slope. .
Identify the y -intercept. .

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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