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3.4 Graph of Linear Equations

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3.4 Graph of Linear Equations

Use the slope-intercept form of the equation of a line.

Slide 3.4-3

In Section 3.3, we found the slope of a line by solving for y. In that form, the slope is the coefficient of x. For, example, the slope of the line with equation y = 2x + 3 is 2. So, what does 3 represent?

Suppose a line has a slope m and y-intercept (0,b). We can find an equation of this line by choosing another point (x,y) on the line as shown. Then we use the slope formula.

Change in y-values

Change in x-values

Subtract in the denominator.

Multiply by x.

Add b to both sides.

Rewrite.

Slide 3.4-4

The result is the slope-intercept form of the equation of a line, because both the slope and the y-intercept of the line can be read directly from the equation. For the line with the equation y = 2x + 3, the number 3 gives the y-intercept (0,3).

Slope-Intercept Form

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

Where m is the slope and b is the y-intercept (0,b).

Slide 3.4-5

Slope:

y-intercept: (0,− 6)

CLASSROOM EXAMPLE 1

Identifying Slopes and y-Intercepts

Identify the slope and y-intercept of the line with each equation.

Solution:

Slope: − 1

y-intercept: (0,0)

Slide 3.4-6

CLASSROOM EXAMPLE 2

Writing an Equation of a Line

Write an equation of the line with slope −1 and y-intercept (0,5).

Solution:

Slide 3.4-7

Graph a line by using its slope and a point on the line.

Slide 3.4-8

Graphing a Line by Using the Slope and y-Intercept

Step 1:Write the equation in slope-intercept form, if necessary, by solving for y.

Step 2:Identify the y-intercept. Graph the point (0,b).

Step 3:Identify slope m of the line. Use the geometric interpretation of slope (“rise over run”) to find another point on the graph by counting from the y-intercept.

Step 4:Join the two points with a line to obtain the graph.

Slide 3.4-9

Slope intercept form

CLASSROOM EXAMPLE 3

Graphing Lines by Using Slopes and y-Intercepts

Graph 3x – 4y = 8 by using the slope and y-intercept.

Solution:

Slide 3.4-10

Graph the line through (2,−3) with slope

CLASSROOM EXAMPLE 4

Graphing a Line by Using the Slope and a Point

Solution:

Make sure when you begin counting for a second point you begin at the given point, not at the origin.

Slide 3.4-11

Write an equation of a line by using its slope and any point on the line.

Slide 3.4-12

We can use the slope-intercept form to write the equation of a line if we know the slope and any point on the line.

Slide 3.4-13

The slope-intercept form is

CLASSROOM EXAMPLE 5

Using the Slope-Intercept Form to Write an Equation

Write an equation, in slope-intercept form, of the line having slope −2 and passing through the point (−1,4).

Solution:

Slide 3.4-14

Slope

Given point

There is another form that can be used to write the equation of a line. To develop this form, let m represent the slope of a line and let (x1,y1) represent a given point on the line. Let (x, y) represent any other point on the line.

Definition of slope

Multiply each side by x − x1.

Rewrite.

Point-Slope Form

The point-slope form of the equation of a line with slope m passing through point (x1,y1) is

Slide 3.4-15

Write an equation of the line through (5,2), with the slope Give the final answer in slope-intercept form.

CLASSROOM EXAMPLE 6

Using the Point-Slope Form to Write Equations

Solution:

Slide 3.4-16

Write an equation of a line by using two points on the line.

Slide 3.4-17

Write an equation of a line by using two points on the line.

Many of the linear equations in Section 3.1−3.3 were given in the form

called standard form, where A,B, andC are real numbers and A and B are not both 0.

Slide 3.4-18

CLASSROOM EXAMPLE 7

Writing the Equation of a Line by Using Two Points

Find an equation of the line through the points (2,5) and (−1,6). Give the final answer in slope-intercept form and standard form.

Solution:

Standard form

Slope-intercept form

The same result would also be found by substituting the slope and either given point in slope-intercept form and then solving for b.

Slide 3.4-19

Slide 3.4-20

Write an equation of a line that fits a data set.

Slide 3.4-21

CLASSROOM EXAMPLE 8

Writing an Equation of a Line That Describes Data

Use the points (3, 4645) and (7, 6185) to write an equation in slope-intercept form that approximates the data of the table. How well does this equation approximate the cost in 2005?

Solution:

The equation gives y = 5415 when x = 5, which is a very good approximation.

Slide 3.4-22