3.4 Graph of Linear Equations

3.4 Graph of Linear Equations

Objective 1

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

Slide 3.4-3

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

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.

Slide 3.4-4

mx b y

mx y b

y bm

x

y mx b

0

y bm

x

Rewrite.

Add b to both sides.

Multiply by x.

Subtract in the denominator.

Change in x-values

Change in y-values

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

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 FormThe 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

y m bx

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

Solution:

y x

Slide 3.4-6

Identifying Slopes and y-Intercepts

36

4y x

Slope: − 1

y-intercept: (0,0)

Slope:

y-intercept: (0,− 6)

3

4

CLASSROOM EXAMPLE 1

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

Solution:

5y x

Slide 3.4-7

Writing an Equation of a LineCLASSROOM EXAMPLE 2

Objective 2

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

Slide 3.4-8

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

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

Solution:

33 34 8xyx x

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

4 3

4 4 4

8xy

32

4y x

Slope intercept form

Slide 3.4-10

Graphing Lines by Using Slopes and y-InterceptsCLASSROOM EXAMPLE 3

Solution:

Graph the line through (2,−3) with slope1.3

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

Slide 3.4-11

Graphing a Line by Using the Slope and a PointCLASSROOM EXAMPLE 4

Objective 3

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

Slide 3.4-12

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

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

Solution:

y m bx

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

4 2 1 b

4 22 2b

2b

2 2y x The slope-intercept form is

Slide 3.4-14

Using the Slope-Intercept Form to Write an EquationCLASSROOM EXAMPLE 5

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

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.

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

11 1

1

m xy

xy

xx

xx

11 yx xm y

1 1 .m xy xy

Slope

Given point

1

1

my y

x x

Slide 3.4-15

Multiply each side by x − x1.

Definition of slope

Rewrite.

Solution:

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

1.3

1 1my xy x

12 5

3y x

1 52

3

6

32

3y x

1 11

3 3xy

Slide 3.4-16

Using the Point-Slope Form to Write EquationsCLASSROOM EXAMPLE 6

Objective 4

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

Slide 3.4-17

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

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

,Ax B Cy

Slide 3.4-18

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

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:

2 1

2 1x

ym

y

x

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

6 5

1 2m

1

3m

1 1my xy x

1

36 1xy

1 17

3 3xy

1 16

3

18

36

3xy

Slide 3.4-19

Writing the Equation of a Line by Using Two Points

173x y

3 17y x

1 17

3 3xy

Standard form

Slope-intercept form

CLASSROOM EXAMPLE 7

Slide 3.4-20

Summary of the forms of linear equations.

Objective 5

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

Slide 3.4-21

Solution:

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?

2 1

2 1x

ym

y

x

6185 4645

7 3m

0

4

154m

1 1my xy x 4645 8 33 5 xy

385 3490xy

4645 385 11554645 4645xy

385m

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

Slide 3.4-22

Writing an Equation of a Line That Describes Data

(5)385 3490y 5415y

CLASSROOM EXAMPLE 8