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Section 3.6

Point-Slope Form

The line with slope m passing through the point (x1, y1) is given by

y – y1 = m(x – x1),

or equivalently, y = m(x – x1) + y1

the point-slope form of a line.

Point-Slope Form15

Example

Find the point-slope form of a line passing through the point (3, 1) with slope 2. Does the point (4, 3) lie on this line? SolutionLet m = 2 and (x1, y1) = (3,1) in the point-slope form.

To determine whether the point (4, 3) lies on the line, substitute 4 for x and 3 for y.

y – y1 = m(x – x1)

y − 1 = 2(x – 3)

3 – 1 ? 2(4 – 3)

2 = 2The point (4, 3) lies on the line because it satisfies the point-slope form.

Page 215

Example

Use the point-slope form to find an equation of the line passing through the points (−2, 3) and (2, 5). SolutionBefore we can apply the point-slope form, we must find the slope.

2 1

2 1

y ym

x x

5 3

2 2

2

4

1

2

Page 216

Example (cont)

We can use either (−2, 3) or (2, 5) for (x1, y1) in the point-slope form. If we choose (−2, 3), the point-slope form becomes the following.

If we choose (2, 5), the point-slope form with x1 = 2 and y1 = 5 becomes

y – y1= m(x – x1)

1)3 ( )

2( 2y x

13 ( 2)

2y x

15 ( 2).

2y x

Page 216

Point-Slope Form

SOLUTIONSOLUTION

Write the point-slope form and then the slope-intercept form of the equation of the line with slope 6 that passes through the point (2,-5).

Substitute the given values 11 xxmyy

265 xy

1265 xy176 xy

Distribute

Subtract 5 from both sides

)2(65 xy Simplify

This is the equation of the line in point-slope form.

This is the equation of the line in slope-intercept form.

Blitzer, Introductory Algebra, 5e – Slide #7 Section 4.5

Point-Slope Form

SOLUTIONSOLUTION

First find the slope of the line. This is done as follows:

5

1

5

)1(2

61

m

Write the point-slope form and then the slope-intercept form of the equation of the line that passes through the points (-2,-1) and (-1,-6).

Point-Slope Form

Use either point provided. Using (-2,-1).

Substitute the given values 11 xxmyy

)2(51 xy

1051 xy

115 xy

Distribute

Subtract 1 from both sides

251 xy Simplify



CONTINUEDCONTINUED

Example

Find the slope-intercept form of the line perpendicular toy = x – 3, passing through the point (4, 6). SolutionThe line y = x – 3 has slope m1 = 1. The slope of the perpendicular line is m2 = −1. The slope-intercept form of a line having slope −1 and passing through (4, 6) can be found as follows. 6 1( 4)y x

6 4y x

10y x

Page 218

Example

A swimming pool is being emptied by a pump that removes water at a constant rate. After 1 hour the pool contains 8000 gallons and after 4 hours it contains 2000 gallons. a. How fast is the pump removing water?

Solution

The pump removes a total of 8000 − 2000 gallons of water in 3 hours, or 2000 gallons per hour.

Page 220similar to Example 7

Example (cont)

b. Find the slope-intercept form of a line that models the amount of water in the pool. Interpret the slope.

The line passes through the points (1,8000) and (4, 2000), so the slope is

2000 80002000

4 1

y – y1= m(x – x1)

y – 8000 = −2000(x – 1)

y – 8000 = −2000x + 2000

y = −2000x + 10,000

A slope of −2000, means that the pump is removing 2000 gallons per hour.


Example (cont)

c. Find the y-intercept and the x-intercept. Interpret each.The y-intercept is 10,000 and indicates that the pool initially contained 10,000 gallons. To find the x-intercept let y = 0 in the slope-intercept form.

The x-intercept of 5 indicates that the pool is emptied after 5 hours.

0 2000 10,000x 2000 10,000x 2000 10,000

2000 2000

x

5x


Example

d. Sketch the graph of the amount of water in the pool during the first 5 hours.

The x-intercept is 5 and the y-intercept is 10,000. Sketch a line passing through (5, 0) and (0, 10,000).

X

Y

1 2 3 4 5 6

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

Wat

er (

gallo

ns)

Time (hours)


Example (cont)

e. The point (2, 6000) lies on the graph. Explain its meaning.

The point (2, 6000) indicates that after 2 hours the pool contains 6000 gallons of water.

X

Y

1 2 3 4 5 6

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

Wat

er (

gallo

ns)

Time (hours)


Group Activity (3.5 on page 214)

Public Tuition: In 2005, the average cost of tuition and fees at public four-year colleges was $6130, and in 2010 it was $7610. Note that the known value for 2008 is $6530.

Solution: The line passes through (2005, 6.1) and (2010, 7.6). Find the slope.

20052010

61307610

in x change

yin change

m 2965

1480

Thus, the slope of the line is 296; tuition and fees on average increased by $296/yr.

Figure not in book

0510

61307610

in x change

yin change

m 2965

1480

Substitute 5 for 2005, 10 for 2010, and 8 for 2008.


Modeling public tuition: Write the slope-intercept form of the of the line shown in the graph. What is the y-intercept and does it have meaning in this situation.

11 xxmyy

20052966130 xy

5934802966130 xy

587350296 xy



2965

1480

Modeling public tuition: Substitute 5 for 2005, 10 for 2010, and 8 for 2008.

11 xxmyy

52966130 xy

14802966130 xy

4650296 xy




Using the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the average cost of tuition and fees at public four-year colleges in 2008.

Substitute 2008 or 8 for x and compute y.

587350)2008(296 y 7018

587350296 xy

4650)8(296 y 7018

The model predicts that the tuition in 2008 will be $7018 and the tuition in 2015 will be $9090.

Use the equation to predict the average cost of tuition and fees at public four-year colleges in 2015.

Substitute 2015 or15 for x and compute y.

587350)2015(296 y 9090

4650)15(296 y 9090

Modeling the Graying of America

Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020.

Solution: The line passes through (10, 30.0) and (30, 35.3). Find the slope.

(10, 30.0)

(30, 35.3)

1030

0.303.35

in x change

yin change

m 265.020

3.5

The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year.



265.020

3.5m


11 xxmyy 10265.00.30 xy

65.2265.00.30 xy

35.27265.0 xy

A linear equation that models the median age of the U.S. population, y, x years after 1970.


This is the equation of the line in slope-intercept form. (10, 30.0)

(30, 35.3)



265.020

3.5m


35.27265.0 xy

A linear equation that models the median age of the U.S. population, y, x years after 1970.

Use the equation to predict the median age in 2020. Because 2020 is 50 years after 1970, substitute 50 for x and compute y.

35.27)50(265.0 y 6.40The model predicts that the median age of the U.S. population in 2020 will be 40.6.

(10, 30.0)

(30, 35.3)

Example 6 Modeling female officers (page 219)

In 1995, there were 690 female officers in the Marine Corps, and by 2010 this number had increased to about 1110. Refer to graph in Figure 3.48 on page 214.

a)The slope of the line passing through (1995, 690) and (2010.1110) is

b)The number of female officers increased, on average by about 28 officers per year.

c)Estimate how many female officers there were in 2006.

19952010

6901110

m 28

Write the slope-intercept form of the of the line shown in the graph.

(1995, 690)

(2010, 1110)

//

11 xxmyy 199528690 xy

5586028690 xy5517028 xy

5517028 xyOR

9985517056168

55170)2006(28

y

y

1995200628690 y 9986901128 y


Modeling public tuition: Substitute 5 for 2005, 10 for 2010, and 8 for 2008.

11 xxmyy

52966130 xy

14802966130 xy

4650296 xy



510

61307610

in x change

yin change

m 2965

1480

6130)3(296 y 7018The model predicts that the tuition in 2008 will be $7018.

Objectives

• Derivation of Point-Slope Form

• Finding Point-Slope Form

• Applications