1. College Algebra Chapter 2 Functions and Graphs
Section 2.5
Applications of Linear Equations and Modeling
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2. Concepts
1. Apply the Point-Slope Formula 2. Determine the Slopes of Parallel and Perpendicular Lines 3. Create Linear Functions to Model Data 4. Create Models Using Linear Regression

3. Concept 1 Apply the Point-Slope Formula

4. Apply the Point-Slope Formula
Point-slope formula for a line:

5. Example 1
Use the point-slope formula to write an equation of the line having the given conditions. Write the answer in slope-intercept form (if possible). Passes through (–4, 2) and m = 2 Solution:

6. Example 2
Use the point-slope formula to write an equation of the line having the given conditions. Write the answer in slope-intercept form (if possible). Passes through (5, 0) and

7. Skill Practice 1
Use the point-slope formula to find an equation of the line passing through the point (-5,2) and having slope -3. Write the answer in slope- intercept form.

8. Example 3
Use the point-slope formula to write an equation of the line having the given conditions. Write the answer in slope-intercept form (if possible). Passes through (–1, –4) and (–2, 1) Solution:

9. Example 4
Use the point-slope formula to write an equation of the line having the given conditions. Write the answer in slope-intercept form (if possible). Passes through (2,3) and the slope is undefined. Solution: Undefined slope → vertical line X=2

10. Skill Practice 2
Write an equation of the line passing through the points (2,-5) and (7,-3).

11. Concept 2 Determine the Slopes of Parallel and Perpendicular Lines

12. Determine the Slopes of Parallel and Perpendicular Lines
Parallel lines have matching slopes.
Perpendicular lines have slopes that are negative reciprocals.

13. Examples 5 – 7 (1 of 2)
The slope of a line is given. Determine the slope of a line parallel and perpendicular to the given line, if possible.

14. Examples 5 – 7 (2 of 2)

15. Example 8
Give the equation of a line that passes through (–1,2) and is parallel to the line defined by y-3x=4. Solution: Y-3x=4 Y=3x+4 Our slope=3 M=3 (-1,2) Y-2=3(x+1) Y=3x+3+2 Y=3x+5

16. Skill Practice 3
Write an equation of the line passing through the point (-3,2) and parallel to the line defined by x+3y=6.Write the answer in slope-intercept form and in standard form.

17. Example 9
Give the equation of a line that passes through (6,8) and is perpendicular to the line defined by 2y+5x=10. Solution:

18. Skill Practice 4
Write an equation of the line passing through the point (-8,-4) and perpendicular to the defined by

19. Concept 3 Create Linear Functions to Model Data

20. Example 10 (1 of 2)
The local hardware store charges $28 to rent a carpet cleaning machine for 24 hours and $10.98 for each medium-sized bottle of rug shampoo.
Write a linear function S that represents the cost of renting the machine for x days along with 2 bottles of rug shampoo.
Solution:
S(x)=28x+2(10.98)
S(x)=28x+21.96

21. Example 10 (2 of 2)
Evaluate S(2) and interpret the meaning in the context of this problem.
Solution:
S(2)=28(2)+21.96
=77.96
It costs $77.96 to vent the machine for 2 days with 2 bottles of shampoo.

22. Skill Practice 5
A speeding ticket is $100 plus $5 for every 1 mph over the speed limit.
Write a linear function to model the cost S(x) (in $) of a speeding ticket for a person caught driving x mph over the speed limit.
Evaluate S(15) and interpret the meaning in the context of this problem.

23. Create Linear Functions to Model Data (1 of 2)
A linear cost function models the cost C(x) to produce x items.
C(x)=mx + b
m is the variable cost per item
b is the fixed cost
A linear revenue function models revenue R(x) for selling x items.
R(x)=px
P is the price per item

24. Create Linear Functions to Model Data (2 of 2)
A linear profit function models the profit for producing and selling x items. P(x)=R(x) - C(x)

25. Example 11 (1 of 4)
Alina is starting a summer business power- washing home driveways and sidewalks. She will charge $35 to pressure-clean a driveway and the sidewalk in front of a house. Her start- up costs include her initial purchase of a power washer for $330 and a fee of $2 per house she must pay to the homeowners association for the use of the water for each house.

26. Example 11 (2 of 4)
Write a linear cost function for power-washing at x homes.
C(x)=2x+330
Write a linear revenue function for power- washing at x homes.
R(x)=35x

27. Example 11 (3 of 4)
Write a linear profit function for power- washing at x homes.
P(x)=35x-(2x+330)=33x-330
How much profit will Alina make if she power- washes at 15 homes?
P(15)=33(15)-330=165
$165 profit

28. Example 11 (4 of 4) How many homes must Alina power-wash to make $330?

29. Skill Practice 6
Repeat Example 6 in the case where the vendor can cut the cost to $0.40 per cup of lemonade, and sell lemonades for $1.50 per cup.

30. Skill Practice 7
Suppose that y represents the average consumer spending on television services per year (in dollars), and that x represent the number of years since
Use the data points (2,308) and (6,408) to write a linear equation relating y to x
Interpret the meaning of the slope in the context of this problem
Interpret the meaning of the y-interpret in the context of this problem
Use the model from part (a) to estimate the average consumer spending on television services for the year 2007.

31. Concept 4 Create Models Using Linear Regression

32. Create Models Using Linear Regression
Creating a Linear Regression Model
Graph the data in a scatter plot.
Inspect the data visually to determine if the data suggest a linear trend.
Invoke the linear regression feature on a calculator, graphing utility, or spreadsheet.
Check the result by graphing the line with the data points to verify that the line passes through or near the data points.

33. Example 12 (1 of 7)
Determine the equation for the least-squares regression line for the given data. X Y
0.5 1
1.3 2
2.9 3
2.4 4 5 6
5.4 7
7.7 8
8.3

34. Example 12 (2 of 7)
Use the STAT button, then EDIT to enter the x and y data in two lists. Exit this screen.

35. Example 12 (3 of 7)
Use the STAT button, then CALC, choose 4:LinReg(ax + b).

36. Example 12 (4 of 7)
Hit CALCULATE.
The equation is y=0.97x+0.3

37. Example 12 (5 of 7) To see the data and the line graphed:
Above the y = key, select STATPLOT.
Turn Plot1 ON and select the type of graph.

38. Example 12 (6 of 7)
Graph

39. Example 12 (7 of 7)
Enter y=0.97x+0.3 into the equation editor and see the line graphed.

40. Skill Practice 8
The data given represent the class averages for individual students based on the number of absences from class.
Find the equation of the least-squares regression line.
Use the model from part (a) to approximate the average for a student who misses 6 classes.
Number of Absences(x) 3 7 1 11 2 14 5
Average in class(y) 88 67 96 62 90 56 97 82