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3. Lesson 5-3 Slope-Intercept Form
Lesson 5-4 Writing Equations in Slope-Intercept Form
Lesson 4-7 Arithmetic Sequences
Lesson 4-8 Writing Equations From Patterns
Lesson 5-6 Geometry: Parallel and Perpendicular Lines
Lesson 5-7 Statistics: Scatter Plots and Lines of Fit
Lesson 4-5 Graphing Linear Equations
Lesson 5-5 Writing Equations in Point-Slope Form
Contents

4. Example 1 Positive Slope Example 2 Negative Slope Example 3 Zero Slope
Example 4 Undefined Slope
Example 5 Find Coordinates Given Slope
Example 6 Find a Rate of Change
Lesson 1 Contents

5. Find the slope of the line that passes through (–3, 2) and (5, 5).
Let and
Substitute.
Example 1-1a

6. Simplify.
Answer: The slope is
Example 1-1b

7. Find the slope of the line that passes through (4, 5) and (7, 6).
Answer:
Example 1-1c

8. Find the slope of the line that passes through (–3, –4) and (–2, –8).
Let and
Substitute.
Example 1-2a

9. Simplify.
Answer: The slope is –4.
Example 1-2b

10. Find the slope of the line that passes through (–3, –5) and (–2, –7).
Answer: –2
Example 1-2c

11. Find the slope of the line that passes through (–3, 4) and (4, 4).
Let and
Substitute.
Example 1-3a

12. Simplify.
Answer: The slope is 0.
Example 1-3b

13. Find the slope of the line that passes through (–3, –1) and (5, –1).
Answer: 0
Example 1-3c

14. Find the slope of the line that passes through (–2, –4) and (–2, 3).
Let and
Answer: Since division by zero is undefined, the slope is undefined.
Example 1-4a

15. Find the slope of the line that passes through (5, –1) and (5, –3).
Answer: undefined
Example 1-4b

16. Find the value of r so that the line through (6, 3) and (r, 2) has a slope of
Slope formula
Substitute.
Subtract.
Example 1-5a

17. Find the cross products.
Simplify.
Add 6 to each side.
Answer:
Simplify.
Example 1-5b

18. Find the value of p so that the line through (p, 4) and (3, –1) has a slope of
Answer: –5
Example 1-5c

19. Find the rates of change for 1991-1995 and 1995-1999.
Travel The graph to the right shows the number of U.S. passports issued in 1991, 1995, and 1999.
Find the rates of change for and
Use the formula for slope.
millions of passports
years
Example 1-6a

20. 1991-1995: Substitute. Simplify.
Answer: The number of passports issued increased by 1.9 million in a 4-year period for a rate of change of 475,000 per year.
Example 1-6b

21. 1995-1999: Substitute. Simplify.
Answer: Over this 4-year period, the number of U.S. passports issued increased by 1.4 million for a rate of change of 350,000 per year.
Example 1-6c

22. Explain the meaning of slope in each case.
Answer: For , on average, 475,000 more passports were issued each year than the last. For , on average, 350,000 more passports were issued each year than the last.
Example 1-6d

23. How are the different rates of change shown on the graph?
Answer: There is a greater rate of change from than from Therefore, the section of the graph for has a steeper slope.
Example 1-6e

24. a. Find the rates of change for 1990-1995 and 1995-2000.
Airlines The graph shows the number of airplane departures in the United States in recent years.
a. Find the rates of change for and
Answer: 240,000 per year; 180,000 per year
Example 1-6f

25. b. Explain the meaning of the slope in each case.
Answer: For , the number of airplane departures increased by about 240,000 flights each year. For , the number of airplane departures increased by about 180,000 flights each year.
Example 1-6g

26. c. How are the different rates of change shown on the graph?
Answer: There is a greater vertical change for than for Therefore, the section of the graph for has a steeper slope.
Example 1-6h

27. End of Lesson 1

28. Example 1 Write an Equation Given Slope and y-Intercept
Example 2 Write an Equation Given Two Points
Example 3 Graph an Equation in Slope-Intercept Form
Example 4 Graph an Equation in Standard Form
Example 5 Write an Equation in Slope-Intercept Form
Lesson 3 Contents

29. Replace m with and b with –6.
Write an equation of the line whose slope is and whose y-intercept is –6.
Slope-intercept form
Replace m with and b with –6.
Answer:
Example 3-1a

30. Write an equation of the line whose slope is 4 and whose y-intercept is 3.
Answer:
Example 3-1b

31. Write an equation of the line shown in the graph.
Step 1 You know the coordinates of two points on the line. Find the slope. Let
Example 3-2a

32. Step 3 Finally, write the equation.
Simplify.
The slope is 2.
Step 2 The line crosses the y-axis at (0, –3). So, the y-intercept is –3.
Step 3 Finally, write the equation.
Slope-intercept form
Replace m with 2 and b with –3.
Answer: The equation of the line is
Example 3-2b

33. Write an equation of the line shown in the graph.
Answer:
Example 3-2c

34. Step 1 The y-intercept is –7. So graph (0, –7).
y = 0.5x – 7
Step 2 The slope is or
From (0, –7), move up 1 unit and right 2 units. Draw a dot.
Step 3 Draw a line connecting the points.
Example 3-3a

35. Graph
Answer:
Example 3-3b

36. Step 1 Solve for y to find the slope-intercept form.
Graph
Step 1 Solve for y to find the slope-intercept form.
Original equation
Subtract 5x from each side.
Simplify.
Divide each side by 4.
Example 3-4a

37. Divide each term in the numerator by 4.
Answer:
Step 2 The y-intercept of is 2. So graph (0, 2).
Example 3-4b

38. From (0, 2), move down 5 units and right 4 units. Draw a dot.
Step 3 The slope is
From (0, 2), move down 5 units and right 4 units. Draw a dot.
5x + 4y = 8
Step 4 Draw a line connecting the points.
Example 3-4c

39. Graph
Answer:
Example 3-4d

40. Health The ideal maximum heart rate for a 25-year-old who is exercising to burn fat is 117 beats per minute. For every 5 years older than 25, that ideal rate drops 3 beats per minute.
Write a linear equation to find the ideal maximum heart rate for anyone over 25 who is exercising to burn fat.
Words The rate drops 3 beats per minute every 5 years, so the rate of change is beats per minute each year. The ideal maximum heart rate for a 25-year-old is 117 beats per minute.
Example 3-5a

41. R a 117 Variables Let R = the ideal heart rate.
Let a = years older than 25.
Equation
ideal rate Ideal rate of years older for 25- rate equals change times than 25 plus year-old. R a
117
Answer:
Example 3-5b

42. The graph passes through (0, 117) with a slope of
Graph the equation.
The graph passes through (0, 117) with a slope of
Answer:
Example 3-5c

43. The age 55 is 30 years older than 25. So,
Find the ideal maximum heart rate for a person exercising to burn fat who is 55 years old.
The age 55 is 30 years older than 25. So,
Ideal heart rate equation
Replace a with 30.
Simplify.
Answer: The ideal heart rate for a 55-year-old person is 99 beats per minute.
Example 3-5d

44. The amount of money spent on Christmas gifts has increased by an average of $150,000 ($0.15 million) per year since Consumers spent $3 million in 1986.
a. Write a linear equation to find the average amount spent for any year since 1986.
Answer: where D is the amount of money spent in millions of dollars, and n is the number of years since 1986
Example 3-5e

45. The amount of money spent on Christmas gifts has increased by an average of $150,000 ($0.15 million) per year since Consumers spent $3 million in 1986.
b. Graph the equation.
Answer:
Example 3-5f

46. c. Find the amount spent by consumers in 1999.
The amount of money spent on Christmas gifts has increased by an average of $150,000 ($0.15 million) per year since Consumers spent $3 million in 1986.
c. Find the amount spent by consumers in 1999.
Answer: $4.95 million
Example 3-5g

47. End of Lesson 3

48. Example 1 Write an Equation Given Slope and One Point
Example 2 Write an Equation Given Two Points
Example 3 Write an Equation to Solve a Problem
Example 4 Linear Extrapolation
Lesson 4 Contents

49. Write an equation of a line that passes through (2, –3) with slope
Step 1 The line has slope To find the y-intercept, replace m with and (x, y) with (2, –3) in the slope-intercept form. Then, solve for b.
Example 4-1a

50. Replace m with , y with –3, and x with 2.
Slope-intercept form
Replace m with , y with –3, and x with 2.
Multiply.
Subtract 1 from each side.
Simplify.
Example 4-1b

51. Step 2 Write the slope-intercept form using
Replace m with and b with –4.
Answer: The equation is
Example 4-1c

52. Check You can check your result by graphing on a graphing calculator.
Use the CALC menu to verify that it passes through (2, –3).
Example 4-1d

53. Write an equation of a line that passes through (1, 4) and has a slope of –3.
Answer:
Example 4-1e

54. x y –3 –4 –2 –8 Multiple-Choice Test Item
The table of ordered pairs shows the coordinates of two points on the graph of a function. Which equation describes the function?
A B
C D x y –3 –4 –2 –8
Read the Test Item The table represents the ordered pairs (–3, –4) and (–2, –8).
Example 4-2a

55. Step 1 Find the slope of the line containing the points. Let and .
Solve the Test Item
Step 1 Find the slope of the line containing the points. Let and .
Slope formula
Simplify.
Example 4-2b

56. Replace m with –4, x with –3, and y with –4.
Step 2 You know the slope and two points. Choose one point and find the y-intercept. In this case, we chose (–3, –4).
Slope-intercept form
Replace m with –4, x with –3, and y with –4.
Multiply.
Subtract 12 from each side.
Simplify.
Example 4-2c

57. Step 3 Write the slope-intercept form using
Replace m with –4 and b with –16.
Answer: The equation is The answer is D.
Example 4-2d

58. x y –1 3 2 6 Multiple-Choice Test Item
The table of ordered pairs shows the coordinates of two points on the graph of a function. Which equation describes the function?
A B
C D x y –1 3 2 6
Answer: B
Example 4-2e

59. Explore You know the cost of regular gasoline in May and June.
Economy In 2000, the cost of many items increased because of the increase in the cost of petroleum. In Chicago, a gallon of self-serve regular gasoline cost $1.76 in May and $2.13 in June. Write a linear equation to predict the cost of gasoline in any month in 2000, using 1 to represent January.
Explore You know the cost of regular gasoline in May and June.
Plan Let x represent the month and y represent the cost of gasoline that month. Write an equation of the line that passes through (5, 1.76) and (6, 2.13).
Example 4-3a

60. Solve Find the slope.
Slope formula
Let and .
Simplify.
Example 4-3b

61. Choose (5, 1.76) and find the y-intercept of the line.
Slope-intercept form
Replace m with 0.37, x with 5, and y with 1.76.
Multiply.
Subtract from each side.
Simplify.
Example 4-3c

62. Write the slope-intercept form using and
Replace m with 0.37 and b with –0.09.
Answer: The equation is
Example 4-3d

63. Replace y with 2.13 and x with 6.
Examine Check your result by substituting the coordinates of the point not chosen, (6, 2.13), into the equation.
Original equation
Replace y with 2.13 and x with 6.
Multiply.
Simplify.
Example 4-3e

64. The average cost of a college textbook in 1997 was $57. 65
The average cost of a college textbook in 1997 was $ In 2000, the average cost was $ Write a linear equation to estimate the average cost of a textbook in any given year since Let x represent years since 1997.
Answer:
Example 4-3f

65. Economy The Yellow Cab Company budgeted $7000 for the July gasoline supply. On average, they use 3000 gallons of gasoline per month. Use the prediction equation where x represents the month and y represents the cost of one gallon of gasoline, to determine if they will have to add to their budget. Explain.
Original equation
Replace x with 7.
Simplify.
Example 4-4a

66. Answer: If gas increases at the same rate, a gallon of gasoline will cost $2.50 in July gallons at this price is $7500, so they will have to add $500 to their budget.
Example 4-4b

67. A student is starting college in 2004 and has saved $400 to use for textbooks. Use the prediction equation where x is the years since 1997 and y is the average cost of a college textbook, to determine whether they will have enough money for 5 textbooks.
Answer: If the cost of textbooks increases at the same rate, the average cost will be $82.15 in Five textbooks at this price is $410.75, so he will not have enough money.
Example 4-4c

68. End of Lesson 4

69. Example 1 Identify Arithmetic Sequences Example 2 Extend a Sequence
Example 3 Find a Specific Term
Example 4 Write an Equation for a Sequence
Lesson 7 Contents

70. Determine whether –15, –13, –11, –9,. is arithmetic
Determine whether –15, –13, –11, –9, ... is arithmetic. Justify your answer.
–15 –13 –11 –9
Answer: This is an arithmetic sequence because the difference between terms is constant.
Example 7-1a

71. Determine whether is arithmetic. Justify your answer.
Answer: This is not an arithmetic sequence because the difference between terms is not constant.
Example 7-1b

72. Determine whether each sequence is arithmetic. Justify your answer. b.
Answer: This is not an arithmetic sequence because the difference between terms is not constant.
Answer: This is an arithmetic sequence because the difference between terms is constant.
Example 7-1c

73. Find the common difference by subtracting successive terms.
Find the next three terms of the arithmetic sequence. –8, –11, –14, –17, ...
Find the common difference by subtracting successive terms.
–3 –3 –3
–8 –11 –14 –17
The common difference is –3.
Add –3 to the last term of the sequence to get the next term in the sequence. Continue adding –3 until the next three terms are found.
–3 –3 –3
–17 –20 –23 –26
Answer: The next three terms are –20, –23, –26.
Example 7-2a

74. Find the next three terms of the arithmetic sequence. 5, 12, 19, 26, ...
Answer: 33, 40, 47
Example 7-2b

75. Find the 9th term of the arithmetic sequence. 7, 11, 15, 19, ...
In this sequence, the first term, a1 , is 7. You want to find the 9th term, Find the common difference.
The common difference is 4.
Example 7-3a

76. Use the formula for the nth term of an arithmetic sequence.
Simplify.
Answer: The 9th term in the sequence is 39.
Example 7-3a

77. Find the 12th term in the arithmetic sequence. 12, 17, 22, 27, ...
Answer: 67
Example 7-3b

78. The common difference is 9.
Consider the arithmetic sequence –8, 1, 10, 19, Write an equation for the nth term of the sequence.
In this sequence, the first term, a1, is –8. Find the common difference. –
The common difference is 9.
Use the formula for the nth term to write an equation.
Formula for nth term
Distributive Property
Simplify.
Example 7-4a

79. Answer: An equation for the nth term in this sequence is .
Check . .
and so on.
Example 7-4a

80. Find the 12th term of the sequence.
Replace n with 12 in the equation written in part a.
Equation for the nth term
Replace n with 12.
Answer:
Simplify.
Example 7-4b

81. Graph the first five terms of the sequence.
(5, 28) 28 5
(4, 19) 19 4
(3, 10) 10 3
(2, 1) 1 2
(1, –8) –8 n
Answer:
Notice the points fall on a line. The graph of an arithmetic sequence is linear.
Example 7-4c

82. b. Find the 18th term in the sequence.
Consider the arithmetic sequence –3, 0, 3, 6, ... a. Write an equation for the nth term of the sequence.
b. Find the 18th term in the sequence.
c. Graph the first five terms in the sequence.
Answer:
Answer: 48
Answer:
Example 7-4d

83. End of Lesson 7

84. Example 1 Extend a Pattern Example 2 Patterns in a Sequence
Example 3 Write an Equation from Data
Example 4 Write an Equation with a Constant
Lesson 8 Contents

85. Study the pattern below. Draw the next three figures in the pattern.
The pattern consists of triangles with one-third shaded. The section that is shaded is rotated in a counterclockwise direction. The next three figures are shown.
Answer:
Example 8-1a

86. Study the pattern below. Draw the 17th triangle in the pattern.
The pattern repeats every third design. Therefore, designs 3, 6, 9, 12, 15, and so on will all be the same. Since 15 is the greatest number less than 17 that is a multiple of 3, the 17th triangle in the pattern will be the same as the second triangle.
Answer:
Example 8-1b

87. Study the pattern below.
a. Draw the next three figures in the pattern.
C. Draw the 19th square in the pattern.
Answer:
Answer:
Example 8-1c

88. Find the next three terms in the sequence –3, –1, 3, 9,
Find the next three terms in the sequence –3, –1, 3, 9, .... Study the pattern in the sequence.
–3 –1 3 9
You can use inductive reasoning to find the next term in the sequence. Notice the pattern 2, 4, 6, .... The difference between each term increases by two in each successive term. To find the next three terms in the sequence, continue adding two to each successive difference. Add 8, 10, and 12.
Example 8-2a

89. Answer: The next three terms are 17, 27, and 39.
–3 –
Answer: The next three terms are 17, 27, and 39.
Example 8-2a

90. Find the next three terms in the sequence. 1, 4, 10, 19, ...
Answer: 31, 46, 64
Example 8-2b

91. The table shows the number of miles driven for each hour of driving.
Hours 1 2 3 4
Miles 50
100
150
200
Graph the data. What conclusion can you make about the relationship between the number of hours driving, h and the numbers of miles driven, m?
Answer: The graph shows a linear relationship between the number of hours driving and the number of miles driven.
Example 8-3a

92. Write an equation to describe this relationship.
Look at the relationship between the domain and the range to find a pattern that can be described as an equation.
Hours 1 2 3 4
Miles 50
100
150
200
Example 8-3b

93.
Hours 1 2 3 4
Miles 50
100
150
200
Since this is a linear relationship, the ratio of the range values to the domain values is constant. The difference of the values for h is 1, and the difference of the values for m is 50. This suggests that m = 50h. Check to see if this equation is correct by substituting values of h into the equation.
Example 8-3b

94. Check The equation checks. Answer:
Since this relation is also a function, we can write the equation as where f(h) represents the number of miles driven.
Example 8-3b

95. The table below shows the number of miles walked for each hour of walking.
Hours 1 2 3 4 5
Miles
1.5
4.5 6
7.5
a. Graph the data. What conclusion can you make about the relationship between the number of miles and the time spent walking?
Example 8-3c

96. Answer:
The graph shows a linear relationship between the number of miles walked m and the time spent walking h.
Example 8-3c

97. b. Write an equation to describe the relationship.
Answer:
Example 8-3c

98. Write an equation in function notation for the relation graphed below.
Make a table of ordered pairs for several points on the graph. x 1 2 3 4 5 y 7 10 13
Example 8-4a

99. The difference in the x values is 1, and the difference in the y values is 3. The difference in y values is three times the difference of the x values. This suggests that Check this equation.
Check If , then or 3. But the y value for is 1. This is a difference of –2. Try some other values in the domain to see if the same difference occurs. x 1 2 3 4 5 3x 6 9 12 15 y 7 10 13
y is always 2 less than 3x.
Example 8-4a

100. This pattern suggests that 2 should be subtracted from one side of the equation in order to correctly describe the relation. Check
Answer: correctly describes this relation. Since the relation is also a function, we can write the equation in function notation as .
Example 8-4a

101. Write an equation in function notation for the relation graphed below.
Answer:
Example 8-4b

102. End of Lesson 8

103. Example 1 Parallel Line Through a Given Point
Example 2 Determine Whether Lines are Perpendicular
Example 3 Perpendicular Line Through a Given Point
Example 4 Perpendicular Line Through a Given Point
Lesson 6 Contents

104. Write the slope-intercept form of an equation for the line that passes through (4, –2) and is parallel to the graph of
The line parallel to has the same slope, Replace m with and (x, y) with (4, -2) in the point-slope form.
Example 6-1a

105. Replace m with y with –2, and x with 4.
Point-slope form
Replace m with y with –2, and x with 4.
Simplify.
Distributive Property
Subtract 2 from each side.
Example 6-1b

106. Write the equation in slope-intercept form.
Answer: The equation is
Example 6-1c

107. Check. You can check your result by graphing both equations
Check You can check your result by graphing both equations. The lines appear to be parallel. The graph of passes through (4, –2).
Example 6-1d

108. Write the slope-intercept form of an equation for the line that passes through (2, 3) and is parallel to the graph of
Answer:
Example 6-1e

109. Geometry The height of a trapezoid is measured on a segment that is perpendicular to a base. In trapezoid ARTP, and are bases. Can be used to measure the height of the trapezoid? Explain.
Example 6-2a

110. Find the slope of each segment.
Example 6-2b

111. Answer: The slope of. and. is 1 and the slope of
Answer: The slope of and is 1 and the slope of is not perpendicular to and , so it cannot be used to measure height.
Example 6-2c

112. The graph shows the diagonals of a rectangle
The graph shows the diagonals of a rectangle. Determine whether is perpendicular to
Answer: The slope of is and the slope of is Since is not perpendicular to
Example 6-2d

113. Step 1 Find the slope of the given line.
Write the slope-intercept form for an equation of a line that passes through (4, –1) and is perpendicular to the graph of
Step 1 Find the slope of the given line.
Original equation
Subtract 7x from each side.
Simplify.
Example 6-3a

114. Divide each side by –2. Simplify.
Step 2 The slope of the given line is So, the slope of the line perpendicular to this line is the opposite reciprocal of or
Example 6-3b

115. Step 3 Use the point-slope form to find the equation.
and
Simplify.
Distributive Property
Example 6-3c

116. Subtract 1 from each side.
Simplify.
Answer: The equation of the line is
Example 6-3d

117. Check You can check your result by graphing both equations on a graphing calculator. Use the CALC menu to verify that passes through (4, –1).
Example 6-3e

118. Write the slope-intercept form for an equation of a line that passes through (–3, 6) and is perpendicular to the graph of
Answer:
Example 6-3f

119. Subtract 5x from each side.
Write the slope-intercept form for an equation of a line perpendicular to the graph of and passes through (0, 6).
Step 1 Find the slope of
Original equation
Subtract 5x from each side.
Simplify.
Example 6-4a

120. Divide each side by 2. Simplify.
Step 2 The slope of the given line is So, the slope of the line perpendicular to this line is the opposite reciprocal of or
Example 6-4b

121. Replace x1 with 0, y1 with 6, and m with
Step 3 Substitute the slope and the given point into the point-slope form of a linear equation. Then write the equation in slope-intercept form.
Point-slope form
Replace x1 with 0, y1 with 6, and m with
Distributive Property
Answer: The equation of the line is
Example 6-4c

122. Write the slope-intercept form for an equation of a line perpendicular to the graph of and passes through the x-intercept of that line.
Answer:
Example 6-4d

123. End of Lesson 6

124. Example 1 Analyze Scatter Plots Example 2 Find a Line of Fit
Example 3 Linear Interpolation
Lesson 7 Contents

125. The graph shows average personal income for U.S. citizens.
Determine whether the graph shows a positive correlation, a negative correlation, no correlation. If there is a positive or negative correlation, describe it.
The graph shows average personal income for U.S. citizens.
Answer: The graph shows a positive correlation. With each year, the average personal income rose.
Example 7-1a

126. Determine whether the graph shows a positive correlation, a negative correlation, no correlation. If there is a positive or negative correlation, describe it.
The graph shows the average students per computer in U.S. public schools.
Answer: The graph shows a negative correlation. With each year, more computers are in the schools, making the students per computer rate smaller.
Example 7-1b

127. a. The graph shows the number of mail-order prescriptions.
Determine whether each graph shows a positive correlation, a negative correlation, no correlation. If there is a positive or negative correlation, describe it.
a. The graph shows the number of mail-order prescriptions.
Answer: Positive correlation; with each year, the number of mail-order prescriptions has increased.
Example 7-1c

128. Answer: no correlation
Determine whether each graph shows a positive correlation, a negative correlation, no correlation. If there is a positive or negative correlation, describe it.
b. The graph shows the percentage of voter participation in Presidential Elections.
Answer: no correlation
Example 7-1d

129. Population (millions)
The table shows the world population growing at a rapid rate.
Year
Population (millions)
1650
500
1850
1000
1930
2000
1975
4000
1998
5900
Example 7-2a

130. Draw a scatter plot and determine what relationship exists, if any, in the data.
Let the independent variable x be the year and let the dependent variable y be the population (in millions).
The scatter plot seems to indicate that as the year increases, the population increases. There is a positive correlation between the two variables.
Example 7-2b

131. Draw a line of fit for the scatter plot.
No one line will pass through all of the data points. Draw a line that passes close to the points. A line is shown in the scatter plot.
Example 7-2c

132. Write the slope-intercept form of an equation for equation for the line of fit.
The line of fit shown passes through the data points (1850, 1000) and (1998, 5900).
Step 1 Find the slope.
Slope formula
Let and
Simplify.
Example 7-2d

133. Answer: The equation of the line is .
Step 2 Use m = 33.1 and either the point-slope form or the slope-intercept form to write the equation. You can use either data point. We chose (1850, 1000).
Point-slope form
Slope-intercept form
Answer: The equation of the line is .
Example 7-2e

134. Check Check your result by substituting (1998, 5900) into
Line of fit equation
Replace x with 1998 and y with 5900.
Multiply.
Subtract.
The solution checks.
Example 7-2f

135. The table shows the number of bachelor’s degrees received since 1988.
Years Since 1988 2 4 6 8 10
Bachelor’s Degrees Received (thousands)
1051
1136
1169
1165
1184
Source: National Center for Education Statistics
Example 7-2g

136. a. Draw a scatter plot and determine what relationship exists, if any, in the data.
Answer: The scatter plot seems to indicate that as the number of years increase, the number of bachelor’s degrees received increases. There is a positive correlation between the two variables.
Example 7-2h

137. b. Draw a line of best fit for the scatter plot.
c. Write the slope-intercept form of an equation for the line of fit.
Answer: Using (4, 1137) and (10, 1184),
Example 7-2i

138. Use the prediction equation where x is the year and y is the population (in millions), to predict the world population in 2010.
Original equation
Replace x with 2010.
Simplify.
Answer: 6,296,000,000
Example 7-2a

139. Use the equation where x is the years since 1988 and y is the number of bachelor’s degrees (in thousands), to predict the number of bachelor’s degrees that will be received in 2005.
Answer: 1,204,000
Example 7-3b

140. End of Lesson 7

141. Example 1 Identify Linear Equations Example 2 Graph by Making a Table
Example 3 Use the Graph of a Linear Equation
Example 4 Graph Using Intercepts
Lesson 5 Contents

142. Subtract z from each side.
Determine whether is a linear equation. If so, write the equation in standard form.
First rewrite the equation so that the variables are on the same side of the equation.
Original equation
Subtract z from each side.
Simplify.
Since has 3 different variables, it cannot be written in the form
Answer: This is not a linear equation.
Example 5-1a

143. Determine whether is a linear equation.
If so, write the equation in standard form.
Rewrite the equation so that both variables are on the same side.
Original equation
Subtract y from each side.
Simplify.
Example 5-1b

144. Multiply each side of the equation by 4.
To write the equation with integer coefficients, multiply each term by 4.
Original equation
Multiply each side of the equation by 4.
Simplify.
The equation is now in standard form where
Answer: This is a linear equation.
Example 5-1b

145. Answer: The equation is now in standard form where
Determine whether is a linear equation. If so, write the equation in standard form.
Since the GCF of 3, 6, and 27 is not 1, the equation is not written in standard form. Divide each side by the GCF.
Original equation
Factor the GCF.
Divide each side by 3.
Simplify.
Answer: The equation is now in standard form where
Example 5-1c

146. Determine whether is a linear equation.
If so, write the equation in standard form.
To write the equation with integer coefficients, multiply each term by 4.
Original equation
Multiply each side of the equation by 4.
Simplify.
Example 5-1d

147. Answer: The equation. can be written as
Answer: The equation can be written as Therefore, it is a linear equation in standard form where
Example 5-1d

148. Answer: linear equation;
Determine whether each equation is a linear equation. If so, write the equation in standard form. a. b. c. d.
Answer: linear equation;
Answer: not a linear equation
Answer: linear equation;
Answer: linear equation;
Example 5-1e

149. In order to find values for y more easily, solve the equation for y.
Graph
In order to find values for y more easily, solve the equation for y.
Original equation
Add x to each side.
Simplify.
Example 5-2a

150. Multiply each side by 2. Simplify.
Select five values for the domain and make a table. Then graph the ordered pairs.
(3, 8) 8 3
(2, 6) 6 2
(0, 2)
(–1, 0) –1
(–3, –4) –4 –3
(x, y) y x
Example 5-2a

151. Answer:
When you graph the ordered pairs, a pattern begins to form. The domain of is the set of all real numbers, so there are an infinite number of solutions of the equation. Draw a line through the points. This line represents all the solutions of .
Example 5-2a

152. .
Answer:
Example 5-2b

153. Shiangtai walks his dog 2. 5 miles around the lake every day. Graph
Shiangtai walks his dog 2.5 miles around the lake every day. Graph where m represents the number of miles walked and d represents the number of days walking.
Select five values for d and make a table. Graph the ordered pairs and connect them to draw a line. d
2.5d t
(d, t)
2.5(0)
(0, 0) 4
2.5(4) 10
(4, 10) 8
2.5(8) 20
(8, 20) 12
2.5(12) 30
(12, 30) 16
2.5(16) 40
(16, 40)
Example 5-3a

154. Suppose Shiangtai wanted to walk 50 miles, how many days would it take him?
Since any point on the line is a solution of the equation, use the graph to estimate the value of the x-coordinate in the ordered pair that contains 50 as the y-coordinate.
Answer: The ordered pair (20, 50) appears to be on the line so it should take Shiangtai 20 days to walk 50 miles. Check this solution algebraically by substituting (20, 50) into the original equation.
Example 5-3b

155. Lily rides her bike 3.5 miles every day.
a. Graph the equation where m represents the number of miles Lily rides and d represents the number of days she rides.
b. Suppose Lily wanted to ride 28 miles, how many days would it take her?
Answer:
Answer: 8 days
Example 5-3c

156. To find the x-intercept, let .
Determine the x-intercept and the y-intercept of Then graph the equation.
To find the x-intercept, let .
Original equation
Replace y with 0.
Divide each side by 4.
To find the y-intercept, let .
Original equation
Replace x with 0.
Divide each side by –1.
Example 5-4a

157. Plot these points. Then draw a line that connects them.
Answer: The x-intercept is 1, so the graph intersects the x-axis at (1, 0). The y-intercept is –4, so the graph intersects the y-axis at (0, –4).
Plot these points. Then draw a line that connects them.
Example 5-4a

158. Answer: x-intercept (5, 0); y-intercept (0, 2)
Determine the x-intercept and the y-intercept of . Then graph the equation.
Answer: x-intercept (5, 0); y-intercept (0, 2)
Example 5-4b

159. End of Lesson 5

160. Example 1 Write an Equation Given Slope and a Point
Example 2 Write an Equation of a Horizontal Line
Example 3 Write an Equation in Standard Form
Example 4 Write an Equation in Slope-Intercept Form
Example 5 Write an Equation in Point-Slope Form
Lesson 5 Contents

161. Answer: The equation is
Write the point-slope form of an equation for a line that passes through (–2, 0) with slope
Point-slope form
Simplify.
Answer: The equation is
Example 5-1a

162. Write the point-slope form of an equation for a line that passes through (4, –3) with slope –2.
Answer:
Example 5-1b

163. Answer: The equation is
Write the point-slope form of an equation for a horizontal line that passes through (0, 5).
Point-slope form
Simplify.
Answer: The equation is
Example 5-2a

164. Write the point-slope form of an equation for a horizontal line that passes through (–3, –4).
Answer:
Example 5-2b

165. Multiply each side by 4 to eliminate the fraction.
Write in standard form.
In standard form, the variables are on the left side of the equation. A, B, and C are all integers.
Original equation
Multiply each side by 4 to eliminate the fraction.
Distributive Property
Example 5-3a

166. Subtract 3x from each side.
Simplify.
Answer: The standard form of the equation is
Example 5-3b

167. Write in standard form.
Answer:
Example 5-3c

168. Write in slope-intercept form.
In slope-intercept form, y is on the left side of the equation. The constant and x are on the right side.
Original equation
Distributive Property
Add 5 to each side.
Example 5-4a

169. Answer: The slope-intercept form of the equation is
Simplify.
Answer: The slope-intercept form of the equation is
Example 5-4b

170. Write in slope-intercept form.
Answer:
Example 5-4c

171. The figure shows trapezoid ABCD with bases and
Write the point-slope form of the lines containing the bases of the trapezoid.
Example 5-5a

172. Step 1 First find the slopes of and
Slope formula
Slope formula
Example 5-5b

173. Step 2 You can use either point for (x1, y1) in the point-slope form.
Method 1 Use (–2, 3).
Method 2 Use (4, 3).
Example 5-5c

174. Method 1 Use (1, –2). Method 2 Use (6, –2).
Answer: The point-slope form of the equation containing The point-slope form of the equation containing
Example 5-5d

175. Write each equation in standard form.
Original equation
Add 3 to each side.
Answer:
Simplify.
Original equation
Subtract 2 from each side.
Answer:
Simplify.
Example 5-5e

176. The figure shows right triangle ABC.
a. Write the point-slope form of the line containing the hypotenuse
b. Write the equation in standard form.
Answer:
Answer:
Example 5-5f

177. End of Lesson 5

178. End of Slide Show