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3. Lesson 3-1 The Distributive Property
Lesson 3-2 Simplifying Algebraic Expressions
Lesson 3-3 Solving Equations by Adding or Subtracting
Lesson 3-4 Solving Equations by Multiplying or Dividing
Lesson 3-5 Solving Two-Step Equations
Lesson 3-6 Writing Two-Step Equations
Lesson 3-7 Using Formulas
Contents

4. Example 1 Use the Distributive Property
Example 2 Use the Distributive Property to Solve a Problem
Example 3 Simplify Algebraic Expressions
Example 4 Simplify Expressions with Subtraction
Lesson 1 Contents

5. Use the Distributive Property to write as an equivalent expression
Use the Distributive Property to write as an equivalent expression. Then evaluate the expression.
Multiply.
Add.
Answer: 52
Example 1-1a

6. Use the Distributive Property to write as an equivalent expression
Use the Distributive Property to write as an equivalent expression. Then evaluate the expression.
Multiply.
Add.
Answer: 30
Example 1-1b

7. Use the Distributive Property to write each expression as an equivalent expression. Then evaluate the expression. a. b.
Answer:
Answer:
Example 1-1c

8. Method 1 Find the cost for 1 person, then multiply by 4.
Recreation North Country Rivers of York, Maine, offers one-day white-water rafting trips on the Kennebec River. The trip costs $69 per person, and wet suits are $15 each.
Write two equivalent expressions to find the total cost of one trip for a family of four if each person uses a wet suit.
Method 1 Find the cost for 1 person, then multiply by 4.
cost for 1 person
Example 1-2a

9. Method 2 Find the cost of 4 trips and 4 wet suits. Then add.
cost of 4 wet suits
cost of 4 trips
Example 1-2a

10. Evaluate either expression to find the total cost.
Distributive Property
Multiply.
Add.
Answer: The total cost is $336.
Check You can check your results by evaluating 4($84).
Example 1-2b

11. Movies The cost of a movie ticket is $7 and the cost of a box of popcorn is $2.
a. Write two equivalent expressions to find the total cost for a family of five to go to the movies if each member of the family gets a box of popcorn.
b. Find the total cost.
Answer:
Answer: $45
Example 1-2c

12. Use the Distributive Property to write as an equivalent algebraic expression.
Simplify.
Answer:
Example 1-3a

13. Use the Distributive Property to write as an equivalent algebraic expression.
Simplify.
Answer:
Example 1-3b

14. Use the Distributive Property to write each expression as an equivalent algebraic expression.
Answer:
Answer:
Example 1-3c

15. Distributive Property
Use the Distributive Property to write as an equivalent algebraic expression.
Rewrite as
Distributive Property
Simplify.
Definition of subtraction
Answer:
Example 1-4a

16. Distributive Property
Use the Distributive Property to write as an equivalent algebraic expression.
Rewrite as
Distributive Property
Simplify.
Answer:
Example 1-4b

17. Use the Distributive Property to write each expression as an equivalent algebraic expression.
Answer:
Answer:
Example 1-4c

18. End of Lesson 1

19. Example 1 Identify Parts of Expressions
Example 2 Simplify Algebraic Expressions
Example 3 Translate Verbal Phrases into Expressions
Lesson 2 Contents

20. Definition of subtraction
Identify the terms, like terms, coefficients, and constants in the expression
Definition of subtraction
Identity Property
Answer: The terms are 4x, –x, 2y, and –3. The like terms are 4x and –x. The coefficients are 4, –1, and 2. The constant is –3.
Example 2-1a

21. Identify the terms, like terms, coefficients, and constants in the expression
Answer: The terms are 5x, 3y, –2y, and 6. The like terms are 3y and –2y. The coefficients are 5, 3, and –2. The constant is 6.
Example 2-1b

22. 5x and 4x are like terms. Simplify . Distributive Property Simplify.
Answer: 9x
Example 2-2a

23. 8n and 4n are like terms. Simplify . Commutative Property
Distributive Property
Simplify.
Answer:
Example 2-2b

24. 6x and –5x are like terms. 4 and –7 are also like terms.
Simplify .
6x and –5x are like terms. 4 and –7 are also like terms.
Definition of subtraction
Commutative Property
Distributive Property
Example 2-2c

25. Simplify.
Answer:
Example 2-2d

26. Distributive Property
Simplify .
Distributive Property
Multiply.
Identity Property
Commutative Property
Example 2-2e

27. Distributive Property
Simplify.
Answer:
Example 2-2f

28. Simplify each expression. a. b. c. d. Answer:
Example 2-2g

29. Variables Let number of hours your friend worked.
Work You and a friend worked in the school store last week. You worked 4 hours more than your friend. Write an expression in simplest form that represents the total number of hours you both worked.
Words Your friend worked some hours. You worked 4 more hours than your friend.
Variables Let number of hours your friend worked.
Let number of hours you worked.
Expression To find the total, add the expressions.
Example 2-3a

30. Distributive Property
Associative Property
Identity Property
Distributive Property
Simplify.
Answer: The expression represents the total number of hours worked, where h is the number of hours your friend worked.
Example 2-3b

31. Library Books You and a friend went to the library
Library Books You and a friend went to the library. Your friend borrowed three more books than you did. Write an expression in simplest form that represents the total number of books you both borrowed.
Answer:
Example 2-3c

32. End of Lesson 2

33. Example 1 Solve Equations by Subtracting
Example 2 Graph the Solutions of an Equation
Example 3 Solve Equations by Adding
Example 4 Use an Equation to Solve a Problem
Example 5 Solve Equations
Lesson 3 Contents

34. Solve . Check your solution.
Write the equation.
Subtract 4 from each side.
Identity Property;
Example 3-1a

35. To check your solution, replace x with –7 in the original equation.
Write the equation.
Check to see whether this sentence is true.
The sentence is true.
Answer: The solution is –7.
Example 3-1b

36. Solve . Check your solution.
Answer: –4
Example 3-1c

37. Graph the solution of on a number line.
Write the equation.
Subtract 8 from each side.
Simplify.
Answer: The solution is –1. To graph the solution, draw a dot at –1 on a number line.
Example 3-2a

38. Graph the solution of on a number line.
Answer:
Example 3-2b

39. Additive Inverse Property;
Solve .
Write the equation.
Rewrite as
Add 3 to each side.
Additive Inverse Property;
Identity Property;
Answer: The solution is –11.
Check your solution.
Example 3-3a

40. Solve .
Answer: –7
Example 3-3b

41. Entertainment Movie A earned $225 million at the box office
Entertainment Movie A earned $225 million at the box office. That is $38 million less than Movie B earned. Write and solve an equation to find the amount Movie B earned.
Words Movie A earned $38 million less than Movie B earned.
Variables Let amount Movie B earned.
Movie A earned $38 million less than Movie B.
Equation
225
Example 3-4c

42. Answer: Movie B earned $263 million at the box office.
Solve the equation.
Think of as
Add 38 to each side.
Simplify.
Answer: Movie B earned $263 million at the box office.
Example 3-4d

43. Construction Board A measures 22 feet
Construction Board A measures 22 feet. That is 9 feet more than the measure of board B. Write and solve an equation to find the measure of board B.
Answer:
Example 3-4e

44. Multiple-Choice Test Item What value of x makes a true statement.
A B C – D –9
Read the Test Item To find the value of x, solve the equation.
Example 3-5a

45. Solve the Test Item Write the equation. Add 1 to each side. Simplify.
Answer: A
Example 3-5b

46. Multiple-Choice Test Item What value of x makes a true statement?
A B – C – D 1
Answer: B
Example 3-5c

47. End of Lesson 3

48. Example 1 Solve Equations by Dividing
Example 2 Use an Equation to Solve a Problem
Example 3 Solve Equations by Multiplying
Lesson 4 Contents

49. Solve . Check your solution and graph it on a number line.
Write the equation.
Divide each side by 7. ,
Identity Property;
Example 4-1a

50. To check your solution, replace x with –8 in the original equation.
Write the equation.
Check to see whether this statement is true.
The statement is true.
Answer: The solution is –8.
Example 4-1b

51. To graph the solution, draw a dot at –8 on a number line.
Example 4-1c

52. Solve . Check your solution and graph it on a number line.
Answer: The solution is –3.
Example 4-1d

53. Hobbies Esteban spent $112 on boxes of baseball cards
Hobbies Esteban spent $112 on boxes of baseball cards. If he paid $14 per box, how many boxes of cards did Esteban buy?
Words $14 times the number of boxes equals the total.
Variables Let x represent the number of boxes.
the number of boxes
The cost per box
times
equals
the total.
Equation
$14 x
$112 •
Example 4-2a

54. Solve the equation. Write the equation. Divide each side by 14.
Simplify.
Example 4-2b

55. Check to see whether this statement is true.
Write the equation.
Check to see whether this statement is true.
The statement is true.
Answer: Therefore, Estaban bought 8 boxes of cards.
Example 4-2c

56. Answer: Drew bought 9 cars.
Toy Cars Drew spent $18 on toy cars. If the cars cost $2 each, how many cars did Drew buy?
Answer: Drew bought 9 cars.
Example 4-2d

57. Solve . Check your solution.
Write the equation.
Multiply each side by –5 to undo the division.
Simplify.
Example 4-3a

58. Check to see whether this statement is true.
Write the equation.
Check to see whether this statement is true.
The statement is true.
Answer: The solution is 60.
Example 4-3a

59. Solve . Check your solution.
Answer: –36
Example 4-3b

60. End of Lesson 4

61. Example 1 Solve Two-Step Equations
Example 2 Use an Equation to Solve a Problem
Example 3 Equations with Negative Coefficients
Example 4 Combine Like Terms Before Solving
Lesson 5 Contents

62. Solve . Check your solution.
Write the equation.
Undo subtraction. Add 4 to each side.
Simplify.
Undo multiplication. Divide each side by 3.
Simplify.
Example 5-1a

63. Check to see whether this statement is true.
Write the equation.
Check to see whether this statement is true.
The statement is true.
Answer: The solution is 7.
Example 5-1b

64. Undo addition. Subtract 8 from each side.
Solve .
Write the equation.
Undo addition. Subtract 8 from each side.
Simplify.
Example 5-1c

65. Undo division. Multiply each side by 5.
Simplify.
Answer: The solution is –25.
Check your solution.
Example 5-1d

66. a. Solve . Check your solution.
b. Solve .
Answer: 4
Answer: 24
Example 5-1e

67. Subtract 32 from each side.
Measurement The formula can be used to convert Fahrenheit degrees to Celsius degrees. Solve the equation to find the equivalent Celsius temperature for 59°F.
Write the equation.
Subtract 32 from each side.
Simplify.
Example 5-2a

68. Answer: The solution is 15.
Divide each side by 1.8.
Simplify.
Answer: The solution is 15.
Therefore, 15° Celsius is the equivalent temperature to 59° Fahrenheit.
Example 5-2b

69. Cell Phones Sue signed up for a cell phone plan that charges $19 per month plus $0.10 per minute used. Her first bill was $ Solve to find out how many minutes Sue used this month.
Answer: 43 minutes
Example 5-2c

70. Definition of subtraction
Solve .
Write the equation.
Identity Property;
Definition of subtraction
Add –5 to each side.
Simplify.
Example 5-3a

71. Answer: The solution is –2.
Divide each side by –1.
Simplify.
Check your solution.
Answer: The solution is –2.
Example 5-3b

72. Solve .
Answer: –13
Example 5-3c

73. Combine like terms, 1b and –3b.
Solve .
Write the equation.
Identity Property;
Combine like terms, 1b and –3b.
Subtract 8 from each side.
Simplify.
Example 5-4a

74. Answer: The solution is –5.
Divide each side by –2.
Simplify.
Answer: The solution is –5.
Example 5-4b

75. Solve .
Answer: –1
Example 5-4c

76. End of Lesson 5

77. Example 1 Translate Sentences into Equations
Example 2 Translate and Solve an Equation
Example 3 Write and Solve a Two-Step Equation
Example 4 Write and Solve a Two-Step Equation
Lesson 6 Contents

78. Translate this sentence into an equation.
Twice a number increased by 5 equals –25.
Answer: The equation is .
Example 6-1a

79. Translate this sentence into an equation.
Four times a number minus 8 equals 28.
Answer: The equation is .
Example 6-1b

80. Translate this sentence into an equation.
When five is added to the product of a number and 8, the result is 12.
Answer: The equation is .
Example 6-1c

81. Translate each sentence into an equation.
a. Five times a number decreased by 9 equals –6.
b. Three times a number increased by 7 equals 18.
c. When seven is subtracted from the product of 2 and a number, the result is 10.
Answer:
Answer:
Answer:
Example 6-1d

82. Nine more than four times a number is 41. Find the number.
Words Nine more than four times a number is 41.
Variable Let the number.
Write the equation.
Equation
Subtract 9 from each side.
Simplify.
Mentally divide each side by 4.
Answer: Therefore, the number is 8.
Example 6-2a

83. Six less than three times a number is 15. Find the number.
Answer: 7
Example 6-2b

84. three times as much as her daughter
Earnings Ms. Blake earns $48,400 per year. This is $4150 more than three times as much as her daughter earns. How much does her daughter earn?
Words Ms. Blake earns $4150 more than three times as much as her daughter.
Variable Let her daughter’s earnings.
Ms. Blake
$4150
more than
earns
three times as much as her daughter
Equation
$48,400 3d
$4150
Example 6-3a

85. Subtract 4150 from each side.
Write the equation.
Subtract from each side.
Simplify.
Divide each side by 3.
Simplify.
Answer: Ms. Blake’s daughter earns $14,750.
Example 6-3b

86. Shopping Tami spent $175 at the grocery store
Shopping Tami spent $175 at the grocery store. That is $25 less than four times as much as Ted spent. How much did Ted spend?
Answer: Ted spent $50.
Example 6-3c

87. Words Together, they collected 128 cans.
Community Service In a canned food drive, Sam collected 12 more cans than Louise. Together, they collected 128 cans. How many cans did Sam collect?
Words Together, they collected 128 cans.
Variables Let number of cans collected by Louise. Then number of cans collected by Sam.
Equation
Write the equation.
Associative Property
Combine like terms.
Example 6-4a

88. Subtract 12 from each side.
Simplify.
Mentally divide each side by 2.
Answer: So, Louise collected 58 cans and Sam collected or 70 cans.
Example 6-4b

89. Answer: Kyle picked 20 tomatoes.
Gardening During the summer, Kyle picked eight more tomatoes from his garden than Matt picked from his garden. Together, they picked 32 tomatoes. How many tomatoes did Kyle pick?
Answer: Kyle picked 20 tomatoes.
Example 6-4c

90. End of Lesson 6

91. Example 1 Use the Distance Formula
Example 2 Find the Perimeter of a Rectangle
Example 3 Find a Missing Length
Example 4 Find the Area of a Rectangle
Example 5 Find a Missing Width
Lesson 7 Contents

92. Replace d with 135 and t with 3.
Travel If you travel 135 miles in 3 hours, what is your average speed in miles per hour?
Write the formula.
Replace d with 135 and t with 3.
Divide each side by 3.
Simplify.
Answer: The average speed is 45 miles per hour.
Example 7-1a

93. Answer: Your average speed is 65 miles per hour.
Vacation If you drive 520 miles in 8 hours to reach your vacation destination, what is your average speed in miles per hour?
Answer: Your average speed is 65 miles per hour.
Example 7-1b

94. Find the perimeter of the rectangle.
Write the formula.
Replace with 20 and w with 15.
Add 20 and 15.
15 cm
20 cm
Simplify.
Answer: The perimeter is 70 centimeters.
Example 7-2a

95. Find the perimeter of the rectangle.
Answer: The perimeter is 40 inches.
Example 7-2b

96. Distributive Property
The perimeter of a rectangle is 60 feet. Its width is 9 feet. Find its length.
Write the formula.
Distributive Property
Replace P with 60 and w with 9.
Simplify.
Subtract 18 from each side.
Simplify.
Mentally divide each side by 2.
Answer: The length is 21 feet.
Example 7-3a

97. Answer: The length is 12 meters.
The perimeter of a rectangle is 36 meters. Its width is 6 meters. Find its length.
Answer: The length is 12 meters.
Example 7-3b

98. Find the area of a rectangle with length 14 feet and width 6 feet.
Write the formula.
Replace with 14 and w with 6.
Simplify.
Answer: The area is 84 square feet.
Example 7-4a

99. Find the area of a rectangle with length 11 yards and width 6 yards.
Answer: The area is 66 square yards.
Example 7-4b

100. Mentally divide each side by 8.
The area of a rectangle is 40 square meters. Its length is 8 meters. Find its width.
Write the formula.
Replace A with 40 and with 8.
Mentally divide each side by 8.
Answer: The width is 5 meters.
Example 7-5a

101. Answer: The width is 3 inches.
The area of a rectangle is 42 square inches. Its length is 14 inches. Find its width.
Answer: The width is 3 inches.
Example 7-5b

102. End of Lesson 7

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