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2.1 Introduction to Variables 43 43 Chapter 2 Understanding Variables and Solving Equations
International Management Managing Across Borders and Cultures Text and Cases
9th Edition Deresky SOLUTIONS MANUAL
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International Management Managing Across Borders and Cultures Text and Cases
9th Edition Deresky TEST BANK
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borders-and-cultures-text-and-cases-9th-edition-deresky-test-bank/
CHAPTER 2 UNDERSTANDING VARIABLES AND SOLVING EQUATIONS
2.1 Introduction to Variables
2.1 Margin Exercises
1.
2.1 Introduction to Variables 44 44 Chapter 2 Understanding Variables and Solving Equations
Value of
Value of
Expression
is
is
is
is
(b)
6. (a) Multiplying any number by gives a product of .
Any
The expression is .
number times zero
The variable is . It represents the class limit.
The constant is .
•
2. (a) Evaluate the expression when is . (b) Changing the grouping of addends , , does not change the sum.
Order books.
Replace c with 25. 7. (a)
can be written as • • • •
is used as a factor times.
(b) Evaluate the expression when is . (b) can be written as • • • •
Replace c with 60. (c)
(d)
8. (a)
can be written as • • • •
can be written as • • • • • •
means
Order books.
3. (a) Evaluate the expression when is feet.
• •
• •
Replace y with 5.
Multiply left to right.
• feet
feet
Replace s with 3 feet. •
The perimeter of the square table is feet. (b) means
(b) Evaluate the expression when is miles. • • • Replace r with 6 and s with 3.
Replace s with 7 miles. • • • Multiply left to right.
• miles
miles
The perimeter of the square park is miles.
• •
•
4. Evaluate the expression
when is .
(c)
means Replace x with 4
Replace a with 40.
Divide.
• • •
• • •
• •
and y with 3.
Multiply left to right.
2.1 Introduction to Variables 45 45 Chapter 2 Understanding Variables and Solving Equations
Add.
•
The approximate systolic blood pressure is .
5. (a) Evaluate the expression when is and
(d)
means
is .
Replace with and with .
Divide.
• • • •
• • • •
• • •
• •
•
Replace c with 2.
Multiply left to right.
Your average score is .
2.1 Introduction to Variables 46 46 Chapter 2 Understanding Variables and Solving Equations
2.1 Section Exercises 14. Expression (rule) for degrees:
1. is the variable; (a) is degrees.
2.
is the constant.
is the variable;
is the constant.
(b) is degrees.
(c) is degrees.
15. Expression (rule) for finding perimeter of an
3. is the variable; equilateral triangle of side length :
is the constant.
4. is the variable;
(a) Evaluate the expression when , the side
length, is inches.
is the constant.
5. is the variable;
•
Replace s with 11.
Follow the rule and multiply.
is the coefficient.
6. is the variable;
is the coefficient.
inches is the perimeter.
(b) Evaluate the expression when , the side
length, is feet.
7. is the variable;
is the coefficient.
•
Replace s with 3.
Follow the rule and multiply.
is the constant.
8. is the variable;
is the coefficient.
is the constant.
feet is the perimeter.
16. Expression (rule) for perimeter:
(a) • meters is meters.
(b) • inches is inches.
9. Both and are variables.
17. Expression (rule) for ordering brushes:
10. Both and are variables.
(a) Evaluate the expression when , the class size,
11.
12.
is the variable;
is the coefficient;
is the constant.
is the variable;
is .
•
Replace c with 12.
Multiply before subtracting.
is the coefficient;
is the constant.
13. Expression (rule) for ordering robes:
brushes must be ordered.
(b) Evaluate the expression when , the class size,
is .
(a) Evaluate the expression when there are
graduates.
•
Replace c with 16.
Multiply before subtracting.
Replace g with 654.
Follow the rule and add.
brushes must be ordered.
(b)
robes must be ordered.
Evaluate the expression when there are
18. Expression (rule) for ordering doughnuts:
(a) • is doughnuts must be ordered.
2.1 Introduction to Variables 47 47 Chapter 2 Understanding Variables and Solving Equations
graduates.
Replace g with 208.
Follow the rule and add.
(b) • is doughnuts must be ordered.
19. Expression (rule) for average test score, where is
the total points and is the number of tests:
(c)
robes must be ordered.
Evaluate the expression when there are
(a) Evaluate the expression when , the total
points, is and , the number of tests, is .
graduates.
Replace g with 95.
Follow the rule and add.
Replace p with 332 and t with 4.
Follow the rule and divide.
robes must be ordered. points is the average test score.
2.1 Introduction to Variables 48 48 Chapter 2 Understanding Variables and Solving Equations
Value
of
Expression
Expression
is
• is
is
• is
is
• is
Value
of
Expression
Expression
is is
, or
is is
, or
is is
, or
Value
of
Value
of
Expression
is , or
is , or
is
, or
Value
of
Value
of
Expression
• • is
• • is
• • is
(b) Evaluate the expression when , the total 27. Multiplying a number by leaves the number
points, is and , the number of tests, is . unchanged. Let represent "a number."
Replace p with 637 and t with 7.
•
or •
Follow the rule and divide. 28. Adding to any number leaves the number
unchanged. Let represent "any number."
points is the average test score. or
20. Expression (rule) for buses:
29. Any number divided by is undefined. Let
represent "any number."(a) is buses.
(b) is buses. is undefined or
is undefined.
21. 30. Multiplication distributes over addition. Let , ,
and represent variables.
• •
22.
31.
32.
33.
34.
written without exponents is
• • • • •
written without exponents is
• • • • • •
written without exponents is
• • • • • •
written without exponents is
• • • • • •
23.
24.
25. A variable is a letter that represents the part of a
rule that varies or changes depending on the
situation. An expression expresses, or tells, the
rule for doing something. For example, is an
expression, and is the variable.
26. The number part in a multiplication expression is
2.1 Introduction to Variables 49 49 Chapter 2 Understanding Variables and Solving Equations
35. can be written as • • • • .
The exponent applies only to the base .
36. can be written as • • • .
The exponent applies only to the base .
37. can be written as • • • .
The exponent applies only to the base .
38. can be written as • • • • • .
The exponent applies only to the base .
39. can be written as • • • • • • .
The exponent applies only to the base .
40. can be written as • • • • .
The exponent applies only to the base .
41. can be written as • • • • • .
The exponent applies only to the base .
The exponent applies only to the base .
42. can be written as • • • • • • • • .
The exponent applies only to the base .
The exponent applies only to the base .
43. Evaluate when is .
means
the coefficient. For example, is the coefficient in
. A constant is a number that is added or
subtracted in an expression. It does not vary. For
example, is the constant in .
•
•
Replace t with 4.
Multiply.
2.1 Introduction to Variables 50 50 Chapter 2 Understanding Variables and Solving Equations
44. • Replace with . 52. Use a calculator. Replace with ,
•
45. Evaluate when is and is .
means
with , and with .
53. Evaluate when is and is , using a
calculator.
• • •
• • •
• •
Replace r with 3 and s with 2.
Multiply left to right.
Replace r with 3
and s with 2.
Use the yx key.
•
46. • • • •
with .
Replace with and
,
Multiply left to right.
• • • •
54.
Use a calculator. Replace with
47. Evaluate when is and is .
means
and with .
,
• •
• •
•
Replace r with 3 and s with 2.
Multiply left to right.
55. Evaluate when is , is , and
is .
Replace x with 4,
48.
Replace with and with .
• •
• • •
y with 2, and
z with 6.
Multiply left to
right within the
abs. value bars.
49. Evaluate when is and is .
means
Replace s with 2
•
Evaluate the
• • • •
and t with 4.
absolute values.
Add.
• • • • Multiply left to right.
• • •
• •
•
56. • •
Replace with , with , and with .
• •
50. • • • • •
and with .
Replace with
• • • • • 57. Evaluate when is and is .
2.1 Introduction to Variables 51 51 Chapter 2 Understanding Variables and Solving Equations
51. Evaluate when is , is , and is ,
using a calculator.
Replace r with 3, s with 2,
and t with 4. x
Replace z with 6
and y with 2.
Follow the order of operations.
,
Use the y key.
Multiply left to right.
Undefined
Numerator: •
Denominator:
Division by 0 is undefined.
2.2 Simplifying Expressions 47 47 Chapter 2 Understanding Variables and Solving Equations
58. Evaluate when is and is .
(b)
The like terms are the constants, and .
•
•
Replace x with 4
and y with 2.
Follow the order of operations.
Numerator: •
Denominator:
•
There are no variable parts.
(c)
The like terms are and since the variable parts match; both are .
The coefficients are and .
(d)
The like terms are and since the variable
parts match; both are .
The coefficients are and .
Undefined Division by 0 is undefined.
Relating Concepts (Exercises 59–60)
2. (a)
These are like terms.
59. (a) Evaluate when is seconds.
Add the coefficients.
The variable part, b,
Replace s with 15.
Divide.
(b)
stays the same.
These are like terms.
Rewrite y3 as 1y3 .
(b)
miles
Evaluate when is seconds.
Add the coefficients.
The variable part, y3 ,
stays the same.
Replace s with 10.
Divide.
(c)
These are like terms.
Rewrite n as 1n.
Change to addition.
(c)
miles
Evaluate when is seconds.
Add the coefficients.
The variable part, n,
stays the same.
Replace s with 5.
Divide.
(d)
These are like terms.
Change to addition.
Add the coefficients.
mile
60. (a) Using part (c) of Exercise 59, the distance
covered in seconds is half of the distance
covered in seconds, or mile.
(e)
The variable part, c,
stays the same.
These are like terms.
Rewrite xy as 1xy
(b) Using part (a) of Exercise 59, the time to Add the coefficients.
cover miles is half the time to cover miles, or
seconds. Or, using parts (b) and (c), find the
number halfway between seconds and
The variable part, xy,
stays the same.
seconds (f)
These are like terms.
2.2 Simplifying Expressions 48 48 Chapter 2 Understanding Variables and Solving Equations
(a) Change to addition.
The like terms are and since the variable
parts match; both are .
Add the coefficients.
Zero times anything is zero.
The coefficients are and .
(c) Using parts (a) and (b) of Exercise 59, find the Change to addition.
number halfway between seconds and
seconds; that is seconds.
Add the coefficients.
The variable part, p2,
2.2 Simplifying Expressions or
stays the same.
2.2 Margin Exercises
1.
(g)
These are like terms.
Rewrite ab as 1ab
2.2 Simplifying Expressions 49 49 Chapter 2 Understanding Variables and Solving Equations
3. (a)
(b)
Rewrite using the
commutative property.
Combine 3b2 7b2
Add the coefficients.
(c) can be written as
• •
•
Change to addition.
Rewrite using the
commutative property.
Rewrite b as 1b.
Add the coefficients
of like terms.
(d)
• •
•
Rewrite x as 1x.
can be written as
(c)
5. (a) can be written as
• •
(d)
Rewrite using the
commutative property.
Add the coefficients
of like terms.
Zero times anything
is zero.
Change to addition.
(b) can be written as
• •
(c) can be written as
• •
• •
•
Rewrite y as 1y.
Rewrite using the
commutative prop .
Add the coefficients
of like terms.
(d)
• •
Multiply.
Change addition
or to subtraction.
(e)
Change to addition.
Rewrite using the
commutative prop.
Add the coefficients
of like terms.
(e)
• •
6. (a)
Multiply.
Change addition
to subtraction.
Distributive property
4. (a) means • • . Using the associative • •
property, it can be rewritten as
• •
•
Rewrite using the
commutative property.
Combine constants.
2.2 Simplifying Expressions 50 50 Chapter 2 Understanding Variables and Solving Equations
(b)
can be written as
or
• •
(b)
• •
Distributive property
Multiply.
• Combine constants.
2.2 Simplifying Expressions 51 51 Chapter 2 Understanding Variables and Solving Equations
(c)
• •
or
(d)
• •
Distributive property
Multiply.
Change to addition.
Rewrite using the
commutative property.
Add the coefficients
of like terms.
Distributive property
Multiply.
7.
8.
9.
These are like terms.
Add the coefficients.
The variable part, r,
stays the same.
These are like terms.
Add the coefficients.
The variable part, t,
stays the same.
These are like terms.
Change to addition.
Rewrite x2 as 1x2 .
Rewrite using the
commutative property.
Combine constants.
Add the coefficients.
The variable part, x2 ,
stays the same.
(e)
10.
• •
Distributive property
Rewrite y as 1y.
Change to addition.
Add the coefficients
of like terms.
11.
These are like terms.
Rewrite p as 1p.
Change to addition.
Add the coefficients.
or
The variable part, p,
2.2 Section Exercises
stays the same.
1.
and are the
12.
only like terms in the expression. The variable
parts match; both are . The coefficients are
and .
2.
and are like 13. These are like terms.
Rewrite a3 as 1a3 .
terms. The variable parts match; both are . The
coefficients are and .
Change to addition.
Add the coefficients.
3. and
are the like terms in the expression. The variable parts match; both are . The coefficients
are and .
The variable part, a3 ,
stays the same.
4.
and 14.
2.2 Simplifying Expressions 52 52 Chapter 2 Understanding Variables and Solving Equations
are like terms. The variable parts match;
both are . The coefficients are and .
5. , , and are
like terms. There are no variable parts; constants
are considered like terms. 15.
Any number minus itself is .
6. , , and
16.
are like terms. There are no variable parts;
constants are considered like terms.
Any number minus itself is .
2.2 Simplifying Expressions 53 53 Chapter 2 Understanding Variables and Solving Equations
17.
These are like terms.
Rewrite xy as 1xy.
Change to addition.
26.
Add the coefficients.
The variable part, xy,
27.
Use the commutative property
to put the constants at the end.
Add the coefficients of like terms.
or stays the same. The only like terms are constants.
18.
28.
or
19.
These are like terms.
Change to addition.
Add the coefficients.
The variable part, t4 ,
stays the same.
29.
,
Write in the understood
coefficients of 1.
Add the coefficients of
like terms.
The variable part, ab 2,
stays the same.
20.
30.
or
21. These are like terms.
Write in the under-
or
stood coefficients
of 1.
Add the coefficients. 2
31.
Write in the understood
coefficients of 1.
Change to addition.
The variable part, y ,
stays the same.
22.
These are like terms.
Rewrite using the
commutative property.
Add the coefficients of
like terms.
23.
Rewrite x as 1x
and x as 1x.
Change to addition.
Add the coefficients.
The variable part, x,
stays the same.
32.
Write in the under-
24.
]
2.2 Simplifying Expressions 54 54 Chapter 2 Understanding Variables and Solving Equations
33.
stood coefficient of 1. Change to
addition. Rewrite using the
commutative property.
25. Use the commutative property
Add the coefficients
to rewrite the expression so that
like terms are next to each other.
Add the coefficients of like terms.
The variable part, a,
stays the same.
•
of like terms.
2.2 Simplifying Expressions 55 55 Chapter 2 Understanding Variables and Solving Equations
34.
44. •
45. By using the associative property, we can write
as
• • • • • .
So, simplifies to .
35. There are no like terms.
46.
•
The expression cannot be simplified.
36. There are no like
terms. The expression cannot be simplified.
37.
Write in the understood coefficients of 1.
Change to addition.
Rewrite using the commutative property.
47. By using the associative property, we can write as
• • • • • • • .
So, simplifies to .
48. •
Write in the understood
Add the coefficients of like terms.
49.
•
•
coefficient of 1.
Rewrite using the
associative property.
38.
50.
•
51.
• •
Distributive property
52.
• •
Distributive property
or
39. By using the associative property, we can write as
• • • .
So, simplifies to .
53.
• •
Distributive property
40. •
54. • •
41. By using the associative property, we can write as
• • • .
55.
• •
Distributive property
2.2 Simplifying Expressions 56 56 Chapter 2 Understanding Variables and Solving Equations
So, simplifies to .
42. •
56. • •
57. Distributive property
43. By using the associative property, we can write as
• • • .
So, simplifies to .
• •
Change addition
to subtraction
of the opposite.
2.2 Simplifying Expressions 57 57 Chapter 2 Understanding Variables and Solving Equations
58. • •
69.
• •
Distributive property
or
59.
• •
Distributive property
Change addition
to subtraction
of the opposite.
Rewrite using the
commutative property.
Add the coefficients
of like terms.
Zero times any
number is 0.
60. • • Zero added to any
or
number is the number
61. Distributive property
• • 70. • •
62. • •
63. Distributive property
• • •
Rewrite using the 71. Distributive property
commutative property.
Combine like terms.
• •
Change to addition.
64. • •
Combine like terms.
Any number plus its
opposite is 0.
65.
• •
Distributive property
Combine like terms.
72. • •
66. • •
73. Distributive property
67.
• •
Distributive property
Change to
addition.
Rewrite using
the
commutative
property. Add
the coefficients
of like terms.
Change addition to subtraction
2.2 Simplifying Expressions 58 58 Chapter 2 Understanding Variables and Solving Equations
• •
Rewrite n as 1n. Change
to addition. Rewrite using
the
commutative property.
Add the coefficients of
like terms.
of the opposite.
or
68.
• •
74. • •
or
Summary Exercises Variables and Expressions 53
75.
• •
Distributive property
Rewrite p as 1p.
Change to addition.
Add the coefficients
of like terms.
Change addition to
subt. of the opposite.
84.
• •
85.
Distributive property
• • • •
Change to addition.
76. Group like terms and add the coefficients.
• •
or
77. A simplified expression usually still has variables,
but it is written in a simpler way. When evaluating
an expression, the variables are all replaced by
specific numbers and the final result is a numerical
answer.
78.
86.
• • • •
• • • •
The answers are equivalent because of the
commutative property of addition.
79. Like terms have matching variable parts, that is,
matching letters and exponents. The coefficients
do not have to match. Examples will vary.
Possible examples: In , the terms
and are like terms. In ,
the terms and are like terms.
80. Add the coefficients of like terms. If no
coefficient is shown, it is assumed to be . Keep
the variable part the same. Examples will vary.
81.
Keep the variable part unchanged when combining
like terms. As shown above, the correct answer is
.
82. In the last step, do not change the sign of the first
term. The correct answer is .
87.
Distributive property
• • • •
•
Change to addition.
Group like terms and add the coefficients.
88.
Summary Exercises
83.
• • •
Distributive prop. Variables and Expressions
Change subtrac-
tion to adding
the opposite.
1.
or 1
is the variable;
1 is the coefficient;
is the constant.
Group like terms
and add the
coefficients.
2.
and are the variables;
is the coefficient.
3. is the variable;
is the coefficient;
is the constant.
54 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 54
4. Expression (rule) for finding the perimeter of an 11. Replace with and with .
octagon of side length :
(a) Evaluate the expression when , the side
length, is yards.
If is multiplied by any number, the result is
Thus, there is no need to make any calculations
since the result is
•
Replace s with 4.
Follow the rule and multiply.
12.
.
Replace with , with , and with
yards is the perimeter.
(b) Evaluate the expression when , the side
length, is inches.
• •
•
Multiply left to right.
•
Replace s with 15.
Follow the rule and multiply.
13. • •
Replace with .
inches is the perimeter.
5. Expression (rule) for finding the total cost of a car
with down payment , monthly payment and
• •
•
Multiply left to right.
number of payments :
(a) Evaluate the expression when the down 14. Replace with and with .
payment is $ , the monthly payment is $ ,
and the number of payments is .
Replace d with $3000, m
• •
•
Multiply left to right.
$ $ •
$ $ ,
with $280, and t with 36.
Multiply before adding.
15. • • •
with .
Replace with and
$ , is the total cost of the car. • • • Multiply left to right.
(b)
Evaluate the expression when the down • •
payment is $ , the monthly payment is $ ,
and the number of payments is .
Replace d with $1750, m
•
with $429, and t with 48. 16. • • • • • • Replace with
$ $ •
$ $ ,
Multiply before adding. and with . Multiply left to right.
• • • • • •
$ , is the total cost of the car.
6. written without exponents is
• • • •
7. written without exponents is
• • • •
8. written without exponents is
• • • • • • • •
55 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 55
• • • • •
• • • •
• • •
• •
•
9. • • •
Replace with . 17.
Use a calculator. Replace with
• • •
• •
•
Multiply left to right. , with , and with .
,
18.
19. 10. Replace with and with .
If is multiplied by any number, the result is
Thus, there is no need to make any calculations
since the result is
56 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 56
20. • •
•
or
21.
22. • •
33. (a) Simplifying the expression correctly:
• •
The student forgot to multiply • .
23.
24. • •
(b)
Simplifying the expression correctly:
• •
• •
Two negative factors give a positive product.
25.
(c) Simplifying the expression correctly:
•
Keep the variable part unchanged; that is, adding
's to 's gives an answer with 's, not 's.
or
26. • •
27.
34. In the last step, do not change the sign of the first
term; keep as . The correct answer is
2.3 Solving Equations Using Addition
2.3 Margin Exercises
28. • •
1. (a)
?
Given equation
Replace c with 95.
110 is more than 80.
No, is not the solution.
29. • •
?
Replace c with 65.
Balances
•
Yes, is the solution.
(No need to check and .)
30. • •
• •
(b)
?
Given equation
Replace c with 28.
57 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 57
No, is not the solution.
31.
• •
?
Replace c with 20.
• • No, is not the solution.
?
Replace c with 24.
or No, is not the solution. ?
32.
• •
Replace c with 32.
Balances
• • Yes, is the solution.
58 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 58
2. (a) Solve for . Combine like terms.
To get by itself, add the opposite of , which is To get r by itself,
. To keep the balance, add to both sides.
add 1 to both sides.
The solution is .
or
Check
The solution is .
Check
Original equation
Replace y with 7.
• •
Replace r with 2.
Balances, so solution is .
(b) Solve for .
Change to addition.
2.3 Section Exercises
Balances
1. Replace with , , , and .
To get by itself add the opposite of , which is
, to both sides.
?
?
Given equation
Replace n with 58.
The solution is .
Yes, is the solution.
(No need to check , , and .)
Check
Original equation 2. Replace with , , , and .
Replace b with 4.
Balances
?
?
?
?
3. (a)
Rewrite both sides by
?
?
changing subtraction to addition. Combine like ? ?
terms.
The check for balances, so is the solution.
To get by itself add the opposite of , which is
, to both sides.
)
3.
and .
?
Replace with , , ,
Given equation
Replace y with 4.
No, is not the solution.
59 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 59
The solution is . ? Replace y with 16.
Check
Original equation
Replace k with 4.
Yes, is the solution.
(No need to check and .)
( ) ( ) 4. Replace with , , ,
Balances and .
(b) Change to ?
?
addition.
Rewrite the left side by using the commutative
?
property. is the solution. (No need to check .)
60 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 60
5. (a) Add to both sides because 12.
gives on the left side. Add 8 to both sides.
(b) Add to both sides
because gives on the right side. The solution is .
6. (a) Add to both sides Check
because gives on the left side. Replace a with 9.
(b) Add to both sides because Balances
gives on the right side. 13.
7.
Add the opposite of
5, 5, to both sides.
Add the opposite of
4, 4, to both sides.
The solution is .
The solution is . Check
Check
Replace p with 4.
Balances
14.
Replace k with 18.
Balances
8.
Add 3 to both sides.
Add 9 to both sides.
The solution is .
The solution is .
Check
Check
Replace a with 9.
Balances
Replace y with 16.
Balances
15.
9.
Change to addition.
Add the opposite of
2, 2, to both sides.
Change to addition.
Add the opposite of
6, 6, to both sides.
The solution is .
The solution is .
Check
Check
10.
Replace r with 10.
Balances
Change to addition. Add 5 to
both sides.
61 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 61
16.
Replace y with 6.
Balances
Change
to
addition.
Add 15
to both
sides.
The solution is .
The solution is .
Check
Check
Replace b with 8.
Balances
17.
Replace k with 15.
Balances
11. Add the opposite of
Add the opposite of
3, 3, to both sides.
The solution is .
13, 13, to both sides.
The solution is .
Check
Replace n with 8.
Check
Replace r with 6.
Balances Balances
62 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 62
18. Correct solution:
Add 19 to both sides.
The solution is .
Change to addition.
Add the opposite of
Check
Replace z with 7.
5, 5, to both sides.
Balances The solution is .
19.
Change to addition.
Add the opposite of
12, 12, to both sides.
Check
Replace z with 8.
Change to addition.
Balances
The solution is .
24.
The given solution is .
Check
Replace x with 0.
Change to addition.
Check
Replace x with 13.
Balances
Balances is the correct solution.
20. 25. The given solution is .
Change to addition.
Add 3 to both sides.
Check
Replace x with 18.
Balances
The solution is .
is the correct solution.
Check
Replace m with 0.
26.
The given solution is .
21.
Balances
Check
Replace k with 5.
Does not balance
Add the opposite of
2, 2, to both sides.
The solution is .
Correct solution:
Add the opposite of
2, 2, to both sides.
Check
Replace t with 3.
The correct solution is .
22.
63 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 63
Balances Add 10 to both sides.
Check
Replace k with 9.
Balances
27. The given solution is .
The solution is .
Check
Check
Replace w with 9.
Balances
Correct solution:
Replace b with 10.
Does not balance
23. The given solution is .
Check
Replace z with 2.
Change to addition.
Does not balance
Add the opposite of
10, 10, to both sides.
The solution is .
64 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 64
Replace b with 12.
Balances
Replace t with 0.
31. Change to addition.
Simplify the left side. Balances
Change to addition. 36.
Add 2 to both sides.
Check
Replace b with 0.
Balances
32.
Add 4 to both sides.
28. The given solution is .
The solution is .
Check
Replace a with 0.
Does not balance
Check
Balances
Correct solution: 33.
Change to addition.
Add.
Add the opposite of Add 10.
14, 14, to both sides.
The correct solution is .
Check
The solution is .
Check
Replace a with 14.
Replace b with 30.
Balances Balances
29.
Simplify the right side.
Change to addition.
34.
Add 4 to both sides. Add 1 to both sides.
The solution is .
The solution is .
Check
Replace c with 6.
Balances
Check
Replace w with 17.
Balances
30. 35.
Change to addition.
Simplify the right side.
Add 2 to both sides.
The solution is .
Check
Check
The solution is .
65 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 65
Check
The solution is .
Add 8 to both sides.
The solution is .
Replace y with 5.
Change to addition.
Balances
Check
Replace p with 0.
Balances
66 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 66
37.
Change to addition.
42.
Check
Combine like terms.
1z is the same as z.
The solution is .
Add 2.
• •
Replace z with 7.
Change to add.
Balances
The solution is .
43.
Change to addition.
38.
The solution is .
44.
Combine like terms.
The solution is .
Check Change to addition.
•
Replace r with 5.
Balances
45.
Combine like terms.
The solution is .
Rearrange and Change to addition.
39.
combine like terms. Add 75 to both sides.
The solution is .
Add 2 to both sides. 46.
Change to addition.
Check
The solution is .
Add 200 to both sides.
• •
40.
Replace w with 3.
Balances
47.
The solution is 1 .
Rearrange and
combine like terms.
Add 3 to both sides.
Add 4 to both sides. The solution is .
67 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 67
The solution is .
48.
Check
Let t = 5.
Balances
Add 4 to both sides.
The solution is .
49.
41. Change to addition. Add 31 to both sides.
Combine like terms.
The solution is .
Add 4 to both sides. 50.
Add 72 to both sides.
1x is the same as x.
The solution is .
The solution is .
68 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 68
51.
Change to
addition.
Rearrange and
60. Check
combine like terms.
Add 9 to both sides.
Does not balance
52.
The solution is .
To correct the errors, change to . Then, add to both sides, not .
The correct solution is .
61.
Add the opposite of
Add 1.
0 0
10, 10, to both sides.
The solution is .
53.
There were graduates this year.
62.
( ) ( ) Change to addition.
0 Add the opposite of 0
Combine like terms. 10, 10, to both sides.
54.
Add 6 to both sides.
The solution is .
There were graduates last year.
63.
Change to addition. Add 37 to both sides.
Combine like terms.
Add 5 to both sides.
55.
The solution is . When the temperature is degrees, a field cricket chirps times (in seconds).
Add 91 to both sides. 64.
Add 37 to both sides.
56.
The solution is .
Change to addition.
When the temperature is degrees, a field cricket
Add 28 to both sides. chirps times (in seconds).
65.
69 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 69
The solution is . Change to addition.
57.
Combine like terms.
Add 65 to both sides.
Add 5 to both sides.
The solution is .
Ernesto's parking fees average $ per month in
winter.
58.
Add 6 to both sides.
66.
Change to addition.
Add 56 to both sides.
The solution is .
59. No, the solution is , the number used to
replace in the original equation. Aimee's parking fees average $ per month in
winter.
70 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 70
67.
Add the opposite of
10, 10, to both sides.
Change all subtractions to additions.
The solution is .
Commutative property
(b) Equations will vary. Some possibilities are:
Combine like terms.
Add 11 to both sides.
Add the opposite of
6, 6, to both sides.
The solution is .
The solution is .
68.
Add the opposite of
5, 5, to both sides.
The solution is .
The solution is . 72. (a) Add the opposite of
69.
Change subtraction within absolute value to
addition and rearrange the terms.
1, 1, to both sides.
The solution is .
Simplify inside absolute value bars. Collect like
terms.
(b)
Change to addition.
Add the opposite of
Evaluate absolute values. 1, 1, to both sides.
Change to addition.
Add 6 to both sides. or The solution is .
The solution is .
(c) $
$
Add the opposite
of $2 50, $2 50,
70. to both sides.
$ $
$
The solution is $ .
(d) Equations will vary. Some possibilities are:
$ $ The solution is $ .
71 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 71
$ $ The solution is $ .
The solution is .
Relating Concepts (Exercises 71–72)
71. (a) Equations will vary. Some possibilities are:
Change to addition.
2.4 Solving Equations Using Division
2.4 Margin Exercises
1. (a) Solve .
Use division to undo multiplication. Divide both
sides by the coefficient of the variable, which is .
Add 1 to both sides.
The solution is .
The solution is .
2.4 Solving Equations Using Division 63 63 Chapter 2 Understanding Variables and Solving Equations
Check
•
Original equation
Replace s with 11.
Check
Balances Replace p with 1.
(b)
Divide both
Balances
The solution is .
Check
sides by 9.
3. (a)
Write in the understood
1 as the coefficient of k.
Divide both
• Replace p with 3.
sides by 1.
Balances The solution is .
(c)
The solution is .
Divide both
sides by 5.
Check
(b)
•
Replace k with 12.
Balances
Check
• Replace x with 8.
Write t as 1t.
Divide both
(d)
Balances
sides by 1.
The solution is .
Divide both
sides by 7.
Check
The solution is .
Check
(c)
•
Replace t with 7.
Balances
•
Replace t with 10.
Balances
Write m as 1m.
Divide both
sides by 1.
2. (a)
Combine like terms.
The solution is .
Divide both
sides by 4.
The solution is .
Check
•
Check
• • Replace n with 7.
2.4 Solving Equations Using Division 64 64 Chapter 2 Understanding Variables and Solving Equations
2.4 Section Exercises
1.
Divide both
Balances
sides by 6.
(b)
Change to addition. Rewrite p as 1p.
The solution is .
Check
• Replace z with 2.
Combine like terms.
2.
Balances
Check
•
Divide both sides by 13.
The solution is .
The solution is .
Balances
2.4 Solving Equations Using Division 65 65 Chapter 2 Understanding Variables and Solving Equations
3. 11.
Divide both Divide both
sides by 12.
The solution is .
sides by 5.
The solution is .
Check
• Replace r with 4.
Check
•
Replace b with 5.
Balances Balances
4.
The solution is .
Check
Balances
•
12.
Check
Balances
•
5.
Divide both
sides by 3.
The solution is .
13.
The solution is .
Combine like terms.
Divide both
Check
sides by 2.
• Replace y with 0. The solution is .
6.
Balances
Check
•
Check
•
Replace r with 3.
Balances
Balances
14.
Check
The solution is . •
7.
Divide both
sides by 7.
Balances
The solution is .
The solution is .
Check
•
Replace k with 10.
Balances
15.
Change to addition.
8. Check
Rewrite p as 1p.
Combine like terms.
•
Divide both
2.4 Solving Equations Using Division 66 66 Chapter 2 Understanding Variables and Solving Equations
The solution is .
9.
Balances
Check
sides by 4.
The solution is .
Divide both
sides by 9.
• Replace p with 3.
The solution is .
Change to addition.
Check
• Replace r with 6. Balances
Balances 16.
10.
Check
•
Change to addition.
Combine like terms.
Divide both
Balances sides by 10.
The solution is . The solution is .
2.4 Solving Equations Using Division 67 67 Chapter 2 Understanding Variables and Solving Equations
Check
•
Replace z with 2.
Change to addition.
24.
Original equation
Change to addition.
Combine like terms.
Divide both
sides by 12.
Balances The solution is .
17.
Original equation
Change to addition.
Combine like terms.
Divide both
25.
Original equation
Divide both
sides by 9.
The solution is .
sides by 5.
The solution is . 26. Original equation
18.
Original equation
Combine like terms.
Divide both
sides by 8.
Divide both
sides by 10.
The solution is .
The solution is . 27.
Original equation
Change to addition.
19. Original equation Combine like terms.
Change to addition.
Rewrite x as 1x.
Divide both
sides by 6.
Combine like terms. The solution is .
20.
Divide both
sides by 8.
The solution is .
Original equation
Change to addition.
Rewrite c as 1c.
Combine like terms.
28.
Original eq
Combine.
Divide both
sides by 7.
The solu-
tion is .
Divide both
sides by 3.
The solution is .
29.
Original equation
Change to addition.
Combine like terms.
21.
Original equation
Change to addition.
Divide both
sides by 4.
Rewrite w as 1w.
Combine like terms. The solution is .
1w is the same as w.
30.
Original equation
The solution is . Change to addition.
2.4 Solving Equations Using Division 68 68 Chapter 2 Understanding Variables and Solving Equations
22.
Original equation
Change to addition.
Combine like terms.
Combine like terms.
Divide both
sides by 11.
1t is the same as t. The solution is .
The solution is .
31.
Original equation
23.
Original equation
Change to addition.
Combine like terms.
Divide both
sides by 12.
• •
To multiply on the left, use
the associative property.
Divide both
sides by 6.
The solution is . The solution is .
2.4 Solving Equations Using Division 69 69 Chapter 2 Understanding Variables and Solving Equations
32. Original equation 40. Original equation
• • To multiply on the left, use Write in the understood 1.
sides by 12.
The solution is .
Divide both
sides by 1.
The solution is .
35. Original equation
• • Associative property 44. Original equation
Divide both
Write in the understood 1.
Divide both
sides by 8.
sides by 1.
The solution is . The solution is .
the associative property.
Divide both
sides by 8.
Divide both
sides by 1.
The solution is .
The solution is . 41. Original equation
33.
• •
Original equation
To multiply on the right,
use the associative prop.
Write in the understood 1.
Divide both
sides by 1.
The solution is .
Divide both
sides by 25.
42.
Original equation
Write in the understood 1.
34.
The solution is .
Original equation
Divide both
sides by 1.
• • To multiply on the right,
use the associative prop. The solution is .
43. Original equation
Divide both Write in the understood 1.
36.
• •
Original equation
Associative property
45.
Original equation
Write in the understood 1.
Divide both
sides by 20.
Divide both
sides by 1.
37.
The solution is .
Original equation
46.
The solution is .
Original equation
• •
Associative property
Write in the understood 1.
Divide both
Divide both sides by 1.
sides by 30. The solution is .
38.
•
•
2.4 Solving Equations Using Division 70 70 Chapter 2 Understanding Variables and Solving Equations
The solution is . Original
equation Associative property
47. Each solution
is the
opposite of
the number in
the equation.
So the rule is:
When you
change the
sign of the
variable from
negative to
positive, then
change the
number in the
equation to its
opposite.
Divide both
sides by 10. In , the opposite of is , so .
39.
The solution is .
Original equation
Write in the understood 1.
48. Equations will vary. Some possibilities are (i) and (ii) .
(i)
Divide both
sides by 1.
Divide both
sides by 5.
The solution is . The solution is .
2.5 Solving Equations with Several Steps 67 67 Chapter 2 Understanding Variables and Solving Equations
(ii)
Change to addition.
Combine like terms.
Divide both
56.
• • • •
Associative property
sides by 2.
The solution is .
Change to addition.
Combine like terms.
49. Divide by the coefficient of , which is , not by Divide both
the opposite of . sides by 50.
The correct solution is .
The solution is .
57.
• • • •
50. You can divide both sides of an equation by the
same nonzero number and keep the equation
balanced.
Assoc. prop. Simplify within the absolute values.
Simplify the absolute values.
Combine like terms.51.
Divide both
sides by 3.
Divide both
sides by 70.
The length of one side is feet.
52.
The solution is .
58.
Divide both
sides by 3.
Simplify within the absolute value.
The length of one side is inches. Simplify the absolute value.
53.
Divide both
Combine like terms.
Divide both
sides by 5. sides by 10.
The length of one side is meters.
The solution is .
54.
Divide both
sides by 5.
2.5 Solving Equations with Several Steps
2.5 Margin Exercises
1. (a) To get 2r by itself,
2.5 Solving Equations with Several Steps 68 68 Chapter 2 Understanding Variables and Solving Equations
The length of one side is yards.
55.
• • • •
7 to both sides.
To solve for r,
Associative property
divide both sides by
the coefficient, 2.
Change to addition.
Combine like terms. The solution is .
Divide both
sides by 1.
Check
•
Replace r with 3.
The solution is . Balances
2.5 Solving Equations with Several Steps 69 69 Chapter 2 Understanding Variables and Solving Equations
(b)
Change to addition.
Add 9 to both sides.
Rewrite p as 1p.
Add 3p.
Divide both Add 6.
sides by 10.
The solution is .
Check Divide both
•
Replace r with 2.
3. (a)
sides by 2.
The solution is .
Balances • • Distribute on the right.
2. (a) Solve, keeping the variable on left side. Change to addition.
Add 2y to both sides.
Add 4 to both sides.
Change to addition.
Add 1 to both sides.
Divide both
sides by 4.
The solution is .
or
The solution is .
Check
Replace y with 2.
Solve, keeping the variable on the right side.
Balances
Add 3y to both sides. (b)
• • Distribute on the left.
Add 7 to both sides. Add 20 to both sides.
Divide both Divide both
sides by 1.
sides by 5.
2.5 Solving Equations with Several Steps 70 70 Chapter 2 Understanding Variables and Solving Equations
The solution is . The solution is .
(b) Solve, keeping the variable on left side.
Rewrite p as 1p.
Check
Replace m with 0.
Balances
Add 1p to both sides.
Change to addition.
Add 2 to both sides.
Divide both
sides by 2.
The solution is .
(c)
• •
Distribute on the left.
Change to addition.
Add 12 to both sides. Divide both
sides by 6.
Solve, keeping the variable on the right side. The solution is .
2.5 Solving Equations with Several Steps 71 71 Chapter 2 Understanding Variables and Solving Equations
Check
Replace t with 5.
Balances
Check
Let p = 1.
Balances
4. (a)
• •
Distribute. 2.
Check
Variables left
Add 2b.
Balances
Add 21.
The solution is .
or
3.
Change to addition.
The solution is . Add 6 to both sides.
Check
•
Divide both
Balances
sides by 8.
(b)
• •
Distribute
Check
The solution is .
Add the
opposite.
Combine
Replace y with 1.
Balances
like terms.
Add 2n.
4.
Check
Balances
Add 6.
The solution is .
Divide both
sides by 6.
The solution is .
5.
To get 9a by itself,
add 10 to both sides.
Check
•
Let n = 2.
Divide both
sides by 9.
The solution is .
2.5 Solving Equations with Several Steps 72 72 Chapter 2 Understanding Variables and Solving Equations
2.5 Section Exercises
Balances
Check
Replace a with 2.
Balances
1.
To get 7p by itself,
add 5 to both sides.
6.
Check
Divide both
Balances
sides by 7.
The solution is .
The solution is .
2.5 Solving Equations with Several Steps 73 73 Chapter 2 Understanding Variables and Solving Equations
7.
To get 3m by itself,
add 1 to both sides.
11. Solve, keeping the variable on the left side.
Change to addition.
Divide both Add 4p to both sides.
sides by 3.
The solution is .
Add 2 to both sides.
Check
Replace m with 0.
Divide both
Balances
sides by 2.
8.
The solution is .
Solve, keeping the variable on the right side.
Change to addition.
The solution is .
Check
Add 6p to both sides.
Add 6 to both sides.
Balances
Divide both
9.
Change to addition.
To get 5x by itself,
add 4 to both sides.
sides by 2.
The solution is .
Check
Divide both
sides by 5.
Balances
The solution is .
Check
Replace x with 4.
Balances
12. Left side: Right side:
2.5 Solving Equations with Several Steps 74 74 Chapter 2 Understanding Variables and Solving Equations
Check
Balances
Balances
10.
The solution is .
Check
The solution is .
2.5 Solving Equations with Several Steps 75 75 Chapter 2 Understanding Variables and Solving Equations
13. Solve, keeping the variable on the left side.
15. simplifies to
.
Change to addition.
Add 6k to
Add 2a to
both sides.
both sides.
Add 6 to
Add 18 to
both sides.
both sides.
Divide both
sides by 8.
The solution is .
Divide both
sides by 5.
Solve, keeping the variable on the right side.
The solution is .
Change to addition.
Add 2k to
both sides.
Check
Balances
Add 10 to
16. simplifies to
.
both sides.
Divide both
sides by 8.
Add 10 to
both sides.
Check
The solution is .
Divide both
sides by 5.
Replace k with 2.
Balances
The solution is .
14. Left side: Right side:
Check
2.5 Solving Equations with Several Steps 76 76 Chapter 2 Understanding Variables and Solving Equations
Balances
17. Neither side can be simplified, so solve the
equation.
Add 3t to
both sides.
The solution is .
Check
Divide both
sides by 11.
The solution is .
Balances
Check
Replace t with 0.
Balances
2.5 Solving Equations with Several Steps 77 77 Chapter 2 Understanding Variables and Solving Equations
23.
Distribute.
Divide both Add 8 to both sides.
sides by 2.
The solution is .
Divide both
sides by 2.
Check The solution is .
24.
Balances Distribute.
Add 18 to both sides.
simplifies to
Distribute.
Add 8 to both sides.
18. Neither side can be simplified, so solve the 21.
equation.
Distribute.
Change to addition.
Add 9z to
both sides.
Divide both
sides by 24.
Add 16 to both sides. Divide both
sides by 8.
Check
The solution is . The solution is .
22.
Replace z with 0. Distribute.
Balances Change to addition.
19. simplifies to .
Add 2 to
Add 36 to both sides.
Divide both
sides by 9.
both sides. The solution is .
20. .
Divide both
sides by 3.
Add 2z to
both sides.
25.
The solution is .
Add 12 to
both sides.
Divide both
sides by 4.
2.5 Solving Equations with Several Steps 78 78 Chapter 2 Understanding Variables and Solving Equations
Divide both
sides by 7.
The solution is .
26.
Distribute.
The solution is . Add 15 to both sides.
Check
Divide both
sides by 5.
Balances The solution is .
2.5 Solving Equations with Several Steps 79 79 Chapter 2 Understanding Variables and Solving Equations
27.
Divide both
Distribute. sides by 9.
Change to addition.
Add 30 to both sides.
The solution is .
33.
Add 18 to b
30.
Distribute.
Change to addition.
Divide both
oth sides.
sides by 6.
The solution is . Divide both
sides by 6.28.
Distribute.
The solution is .
Change to addition. 34.
Add 49 to both sides. Change to addition.
Divide both
sides by 7.
Add 40 to both sides.
Divide both
The solution is .
sides by 8.
29.
Distribute.
35.
The solution is .
Change to addition. Change to addition.
Add 24 to both sides.
Add 12 to both sides.
Divide both
Divide both
sides by 12.
sides by 9.
The solution is . The solution is .
36.
Add 33 to both sides.
Add 24 to both sides.
Divide both
Divide both sides by 8.
sides by 11. The solution is .
The solution is .
37.
31. Add 3x to both sides.
2.5 Solving Equations with Several Steps 80 80 Chapter 2 Understanding Variables and Solving Equations
Divide both
sides by 2.
The solution is .
38.
Change to addition.
Add 2n to both sides
Distribute.
Divide both
sides by 9.
The solution is .
Distribute.
Change to addition.
Add 4 to both sides.
Divide both
sides by 2.
The solution is .
32.
.
Change to addition.
Add 9 to both sides.
2.5 Solving Equations with Several Steps 81 81 Chapter 2 Understanding Variables and Solving Equations
39.
Change to addition.
Divide both
sides by 5.
Add 2a. The solution is .
Add 7 to both sides.
44.
Change to addition.
Combine like terms.
Add 1y.
Divide both Divide both
sides by 6. sides by 4.
The solution is . The solution is .
45. 40. Distribute.
Change to addition. Change to add.
Add 1r. Group terms.
Combine terms.
Add c.
Add 8.
Divide both
sides by 9.
Add 56.
The solution is .
41.
Divide
both sides
by 9.
Change to addition.
Add 5b to both sides.
The solution is .
46.
Add 28 to both sides.
Divide both
sides by 7.
The solution is .
2.5 Solving Equations with Several Steps 82 82 Chapter 2 Understanding Variables and Solving Equations
42.
Change to add.
Add 8t.
The solution is .
Add 9.
47.
Distribute.
Divide both
sides by 5.
The solution is .
Add the
opposites.
Group like
terms.
Combine
43.
like terms.
Change to addition. Add 13y.
Combine like terms.
Add 2k.
Add 5.
2.5 Solving Equations with Several Steps 83 83 Chapter 2 Understanding Variables and Solving Equations
52.
Divide
by 2.
Add 3x.
The solution is .
48.
Add 8.
Divide both
sides by 7.
The solution is .
53.
Add 6p.
The solution is .
49.
Add 17.
Change to add. Divide
Combine terms. by 16.
Add 6.
Divide both
sides by 1.
The solution is .
54.
The solution is .
50.
Change to add.
Combine terms.
Add 5.
Divide both
2.5 Solving Equations with Several Steps 84 84 Chapter 2 Understanding Variables and Solving Equations
sides by 1.
The solution is .
The solution is .
51.
Distribute.
55.
Change to add. Add 4.
Add 6c.
Add b.
Add 42.
The solution is .
The solution is .
2.5 Solving Equations with Several Steps 85 85 Chapter 2 Understanding Variables and Solving Equations
Add 2t to both sides
(addition property). 64.
Add 5 to both sides
(addition property).
56.
(b) The sum of and a positive number is
negative, so must be negative.
62. (a) It must be positive, because the sum of two
Add 1. negative numbers is always negative.
(b) The sum of and a negative number is positive, so must be positive.
The solution is .
57. The series of steps may vary. One possibility is:
63. (a) It must be positive. When the signs are the
same, the product is positive, and when the signs
are different, the product is negative.
Change to addition.
Divide both sides by 5
(b) The product of and a negative number is
negative, so must be positive.
(a) It must be negative also. When the signs are
different, the product is negative, and when the
signs match, the product is positive.
(b) The product of and a negative number is
(division
property).
positive, so must be negative.
The solution is .
58. Multiplication distributes over both addition and
Chapter 2 Review Exercises
1. (a) In the expression , is the variable,
subtraction. Examples will vary. Some is the coefficient, and is the constant term.
possibilities are is and
is .
59. Check
(b) The term that has as the constant term and
as the coefficient is .
2. (a) Evaluate when is .
The check does not balance, so is not the correct solution. The student added to on the left side, instead of adding to . The correct
•
Replace c with 15. Order test tubes.
solution, obtained using , ,
is .
(b) Evaluate when is .
60. Check
•
Replace c with 24. Order test tubes.
The check does not balance, so is not the 3. (a) means • • • • •
correct solution.
2.5 Solving Equations with Several Steps 86 86 Chapter 2 Understanding Variables and Solving Equations
Student did not
distribute the
over the .
(
b
)
4.
(a)
means
• • •
•
means
•
•
Repla
ce n
with
3.
(b)
means
The correct solution is .
Relating Concepts (Exercises 61–64)
61. (a) It must be negative, because the sum of two
positive numbers is always positive.
• •
• •
•
Replace n with 3.
8. Associative property
• •
•
9.
• •
Distribute.
Chapter 2 Review Exercises 77
(c) means
• • •
Replace m with 2
12.
Distribute.
• • • and p with 4.
or
• •
•
13. Expressions will vary. One possibility is .
14. Add 16 to both sides.
(d)
means
• • • • • •
• • • • • •
Replace m with 2
and n with 3.
The solution is .
Check
Replace n with 11.
• • • • • Balances
• • • •
• • •
• •
15.
•
5.
Combine like terms.
The solution is .
or Check
6.
Rewrite x as 1x.
Change to addition.
Combine like terms.
16.
Balances
7.
• •
Associative property
Divide both
sides by 6.
• The solution is .
17.
Combine like terms.
Divide both
sides by 4.
10.
Distribute.
18.
The
solution is
.
Divide both
• •
sides by 7.
or The solution is .
11.
• •
Distribute.
19.
Divide both
sides by 3.
The solution is .
78 Chapter 2 Understanding Variables and Solving Equations
20.
Divide both
sides by 3.
Divide both The solution is .
sides by 15. 26. [2.5]
The solution is . Distribute.
21.
Add 6 to
both sides.
Divide both
sides by 1.
The solution is .
Divide both
sides by 2.
22. The solution is .
27. [2.5]
Add 8w to
both sides.
Add 6y to
both sides.
Divide both
Add 4 to
both sides.
sides by 4.
The solution is .
The solution is .
Divide both
sides by 4.
28. [2.5]
Add the
opposites.
Combine
like terms.
Divide both
sides by 5.
23.
• • Distribute.
The solution is .
Add 8 to 29. [2.5]
both sides. Divide both
sides by 4.
The solution is .
Add 7 to
both sides.
Divide both
sides by 3.
The solution is .
24. Add 4 to
30. [2.5] Add 3b to
79 Chapter 2 Understanding Variables and Solving Equations
both sides.
both sides.
Divide both
sides by 2.
Add 6 to
both sides.
The number of employees is .
Divide both
25. [2.5]
sides by 2.
Add 4a to
both sides.
The solution is .
31. [2.3]
Add 3 to
both sides.
The solution is .
Chapter 2 Test 79
32. [2.5]
Distribute.
Add 3n to
Chapter 2 Test
1. In the expression , is the coefficient,
both sides. is the variable, and is the constant term.
Add 3 to
both sides.
Divide both
sides by 3.
2. Evaluate the expression when is and
is .
• •
The solution is .
33. [2.5]
Buy hot dogs.
Distribute. 3. means • • • • • • •
Add 42 to
4. means • • • • •
both sides. 5. means
• • •
Divide both
sides by 21.
• • • Replace s with 5
and t with 4.
34. [2.5]
The solution is . • •
•
6.
Add 6d.
The solution is .
Add 13 to
both sides.
Divide
by 4.
7.
35. [2.5]
8.
or
Add 12b to
both sides.
9.
There are no like terms.
The expression cannot be simplified.
Divide 10.
The solution is .
by 18.
• •
Associative property
of multiplication
80 Chapter 2 Understanding Variables and Solving Equations Cumulative Review Exercises (Chapters 1–2) 80
11.
• •
12.
Associative property
of multiplication
Distributive property
19.
Add 6 to both sides.
Divide both
sides by 3.
13.
• • Distributive property
The solution is . Check
Replace a with 3.
14.
Distributive property
20.
Balances
Combine like terms.
or
15.
Combine
The solution is .
Add 8.
16.
like terms.
21.
Add 9 to both sides.
Divide both
sides by 1.
The solution is .
Check
Replace x with 5.
The solution is .
Balances 22.
17.
Add 2x to both sides.
Divide both
sides by 7.
Add 9 to both sides.
Divide both
sides by 7.
The solution is .
Check
• )
Replace w with 11.
The solution is .
81 Chapter 2 Understanding Variables and Solving Equations Cumulative Review Exercises (Chapters 1–2) 81
Balances 23.
18.
Add 3m to
both sides.
Divide both
sides by 1.
The solution is . both sides.
Check
•
Replace p with 14.
Divide both
sides by 4.
Balances The solution is .
82 Chapter 2 Understanding Variables and Solving Equations Cumulative Review Exercises (Chapters 1–2) 82
24.
Add 6b to
(b) lies to the left of on the number line, so
.
4. (a) Commutative
both sides.
property of addition: Changing the order of the
Add 42 to
both sides.
addends does not change the sum.
(b) • Multiplication property of
zero: Multiplying any number by gives a
product of .
(c) • •
Distributive
The solution is .
25.
Distribute.
property: Multiplication distributes over addition.
5. (a)
Underline the hundreds place:
Add 24 to
both sides.
Divide both
sides by 3.
The next digit is or less, so leave as . Change
and to .
(b) , ,
Underline the thousands place: ,
The next digit is or more, so add to , write the
and add to the ten-thousands place. Change
and to .The solution is .
26. Addition property of equality: Start with a possible
solution, for example, . Now add an
abitrary number, say , to both sides, to give us
the equation .
6.
7.
Change to addition.
Division property of equality: Start with a possible
solution, for example, . Now multiply both
sides by an abitrary number, say , to give us the
equation .
8.
6 is 6 units from 0.
4 is 4 units from 0.
Thus, equations will vary. Two possibilities are Same sign, positive product
Solving:
and
. 9.
•
Same sign, positive product
Add 5 to both sides.
Divide both
sides by 6.
10.
Same sign, positive quotient
11.
Addition of a number
Cumulative Review Exercises
12.
and its opposite is zero.
83 Chapter 2 Understanding Variables and Solving Equations Cumulative Review Exercises (Chapters 1–2) 83
(Chapters 1–2)
1. , , , in words is three hundred six
billion, four thousand, two hundred ten.
• •
•
Exponent
Multiply left to right.
2. Eight hundred million, sixty-six thousand:
, ,
13. is undefined. Division by is undefined.
3. (a) lies to the right of on the number line,
so .
14. •
Different signs, negative product
84 Chapter 2 Understanding Variables and Solving Equations Cumulative Review Exercises (Chapters 1–2) 84
15.
Change to addition.
22. shares rounds to .
$ rounds to $ .
$ stays $ (it's a single digit number).
16.
Different signs, negative quotient
17.
18.
Each stock dropped in value by $ and Doug
owned shares. Multiply to find out how much
money he lost. Then, subtract this amount from
the original total value.
Estimate: $ • $
Change to addition. Exact: $ • $
Multiply.
Add.
His shares are now worth $ .
23. $ rounds to $ .
$ rounds to $ .
19.
Numerator:
Denominator:
Multiply.
Change to addition.
Add left to right.
months (in one year) rounds to .
Estimate: $ $
$ $
Exact: $ $ $ $
She will spend $ for rent and parking in one
year.
24. means • • • • • •
25. means
• • • •
Replace x with 5
20.
• • Exponents
Last step is division:
days rounds to .
26.
• • • •
• • •
• •
•
and y with 2.
Multiply left to right.
miles rounds to . Change to addition.
Average distance "per" day implies division.
Estimate: miles
miles per day days
miles
or
27.
Combine like terms.
Exact:
days miles per day
Write the understood
coefficients of .
The average distance the tiger traveled each day
was miles.
21.
degrees
rounds
to .
"Rise"
85 Chapter 2 Understanding Variables and Solving Equations Cumulative Review Exercises (Chapters 1–2) 85
of degrees rounds to .
•
Change to addition.
Combine like terms.
A start temperature of degrees followed by a
rise of degrees implies addition.
28.
Estimate: degrees
Exact: degrees
The daytime temperature was degrees. or
86 Chapter 2 Understanding Variables and Solving Equations Cumulative Review Exercises (Chapters 1–2) 86
33.
Add 2 to both sides.
37.
29.
• Associative property
35.
Add 2m to both sides.
30. Add 6 to both sides.
• • Distribute.
Divide both
31.
The solution is .
sides by 3.
or
32.
Check
Add x to both sides.
Replace m with 4.
Divide both Balances
sides by 2.
36.
The solution is .
Check
Add 4x to both sides.
Replace x with 4. Add 18 to both sides.
Balances Divide both
sides by 14.
The solution is .
Divide both
sides by 7.
Divide both
sides by 1.
The solution is . The solution is .
Check
38.
34.
87 Chapter 2 Understanding Variables and Solving Equations Cumulative Review Exercises (Chapters 1–2) 87
Replace y with 6.
Balances
Add 1b
to both
sides.
Add 1 to
Divide both
sides by 3.
both sides.
Divide both
The solution is .
Check
Replace k with 7.
Balances
The solution is .
sides by 2.
88 Chapter 2 Understanding Variables and Solving Equations Cumulative Review Exercises (Chapters 1–2) 88
39.
Add 2 to
both sides.
40.
Add 18.
Add 8t to
both sides.
Add 10y
Divide
Divide both
by 4.
The solution is .
89 Chapter 2 Understanding Variables and Solving Equations Cumulative Review Exercises (Chapters 1–2) 89
sides by 6.
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The solution is .