43

Chapter 2 Understanding Variables and Solving Equations

2.1 Introduction to Variables

43

(b)

CHAPTER 2 UNDERSTANDING VARIABLES AND SOLVING EQUATIONS

Value of

is is is is

2.1 Introduction to Variables

6.

2.1 Margin Exercises

(a) Multiplying any number of .

1.

by gives a product

Any The expression is The variable is limit.

times

number

.

zero

. It represents the class •

The constant is . 2.

Expression

(a) Evaluate the expression

when is

(b) Changing the grouping of addends does not change the sum.

.

, ,

Replace c with 25. 7. Order

can be written as • • • • is used as a factor times.

(a)

books.

(b) Evaluate the expression

when is

can be written as • • • •

(b)

.

(c)

Replace c with 60.

can be written as

Order 3.

books. when is

feet.

• •

Replace s with 3 feet.

means

(a) •

(a) Evaluate the expression

can be written as • • • • • •

(d) 8.

Replace y with 5. Multiply left to right.

feet feet

The perimeter of the square table is (b) Evaluate the expression

(b)

feet.

when is

miles.

• • •

means Replace r with 6 and s with 3.

44

Chapter 2 Understanding Variables and Solving Equations •

Replace s with 7 miles. •

• •

miles miles

Multiply left to right.

• •

The perimeter of the square park is 4.

2.1 Introduction to Variables

Evaluate the expression

miles.

when is

.

(c)

means Replace x with 4

Replace a with 40. Divide.

• •

and y with 3. Multiply left to right.

The approximate systolic blood pressure is 5.

(a) Evaluate the expression

when is

is .

. and

means

(d)

• • • •

Replace with

and with .

Divide.

• •

.

• •

Replace c with 2. Multiply left to right.

44

45

Chapter 2 Understanding Variables and Solving Equations

14.

2.1 Section Exercises 1.

2.

Expression (rule) for degrees:

is the variable;

(a)

is

degrees.

is the constant.

(b)

is

degrees.

is the variable; is the constant.

(c)

is

degrees.

3.

15.

Expression (rule) for finding perimeter of an equilateral triangle of side length :

is the variable; is the constant.

4.

(a) Evaluate the expression when , the side length, is inches.

is the variable; is the constant.

5.

is the variable;

7.

(b) Evaluate the expression when , the side length, is feet.

is the variable; is the coefficient. is the variable; is the coefficient.

9. 10. 11.

12.

is the variable; is the coefficient. is the constant.

16.

Both

17.

Both

and

and

are variables.

are variables.

Expression (rule) for perimeter: (a)

(b)

meters is inches is

meters. inches.

Expression (rule) for ordering brushes: (a) Evaluate the expression when , the class size, is

is the variable; is the coefficient; is the constant.

. Replace c with 12. Multiply before subtracting.

is the variable; is the coefficient; is the constant.

13.

Replace s with 3. Follow the rule and multiply. feet is the perimeter.

is the constant. 8.

Replace s with 11. Follow the rule and multiply. inches is the perimeter.

is the coefficient. 6.

2.1 Introduction to Variables

brushes must be ordered. (b) Evaluate the expression when , the class size, is .

Expression (rule) for ordering robes: (a) Evaluate the expression when there are graduates.

Replace c with 16. Multiply before subtracting.

Replace g with 654. Follow the rule and add. robes must be ordered. (b) Evaluate the expression when there are

brushes must be ordered. 18.

Expression (rule) for ordering doughnuts: (a)

is

doughnuts must be ordered.

45

46

Chapter 2 Understanding Variables and Solving Equations graduates.

(b) Replace g with 208. Follow the rule and add. robes must be ordered.

(c) Evaluate the expression when there are graduates.

19.

â&#x20AC;˘

2.1 Introduction to Variables is

doughnuts must be ordered.

Expression (rule) for average test score, where the total points and is the number of tests: (a) Evaluate the expression when , the total points, is and , the number of tests, is . Replace p with 332 and t with 4.

Replace g with 95. Follow the rule and add. robes must be ordered.

Follow the rule and divide. points is the average test score.

is

46

47

Chapter 2 Understanding Variables and Solving Equations

(b) Evaluate the expression when , the total points, is

27.

and , the number of tests, is .

2.1 Introduction to Variables

Multiplying a number by leaves the number unchanged. Let represent "a number." •

Replace p with 637 and t with 7. 28. Follow the rule and divide.

Adding to any number leaves the number unchanged. Let represent "any number."

21.

or

Expression (rule) for buses: (a)

is

buses.

(b)

is

buses.

Value of

29.

Any number divided by is undefined. Let represent "any number." is undefined or

Expression

Expression •

is

30.

is

Multiplication distributes over addition. Let , , and represent variables. •

31.

written without exponents is

is

32.

written without exponents is •

Expression

Expression

is Value of

is

24.

Value of

Value of

Expression •

• •

25.

, or

, or is , or

is is is

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.

written without exponents is • •

26.

Expression is

written without exponents is •

34.

, or is , or is , or

is

Value of

33.

is

is

23.

• • • • • •

is Value of

is undefined.

is •

is

22.

or

points is the average test score. 20.

47

The number part in a multiplication expression is

48

Chapter 2 Understanding Variables and Solving Equations

2.1 Introduction to Variables

35.

can be written as • • • • . The exponent applies only to the base .

40.

can be written as • • • • . The exponent applies only to the base .

36.

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 .

37.

can be written as • • • . The exponent applies only to the base .

42.

38.

can be written as • • • • • . The exponent applies only to the base .

can be written as • • • • • • • • . The exponent applies only to the base . The exponent applies only to the base .

43.

Evaluate

39.

can be written as • • • • • • . The exponent applies only to the base . 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 .

when is

.

means • •

Replace t with Multiply.

4.

48

49

Chapter 2 Understanding Variables and Solving Equations

44.

Replace with

.

Evaluate

with , and with

when is

53.

• •

,

.

and is .

means • • •

Use a calculator. Replace with

52.

45.

2.1 Introduction to Variables

Evaluate calculator.

when is

Replace r with 3 and s with 2. Multiply left to right.

and is , using a

Replace r with and s with 2.

3

Use the yx key. •

Multiply left to right. • • • •

46. with

, •

47.

Replace with and

.

Evaluate

Use a calculator. Replace with

54.

when is

and with

and is .

. ,

means • • •

Replace r with 3 and s with 2. Multiply left to right.

55.

Evaluate is .

when

is ,

, and

Replace x with 4,

Replace with and with

48.

49.

is

Evaluate

y with 2, and z with 6. Multiply left to right within the abs. value bars.

. •

when is and is

.

means

Evaluate the

Replace s with 2 • • • •

and t with

4.

Multiply left to right.

• •

56.

Replace

with , •

• • • • •

50.

with

, and with

.

Replace with

and with . •

57.

Evaluate

when is

and

is

.

49

50 51.

Chapter 2 Understanding Variables and Solving Equations Evaluate when is using a calculator.

, is , and is

Replace r with and t with 4.

,

2.1 Introduction to Variables Replace z with 6 and y with 2.

3, s with 2,

x

Use the y key.

Numerator: â&#x20AC;˘

Multiply left to right.

Denominator: ,

Undefined

Division by 0 is undefined.

50

47

Chapter 2 Understanding Variables and Solving Equations

2.2 Simplifying Expressions

47

(b) 58.

Evaluate

when

is

and

is

The like terms are the constants,

.

There are no variable parts.

Replace x with 4 and y with 2.

(c) The like terms are and parts match; both are .

The coefficients are Numerator:

and

since the variable .

(d) •

Denominator: •

Undefined

and .

Division by 0 is undefined.

2.

The like terms are and parts match; both are .

since the variable

The coefficients are

and

.

(a)

These are like terms.

Relating Concepts (Exercises 59–60) 59.

(a) Evaluate

when is

Add the coefficients. The variable part, b,

seconds.

stays the same.

Replace s with 15. (b) Divide. miles (b) Evaluate

when is

Add the coefficients. The variable part, y3 , stays the same.

seconds.

Replace s with 10.

(c)

Divide. miles (c) Evaluate

when is

(d)

Divide. mile 60.

(a) Using part (c) of Exercise 59, the distance covered in seconds is half of the distance covered in seconds, or mile.

These are like terms. Rewrite n as 1n. Change to addition. Add the coefficients. The variable part, n, stays the same.

seconds.

Replace s with 5.

These are like terms. Rewrite y3 as 1y3 .

These are like terms. Change to addition. Add the coefficients. The variable part, c, stays the same.

(e)

These are like terms. Rewrite xy as 1xy

(b) Using part (a) of Exercise 59, the time to

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.

48

Chapter 2 Understanding Variables and Solving Equations

2.2 Simplifying Expressions

48

(c) Using parts (a) and (b) of Exercise 59, find the

number halfway between seconds and seconds; that is seconds.

Add the coefficients. The variable part, p 2, or

2.2 Simplifying Expressions (g)

2.2 Margin Exercises

stays the same. These are like terms. Rewrite ab as 1ab

1.

(a) The like terms are and since the variable parts match; both are . The coefficients are

and

.

49

3.

Chapter 2 Understanding Variables and Solving Equations

(a)

2.2 Simplifying Expressions

can be written as

(c) Rewrite using the commutative property. Combine 3b2 7b2 Add the coefficients.

• •

(b) (d)

Change to addition. Rewrite using the commutative property. Rewrite b as 1b. Add the coefficients of like terms.

Rewrite x as 1x. can be written as •

• •

5.

(a)

(c) Rewrite using the commutative property. Add the coefficients of like terms.

(b)

Zero times anything is zero.

(c)

can be written as •

can be written as

can be written as •

(d)

• •

Change to addition. Rewrite y as 1y. Rewrite using the commutative prop . Add the coefficients of like terms.

(d) •

or

to subtraction.

(e)

(e)

Change to addition. Rewrite using the commutative prop. Add the coefficients of like terms.

Multiply. Change addition to subtraction. 6.

4.

(a)

means •

. Using the associative

property, it can be rewritten as •

• •

(a)

Distributive property •

Rewrite using the commutative property. Combine constants.

49

50

Chapter 2 Understanding Variables and Solving Equations

2.2 Simplifying Expressions or

(b)

can be written as (b) •

Distributive property Multiply. Combine constants.

50

51

Chapter 2 Understanding Variables and Solving Equations

(c) •

Distributive property Multiply.

7.

Change to addition. Rewrite using the commutative property.

These are like terms. Add the coefficients. The variable part, r, stays the same.

8.

Add the coefficients of like terms.

2.2 Simplifying Expressions

These are like terms. Add the coefficients.

or (d)

The variable part, t, stays the same.

Distributive property •

Multiply.

9.

These are like terms. Rewrite x2 as 1x2 .

Rewrite using the commutative property. Combine constants.

Add the coefficients. The variable part, x2 , stays the same. 10.

(e)

Distributive property Rewrite y as 1y.

11.

The variable part, p,

2.2 Section Exercises 1.

These are like terms. Rewrite p as 1p. Change to addition. Add the coefficients.

stays the same. and

are the

12.

only like terms in the expression. The variable parts match; both are . The coefficients are and . 13. 2.

and terms. The variable parts match; both are coefficients are and .

3.

are like . The

and are the like terms in the expression. The variable parts match; both are . The coefficients are and .

4.

These are like terms. Rewrite a3 as 1a3 .

and

The variable part, a3 , stays the same. 14.

51

52

Chapter 2 Understanding Variables and Solving Equations

2.2 Simplifying Expressions

are like terms. The variable parts match; both are 5.

. The coefficients are

and

, , and

. are

like terms. There are no variable parts; constants are considered like terms. 6.

, , and are like terms. There are no variable parts; constants are considered like terms.

15. Any number minus itself is . 16. Any number minus itself is .

52

53

Chapter 2 Understanding Variables and Solving Equations

17.

or

These are like terms. Rewrite xy as 1xy. Change to addition.

26.

27.

2.2 Simplifying Expressions

The variable part, xy,

Use the commutative property to put the constants at the end. Add the coefficients of like terms.

stays the same.

The only like terms are constants.

18.

28. or

19.

Write in the understood coefficients of 1. Add the coefficients of like terms.

29.

The variable part, t 4 , stays the same.

,

20.

The variable part, ab 2, stays the same.

or 30.

21.

These are like terms. Write in the understood coefficients of 1. Add the coefficients.

or

31.

Write in the understood coefficients of 1. Change to addition.

2

The variable part, y , stays the same.

Rewrite using the commutative property. Add the coefficients of like terms.

22.

These are like terms. 23.

Rewrite x as 1x and x as 1x. Change to addition. Add the coefficients. The variable part, x, stays the same.

24. ]

32.

Write in the under-

53

54 33.

25.

Chapter 2 Understanding Variables and Solving Equations

2.2 Simplifying Expressions

stood coefficient of 1. Change to addition. Rewrite using the commutative property.

Use the commutative property

to rewrite the expression so that like terms are next to each other. Add the coefficients of like terms.

of like terms.

The variable part, a, stays the same.

â&#x20AC;˘

54

55

Chapter 2 Understanding Variables and Solving Equations

2.2 Simplifying Expressions

44.

34.

45.

By using the associative property, we can write as •

• •

So, 35.

There are no like terms.

55

• •

simplifies to

. .

46.

The expression cannot be simplified. 36.

There are no like terms. The expression cannot be simplified.

47.

By using the associative property, we can write as

37.

Write in the understood coefficients of 1. Change to addition.

• •

So,

simplifies to

• •

. .

48. Rewrite using the commutative property.

Write in the understood 49.

coefficient of 1. Rewrite using the associative property.

Add the coefficients of like terms. • •

50. •

38.

51.

Distributive property •

52.

Distributive property

or 39.

By using the associative property, we can write as •

So, 40.

41.

simplifies to

.

Distributive property

. •

54.

By using the associative property, we can write as •

53.

.

55.

Distributive property •

56

Chapter 2 Understanding Variables and Solving Equations So,

simplifies to

2.2 Simplifying Expressions •

56.

.

42.

57.

Distributive property •

43.

By using the associative property, we can write as •

So,

simplifies to

. .

Change addition to subtraction of the opposite.

56

57

Chapter 2 Understanding Variables and Solving Equations

58.

2.2 Simplifying Expressions

69.

Distributive property •

or 59.

Rewrite using the commutative property. Add the coefficients of like terms.

Distributive property •

Change addition to subtraction of the opposite.

60.

Zero times any number is 0. Zero added to any

number is the number or Distributive property

61. •

70.

62.

Distributive property

63. •

Rewrite using the commutative property. Combine like terms.

64.

71.

Distributive property •

Change to addition. Combine like terms. Any number plus its

opposite is 0.

65.

Distributive property •

Combine like terms.

66.

72.

73. 67.

Distributive property Distributive property

Change to addition. Rewrite using the

commutative property. Add the coefficients of like terms. Change addition to subtraction

57

58

Chapter 2 Understanding Variables and Solving Equations •

2.2 Simplifying Expressions

Rewrite n as 1n. Change to addition. Rewrite using the commutative property. Add the coefficients of like terms.

of the opposite.

68.

or

74.

or

58

Summary Exercises Variables and Expressions 75.

Distributive property •

84. •

Rewrite p as 1p. Change to addition. Add the coefficients of like terms.

85. Distributive property •

Change addition to subt. of the opposite.

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.

86. •

78. •

87.

79.

80.

81.

82.

Distributive property

The answers are equivalent because of the commutative property of addition. 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. 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.

• •

88. Keep the variable part unchanged when combining like terms. As shown above, the correct answer is . In the last step, do not change the sign of the first term. The correct answer is .

83.

Distributive prop. •

Change subtraction to adding the opposite.

Summary Exercises Variables and Expressions 1. or

1

is the variable; 1 is the coefficient; is the constant.

53

Group like terms and add the coefficients.

2.

3.

and are the variables; is the coefficient. is the variable; is the coefficient; is the constant.

54

4.

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition

Expression (rule) for finding the perimeter of an octagon of side length :

Replace

12.

Replace s with 4. Follow the rule and multiply.

with ,

with

, and

with

Multiply left to right.

(b) Evaluate the expression when , the side length, is inches. Replace s with 15. Follow the rule and multiply.

13.

Replace

inches is the perimeter. 5.

and with .

.

yards is the perimeter.

with

If is multiplied by any number, the result is Thus, there is no need to make any calculations since the result is

(a) Evaluate the expression when , the side length, is yards.

Replace

11.

with

.

Multiply left to right.

Expression (rule) for finding the total cost of a car with down payment , monthly payment and number of payments : 14.

(a) Evaluate the expression when the down payment is \$ , the monthly payment is \$ and the number of payments is .

Replace

,

with and

with

.

Multiply left to right.

Replace d with \$3000, m

\$ \$ \$

\$ ,

with \$280, and t with 36. Multiply before adding.

\$

15. with

,

is the total cost of the car.

payment is \$ , the monthly payment is \$ and the number of payments is .

(b) Evaluate the expression when the down

Replace

with

and

. •

Multiply left to right.

,

Replace d with \$1750, m with \$429, and t with 48. \$

\$

\$

\$

\$ 6.

,

16. and

is the total cost of the car.

with

,

7.

Replace

. Multiply left to right. •

written without exponents is • • • •

written without exponents is •

with

8.

written without exponents is •

• • • • • • •

54

55

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition •

• •

9.

Replace

with .

17.

Multiply left to right.

Use a calculator. Replace

18. 19. 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

with

, and

with

with

. ,

• •

10.

, •

55

56

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition

20.

56

• •

or 21.

33.

(a) Simplifying the expression correctly: •

22.

The student forgot to multiply • . 23.

(b) Simplifying the expression correctly: • •

24.

• •

Two negative factors give a positive product. (c) Simplifying the expression correctly:

25.

Keep the variable part unchanged; that is, adding 's to 's gives an answer with 's, not 's.

or 26.

34.

In the last step, do not change the sign of the first term; keep as . The correct answer is

27.

2.3 Solving Equations Using Addition 2.3 Margin Exercises 1. •

28.

Given equation

(a)

?

No, •

29.

is not the solution. ?

• •

.)

Given equation ?

Replace c with 65. Balances

Yes, is the solution. (No need to check and (b)

30.

Replace c with 95. 110 is more than 80.

Replace c with 28.

57

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition No,

is not the solution. ?

31.

• •

No,

is not the solution. ?

No,

or

Replace c with 20.

Replace c with 24.

is not the solution. ?

Replace c with 32. Balances

32. •

• •

Yes,

is the solution.

57

58

2.

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition

(a) Solve

Combine like terms.

for .

To get r by itself,

To get by itself, add the opposite of , which is . To keep the balance, add

to both sides.

or Check

The solution is . Check

The solution is .

â&#x20AC;˘

Original equation Replace y with 7.

â&#x20AC;˘

Replace r with 2.

Balances, so solution is . (b) Solve

Balances

for .

2.3 Section Exercises

1.

To get by itself add the opposite of , to both sides.

The solution is Check

Replace

with

,

,

, and .

Given equation

, which is

?

Replace n with 58.

?

Yes, is the solution. (No need to check , , and .)

.

Original equation Replace b with

2.

Replace with ,

4.

,

, and

?

?

?

?

.

Balances 3.

Rewrite both sides by

(a)

changing subtraction to addition. Combine like terms.

?

?

?

?

The check for To get by itself add the opposite of , to both sides.

, which is

3.

balances, so Replace

and

)

with

,

. ?

No,

is the solution.

Given equation Replace y with 4.

is not the solution.

,

,

58

59

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition The solution is

.

?

Replace y with

16.

Check Yes, is the solution. (No need to check and

Original equation Replace k with 4. (

)

(

)

4. Balances

(b)

Change to

Replace and

.) with

,

,

,

. ?

?

Rewrite the left side by using the commutative property.

is the solution. (No need to check

.)

59

60

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition

5.

(a)

6.

(a)

because

12.

on the left side.

(b) because

to both sides because

gives

(b)

60

8 to both sides.

to both sides The solution is

on the right side. to both sides

Check

on the left side.

Replace a with

9.

Balances

to both sides because

on the right side.

.

13. Add the opposite of 4, 4, to both sides.

7. Add the opposite of 5, 5, to both sides.

The solution is The solution is .

.

Check Replace k with 18. Balances

Check Replace p with 4. Balances

14. Add 9 to both sides.

3 to both sides. The solution is

The solution is .

.

Check Replace y with 16. Balances

Check Replace a with 9. Balances

15.

9.

Change to addition. Add the opposite of 6, 6, to both sides.

Change to addition. Add the opposite of 2, 2, to both sides.

The solution is . The solution is

.

Check

Check Replace r with 10. Balances 10.

61

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition Replace y with 6. Balances 16.

Change to addition. Add 15 to both sides. The solution is The solution is .

Check

Check

Replace k with 15. Balances

Replace b with 8. Balances

17.

11.

13,

3, to both sides.

The solution is

.

13, to both sides.

The solution is

.

.

Check

Check Replace n with Balances

8.

Replace r with Balances

6.

61

62

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition

Correct solution:

19 to both sides.

The solution is

.

5, 5, to both sides.

Check Replace z with

7. The solution is .

Balances Check

19.

Replace z with 8. Change to addition. Balances

Change to addition. Add the opposite of 12, 12, to both sides.

The given solution is

24.

.

The solution is . Check Check

Replace x with 13. Balances

Replace x with 0. Change to addition.

is the correct solution.

Balances

The given solution is

25.

.

Check Replace x with Balances

The solution is .

is the correct solution.

Check

The given solution is

26.

Replace m with 0.

.

Check Replace k with 5. Does not balance

Balances 21. Correct solution:

Add the opposite of 2, 2, to both sides. The solution is

Add the opposite of 2, 2, to both sides.

.

Check Replace t with

3.

The correct solution is 22.

.

18.

62

63

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition Balances

Check Replace k with Balances

The given solution is

27. The solution is .

Replace w with 9. Balances The given solution is

.

Check

Check

23.

9.

Replace b with 10. Does not balance Correct solution:

.

Check Replace z with 2. Change to addition. Does not balance

Add the opposite of 10, 10, to both sides. The solution is .

63

64

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition

32.

Check Replace b with 0. Balances 28.

64

The given solution is .

The solution is .

Check

Check Replace a with 0. Does not balance

Balances 33.

Correct solution:

10.

14, 14, to both sides. The solution is The correct solution is

.

.

Check

Check

Replace b with

Replace a with 14.

30.

Balances

Balances 29.

34. Simplify the right side. Change to addition.

Check

The solution is .

The solution is Check

Replace b with 12. Replace c with 6. Balances Balances

Replace w with 17. Replace t with 0. Balances Change to addition. Balances

31. Simplify the left side. Change to addition. Add 2 to both sides.

30.

The solution is

.

35. 36.

Change to addition. Simplify the right side. Add 2 to both sides.

. The solution is .

Check Check

65

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition

The solution is .

Check

The solution is . Replace y with 5. Change to addition. Balances

Check Replace p with 0. Balances

65

66

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition

42.

37. Change to addition. Combine like terms. 1z is the same as z. The solution is .

Check

Replace z with 7. The solution is Change to add. Balances

.

38.

Combine like terms. The solution is . The solution is

.

Check

Combine like terms. The solution is .

Replace r with 5. Balances

Rearrange and 39.

combine like terms.

The solution is Add 2 to both sides.

.

The solution is . Check

The solution is 1 •

Replace w with 3. 47. Balances

.

Rearrange and combine like terms.

40. Add 3 to both sides.

The solution is

.

66

67

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 48. The solution is

.

Check Add 4 to both sides. Let t =

5. The solution is

Balances 41.

.

49. Add 31 to both sides.

Change to addition. 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

.

67

68

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition

51.

60.

68

Check

Change to addition. Rearrange and combine like terms. Add 9 to both sides.

The solution is

Does not balance To correct the errors, change to . Then, add to both sides, not The correct solution is .

.

.

52. 61. Add the opposite of 0

0

There were

The solution is .

10,

10, to both sides.

62.

53. (

)

(

)

0

0

Combine like terms.

10, to both sides.

Add 6 to both sides. The solution is

.

54.

There were

37 to both sides.

Combine like terms. Add 5 to both sides.

The solution is

When the temperature is degrees, a field cricket chirps times (in seconds).

.

91 to both sides.

The solution is

37 to both sides.

.

56. Change to

When the temperature is

chirps

degrees, a field cricket

65.

times (in

seconds).

69

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition The solution is

.

69

5 to both sides.

The solution is

Ernesto's parking fees average \$ winter.

.

58.

6 to both sides.

The solution is 59.

per month in

.

No, the solution is , the number used to replace in the original equation.

Aimee's parking fees average \$ winter.

per month in

70

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition

67. Add the opposite of 10,

10, to both sides.

The solution is

.

(b) Equations will vary. Some possibilities are: Commutative property 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.

69.

1, Change subtraction within absolute value to addition and rearrange the terms.

1, to both sides.

The solution is . (b)

Simplify inside absolute value bars. Collect like terms.

Evaluate absolute values.

1, 1, to both sides.

or

6 to both sides. (c)

The solution is .

\$

\$ Add the opposite of \$2 50, \$2 50,

The solution is . 70.

to both sides. \$

\$ \$

The solution is \$

.

(d) Equations will vary. Some possibilities are: \$

\$

The solution is \$

.

70

71

Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition \$

The solution is \$

2.4 Solving Equations Using Division

The solution is .

2.4 Margin Exercises

Relating Concepts (Exercises 71â&#x20AC;&#x201C;72) 71.

\$

(a) Equations will vary. Some possibilities are:

1.

(a) Solve

.

Use division to undo multiplication. Divide both sides by the coefficient of the variable, which is .

.

The solution is

.

71 .

63

Chapter 2 Understanding Variables and Solving Equations 2.4 Solving Equations Using Division

Original equation Replace s with 11.

Check •

63

Check

Replace p with

Balances

1.

(b) Divide both sides by

Balances

9. 3.

The solution is

(a) Write in the understood 1 as the coefficient of k. Divide both

.

Check •

Replace p with

3.

sides by

Balances

The solution is

(c)

.

Check Divide both sides by 5.

The solution is .

Replace k with 12. Balances

(b) Write t as 1t. Divide both

Check •

Replace x with 8.

sides by 1. The solution is

Balances (d)

The solution is

.

Check

Divide both sides by 7.

Replace t with Balances

. (c)

Check •

Replace t with Balances

Write m as Divide both

10.

sides by 2.

1.

(a)

1.

The solution is Combine like terms. Divide both sides by 4. The solution is

Check

Check .

Replace n with •

7.

1m.

.

7.

64

Chapter 2 Understanding Variables and Solving Equations 2.4 Solving Equations Using Division 2.4 Section Exercises

1. Divide both sides by 6.

Balances

The solution is .

(b) Check â&#x20AC;˘

Change to addition. Rewrite p as 1p.

Replace z with 2. Balances

Combine like terms.

2.

Check â&#x20AC;˘

Divide both sides by

13. Balances

The solution is

.

The solution is .

64

65

Chapter 2 Understanding Variables and Solving Equations 2.4 Solving Equations Using Division

3.

11. Divide both

Divide both

sides by 12. The solution is .

sides by 5. The solution is

Check •

Check

Replace r with 4.

Balances 4.

.

Replace b with

5.

Balances

Check 12.

Check •

Balances Balances

The solution is . 5.

The solution is Divide both sides by 3.

.

13. Combine like terms. Divide both

The solution is . Check

sides by 2. •

Replace y with 0.

The solution is .

Balances 6.

Check

Check

Balances

14.

Replace r with 3. Balances Check

The solution is .

7. Divide both sides by 7. The solution is

Balances

.

The solution is .

Check •

8.

Replace k with Balances

10.

15. Change to addition. Rewrite p as 1p. Combine like terms.

Check •

Divide both

65

66

Chapter 2 Understanding Variables and Solving Equations 2.4 Solving Equations Using Division sides by 4. The solution is

Balances The solution is

.

.

Check

9. Divide both sides by 9.

The solution is .

Balances

Replace r with 6. Balances

10.

3.

Check •

Replace p with

16.

Check •

Change to addition. Combine like terms. Divide both

Balances The solution is . sides by 6.

sides by

10.

The solution is

.

66

67

Chapter 2 Understanding Variables and SolvingCumulative Equations Review Exercises (Chapters 1–2)

The solution is

.