Chapter 2

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2.1 Study Guide For use with pages 63–68 GOAL

Use properties of addition and multiplication.

VOCABULARY

Lesson 2.1

Commutative Property of Addition: In a sum, you can add the numbers in any order. Associative Property of Addition: Changing the grouping of the numbers in a sum does not change the sum. Commutative Property of Multiplication: In a product, you can multiply the numbers in any order. Associative Property of Multiplication: Changing the grouping of the numbers in a product does not change the product. Identity Property of Addition: The sum of a number and the additive identity, 0, is the number. Identity Property of Multiplication: The product of a number and the multiplicative identity, 1, is the number.

EXAMPLE

1 Using Properties of Addition You listened to your radio for 27 minutes on Monday, 9 minutes on Tuesday, and 13 minutes on Wednesday. Find the total time you spent listening to your radio.

Solution The total time is the sum of the three times. Use properties of addition to group together times that are easy to add mentally. 27 ⫹ 9 ⫹ 13 ⫽ (27 ⫹ 9) ⫹ 13 ⫽ (9 ⫹ 27) ⫹ 13 ⫽ 9 ⫹ (27 ⫹ 13) ⫽ 9 ⫹ 40 ⫽ 49

Use order of operations. Commutative property of addition Associative property of addition Add 27 and 13. Add 9 and 40.

Answer: The total time is 49 minutes.

EXAMPLE

2 Using Properties of Multiplication Evaluate 2xy when x ⫽ 8 and y ⫽ ⫺35. 2xy ⫽ 2(8)(⫺35) ⫽ [2(8)](⫺35) ⫽ [8(2)](⫺35) ⫽ 8[2(⫺35)] ⫽ 8(⫺70) ⫽ ⫺560

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Pre-Algebra Chapter 2 Resource Book

Substitute 8 for x and ⫺35 for y. Use order of operations. Commutative property of multiplication Associative property of multiplication Multiply 2 and ⫺35. Multiply 8 and ⫺70.

Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved.

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2.1 Study Guide Continued

For use with pages 63–68 Exercises for Examples 1 and 2 Evaluate the expression. Justify each of your steps.

1. 64 ⫹ 15 ⫹ 6

2. 34 ⫹ 75 ⫹ 26

3. 48 ⫹ 36 ⫹ 22

4. Evaluate 15xy when x ⫽ 9 and y ⫽ 2. Lesson 2.1

EXAMPLE

3 Using Properties to Simplify Variable Expressions Simplify the expression. 21 ⫹ 3(8y) ⫹ 30 ⫽ 21 ⫹ (3 p 8)y ⫹ 30 ⫽ 21 ⫹ 24y ⫹ 30 ⫽ (21 ⫹ 24y) ⫹ 30 ⫽ (24y ⫹ 21) ⫹ 30 ⫽ 24y ⫹ (21 ⫹ 30) ⫽ 24y ⫹ 51

Associative property of multiplication Multiply 3 and 8. Use order of operations. Commutative property of addition Associative property of addition Add 21 and 30.

Exercises for Example 3 Simplify the expression.

5. ⫺8 ⫹ x ⫹ 3

EXAMPLE

6. 3(21x)

4 Multiplying by a Conversion Factor The African Elephant is the largest living land animal. Its average weight is 6 tons. What is the African Elephant’s average weight in pounds?

Solution (1) Find a conversion factor that converts tons to pounds. The statement 1 ton ⫽ 2000 pounds gives you two conversion factors. Unit analysis shows that a conversion factor that converts tons to pounds has pounds in the numerator and tons in the denominator: pounds

tons p ᎏᎏ ⫽ pounds 冫冫 tons

(2) Multiply the African Elephant’s weight by the conversion factor from Step 1. 2000 pounds

6 tons ⫽ 6 冫 tons p ᎏᎏ ⫽ 12,000 pounds ton 1冫

Answer: The average weight of the African Elephant is 12,000 pounds.

Exercise for Example 4 7. Use a conversion factor to convert 3 years to months.


Chapter 2

Pre-Algebra Resource Book

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Use the distributive property.

VOCABULARY Two numerical expressions that have the same value are called equivalent numerical expressions. Two variable expressions that have the same value for all values of the variable(s) are called equivalent variable expressions.

EXAMPLE

1 Evaluating Numerical Expressions You are raising money for a field trip. The school matches your earnings. You earn $125 selling sandwiches and $95 at a car wash. Find the amount you have toward your field trip with the school’s contribution.

Lesson 2.2

Solution Method 1: Find the amount you earned. Then multiply the result by 2, because your earnings are matched by the school. Total amount toward trip ⫽ 2(125 ⫹ 95) ⫽ 2(220) ⫽ 440 Method 2: Find the amount earned and matched for selling sandwiches and the amount earned and matched for washing cars. Then add the amounts. Total amount toward trip ⫽ 2(125) ⫹ 2(95) ⫽ 250 ⫹ 190 ⫽ 440 Answer: You have $440 for your field trip.

Exercise for Example 1 1. You and a friend each spend $5 on a movie ticket and $4 on snacks. Write and evaluate two expressions that can be used to find the amount you both spent.

EXAMPLE

2 Using the Distributive Property You buy 5 shorts for $15.02 each. Use the distributive property and mental math to find the total cost of the shorts. Total cost ⫽ 5(15.02) ⫽ 5(15 ⫹ 0.02) ⫽ 5(15) ⫹ 5(0.02) ⫽ 75 ⫹ 0.10 ⫽ 75.10

Write expression for total cost. Rewrite 15.02 as 15 ⫹ 0.02. Distributive property Multiply, then add, using mental math.

Answer: The total cost of the shorts is $75.10. 20



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EXAMPLE

For use with pages 71–75

3 Writing Equivalent Variable Expressions Use the distributive property to write an equivalent variable expression. a. 15(3y ⫺ 4) ⫽ 15(3y) ⫺ 15(4) ⫽ 45y ⫺ 60

Distributive property Multiply.

b. ⫺6(3x ⫺ 1) ⫽ ⫺6(3x) ⫺ (⫺6)(1) ⫽ ⫺18x ⫺ (⫺6) ⫽ ⫺18x ⫹ 6

Distributive property Multiply. Definition of subtraction

c. (2z ⫹ 5)(⫺11) ⫽ 2z(⫺11) ⫹ 5(⫺11) ⫽ ⫺22z ⫹ (⫺55) ⫽ ⫺22z ⫺ 55

Distributive property Multiply. Definition of subtraction

Exercises for Examples 2 and 3 Evaluate the expression using the distributive property and mental math.

2. 5(197)

3. 35(11)

4. 4(13.04)

5. 7(8.98)

6. 12(7x ⫹ 8)

EXAMPLE

7. 3(9y ⫺ 1)

8. ⫺5(9z ⫹ 6)

9. ⫺8(11m ⫺ 9)

4 Finding Areas of Geometric Figures Find the area of the rectangle or triangle. a.

b.

2y ⫹ 1

2 3x ⫺ 8

16

Solution a. Use the formula for the area of a rectangle.

b. Use the formula for the area of a triangle. 1 2

A ⫽ lw

1 2

A ⫽ ᎏᎏ bh ⫽ ᎏᎏ (16)(2y ⫹ 1)

⫽ (3x ⫺ 8)(2)

⫽ 8(2y ⫹ 1)

⫽ 3x(2) ⫺ 8(2) ⫽ (6x ⫺ 16) square units

⫽ 8(2y) ⫹ 8(1) ⫽ (16y ⫹ 8) square units

Exercises for Example 4 Find the area of the rectangle or triangle.

10.

11. 4x ⫺ 3

7 3x ⫹ 5 Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved.

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Chapter 2


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Lesson 2.2

Use the distributive property to write an equivalent variable expression.

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Simplify variable expressions.

VOCABULARY The parts of an expression that are added together are called terms. The coefficient of a term with a variable is the number part of the term. A constant term, such as 7, has a number but no variable. Like terms are terms that have identical variable parts.

EXAMPLE

1 Identifying Parts of an Expression Identify the terms, like terms, coefficients, and constant terms of the expression 2y 3 ⫺ 7 ⫹ 2y ⫺ y 3 ⫹ 3.

Solution (1) Write the expression as a sum: 2y 3 ⫹ (–7) ⫹ 2y ⫹ (⫺y 3) ⫹ 3. (2) Identify the parts of the expression. Note that because ⫺y 3 ⫽ ⫺1y 3, the coefficient of ⫺y 3 is ⫺1. Terms: 2y 3, ⫺7, 2y, ⫺y 3, 3

Like terms: 2y 3 and ⫺y 3; ⫺7 and 3

Coefficients: 2, 2, ⫺1

Constant terms: ⫺7, 3

Exercises for Example 1 For the given expression, identify the terms, like terms, coefficients, and constant terms.

1. 9t 2 ⫺ 12t ⫹ t 2 ⫺ 1

3. 5y ⫺ 3 ⫹ 2y

2 Simplifying an Expression 17x 2 ⫹ 2 ⫹ x 2 ⫺ 5 ⫽ 17x 2 ⫹ 2 ⫹ x 2 ⫹ (–5) ⫽ 17x 2 ⫹ x 2 ⫹ 2 ⫹ (–5) ⫽ 17x 2 ⫹ 1x 2 ⫹ 2 ⫹ (–5) ⫽ (17 ⫹ 1)x 2 ⫹ 2 ⫹ (–5) ⫽ 18x 2 ⫺ 3

Lesson 2.3

EXAMPLE

2. 11m 4 ⫹ 4m ⫺ 5 ⫺ 15m

Write as a sum. Commutative property Coefficient of x 2 is 1. Distributive property Simplify.


4. 3x ⫺ 21 ⫺ 7x ⫹ 20 5. 2y 5 ⫹ 5y ⫺ y 5 ⫹ 5 6. 11z3 ⫺ 3z ⫺ 3 ⫹ z3 ⫹ 2z


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EXAMPLE

For use with pages 78 –83

3 Simplifying Expressions with Parentheses a. 5(x ⫺ 2) ⫺ 9x ⫹ 11 ⫽ 5x ⫺ 10 ⫺ 9x ⫹ 11 ⫽ 5x ⫺ 9x ⫺ 10 ⫹ 11 ⫽ ⫺4x ⫹ 1

Distributive property Group like terms. Combine like terms.

b. 6x ⫺ (8 ⫺ 14x) ⫹ 1 ⫽ 6x ⫺ 1(8 ⫺ 14x) ⫹ 1 ⫽ 6x ⫺ 8 ⫹ 14x ⫹ 1 ⫽ 6x ⫹ 14x ⫺ 8 ⫹ 1 ⫽ 20x ⫺ 7

Identity property Distributive property Group like terms. Combine like terms.


7. 5y ⫹ 7(2y ⫹ 1) ⫺ 5 EXAMPLE

8. 8k ⫺ 5 ⫹ 5(2k ⫺ 3) ⫺ 7

9. 11n ⫺ (n ⫺ 5) ⫹ 3n

4 Writing and Simplifying an Expression You spend a total of 50 minutes talking long-distance to your friend and grandparents. It costs 4 cents per minute to call your friend and 6 cents per minute to call your grandparents. a. Let t be the time you talk with your friend (in minutes). Write an expression in terms of t for the cost of both phone calls. b. Find the cost of the phone calls if you talk with your friend for 25 minutes.

Solution a. Write a verbal model for the cost of the phone calls. Lesson 2.3

Length of Long-distance Length of Long-distance call with rate for calling p call with ⫹ rate for calling p grandparents friend friend grandparents 0.04t ⫹ 0.06(50 ⫺ t) ⫽ 0.04t ⫹ 3 ⫺ 0.06t ⫽ 3 ⫺ 0.02t

Distributive property Combine like terms.

b. Evaluate the expression in part (a) when t ⫽ 25. 3 ⫺ 0.02t ⫽ 3 ⫺ 0.02(25) ⫽ $2.50

Exercise for Example 4 10. You spend a total of 25 minutes typing an e-mail to your friend and writing a letter to your aunt. You can type 60 words per minute and handwrite 20 words per minute. Let m be the number of minutes you type the e-mail to your friend. Write an expression in terms of m for the total number of words you wrote. Evaluate the expression if you spend 10 minutes typing the e-mail to your friend.

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Solve equations with variables.

VOCABULARY An equation is a mathematical sentence formed by placing an equal sign, ⫽, between two expressions. A solution of an equation with a variable is a number that produces a true statement when it is substituted for the variable. Finding all solutions of an equation is called solving the equation.

EXAMPLE

1 Writing Verbal Sentences as Equations Verbal Sentence a. The sum of 3y and 1 is 10.

Equation 3y ⫹ 1 ⫽ 10

b. The difference of t and 2 is 11.

t ⫺ 2 ⫽ 11

c. The product of ⫺1 and m is ⫺5.

⫺m ⫽ ⫺5

d. The quotient of 2x and 5 is 10.

2x ᎏᎏ ⫽ 10 5

Exercises for Example 1 Write the verbal sentence as an equation.

1. 7x divided by 3 equals 2.

EXAMPLE

2. The difference of 3 and 2x is 5.

2 Checking Possible Solutions Tell whether 3 or ⫺3 is a solution of 6y ⫺ 5 ⫽ 13. a. Substitute 3 for y.

b. Substitute ⫺3 for y.

6y ⫺ 5 ⫽ 13 6(3) ⫺ 5 ⱨ 13

6y ⫺ 5 ⫽ 13 6(⫺3) ⫺ 5 ⱨ 13

18 ⫺ 5 ⱨ 13

⫺18 ⫺ 5 ⱨ 13

13 ⫽ 13 ✓ Answer: 3 is a solution.

⫺23 ⫽ 13 Answer: ⫺3 is not a solution.

Exercises for Example 2 Tell whether the given value of the variable is a solution of the equation. y 2

5. ᎏᎏ ⫹ 1 ⫽ 7; y ⫽ 6


Lesson 2.4

3. 8 ⫺ 3m ⫽ 17; m ⫽ ⫺3

4. 5x ⫺ 7 ⫽ 13; x ⫽ ⫺4 k 3

6. ᎏᎏ ⫺ 3 ⫽ ⫺36; k ⫽ ⫺99

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EXAMPLE


3 Solving Equations Using Mental Math Equation

Question

Solution

a. ⫺3t ⫽ 39

⫺3 times what number equals 39?

⫺13

b. 11 ⫺ m ⫽ 4

11 minus what number equals 4?

7

c. 25 ⫹ k ⫽ 13

25 plus what number equals 13?

70 m

d. ᎏᎏ ⫽ 35

⫺12

70 divided by what number equals 35?

2

Check ⫺3(⫺13) ⫽ 39 ✓ 11 ⫺ 7 ⫽ 4 ✓ 25 ⫹ (⫺12) ⫽ 13 ✓ 70 ᎏᎏ ⫽ 35 ✓ 2

Exercises for Example 3 Solve the equation using mental math. 84 x

7. ᎏᎏ ⫽ ⫺7

EXAMPLE

8. 5x ⫽ 100

9. 20 ⫺ n ⫽ 3

10. m ⫹ 3 ⫽ ⫺1

4 Writing and Solving an Equation You divide an 8-quart bag of potting soil into 4 portions for flowers you are planting. Find the size of each portion of potting soil.

Solution First write a verbal model for this situation. Number of portions p Size of each portion ⫽ Total amount in bag Let p represent the size of each portion. 4p ⫽ 8 4(2) ⫽ 8

Substitute for quantities in verbal model. Use mental math to solve for p.

Answer: Because p ⫽ 2, each portion is 2 quarts.

Exercises for Example 4 11. Your 19-year-old sister is 4 years older than you. Write and solve an equation to find your age.

Lesson 2.4

12. You earn $6 per lawn you mow. Yesterday you earned $24. Write and solve an equation to find the number of lawns you mowed yesterday.

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Solve equations using addition or subtraction.

VOCABULARY Inverse operations are two operations that undo each other, such as addition and subtraction. Equivalent equations are equations that have the same solution(s).

EXAMPLE

1 Solving an Equation Using Subtraction Solve m 12 7. m 12 7 m 12 12 7 12 m 5

Write original equation. Subtract 12 from each side. Simplify.

Answer: The solution is 5. ✓ Check m 12 7 5 12 ⱨ 7 77 ✓ EXAMPLE

Write original equation. Substitute 5 for m. Solution checks.

2 Solving an Equation Using Addition Solve 2 x 9. 2 x 9 2 9 x 9 9 7x

Write original equation. Add 9 to each side. Simplify.

Answer: The solution is 7.

Exercises for Examples 1 and 2 Solve the equation. Check your solution.

1. 7 k 42

2. 21 y 14

3. m 9 13

4. 3 n 7

5. j 13 2

6. 1 x 5

7. f 11 2

8. y 12 8

9. z 5 7

11. k 2 15

12. j 17 13

10. x 1 0


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Lesson 2.5


EXAMPLE


3 Writing and Solving an Equation You are traveling to Louisville, Kentucky. You have already traveled 122 miles, and you just passed a road sign that said Louisville is 76 miles away. How far is Louisville from the start of your trip?

Solution Let d represent the distance from the start of your trip to Louisville. Write a verbal model. Then use the verbal model to write an equation. Distance from the start to Louisville

Distance traveled

Remaining distance

d 122 76 d 122 122 76 122 d 198

Substitute. Add 122 to each side. Simplify.

Answer: Louisville is 198 miles from the start of your trip.

Exercises for Example 3 13. You have $37 left after shopping. You started with $85. How much money did you spend? 14. You are in a 50-kilometer bike race. You have to bike 21 kilometers until you reach the finish line. How far have you already biked?

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EXAMPLE

Solve equations using multiplication or division.

1 Solving an Equation Using Division Solve 9x ⫽ ⫺108. Write original equation.

9x ⫺108 ᎏᎏ ⫽ ᎏᎏ 9 9

Divide each side by 9.

x ⫽ ⫺12

Lesson 2.6

9x ⫽ ⫺108

Simplify.

Answer: The solution is ⫺12. ✓ Check

EXAMPLE

9x ⫽ ⫺108 9(⫺12) ⱨ ⫺108 ⫺108 ⫽ ⫺108 ✓

Write original equation. Substitute ⫺12 for x. Solution checks.

2 Solving an Equation Using Multiplication k 25

Solve ᎏᎏ ⫽ 5. k ᎏᎏ ⫽ 5 25 k 25 p ᎏᎏ ⫽ 25 p 5 25

k ⫽ 125

Write original equation. Multiply each side by 25. Simplify.

Answer: The solution is 125.


1. 3x ⫽ 3

2. 8a ⫽ 32

3. ⫺m ⫽ 5

4. 7n ⫽ ⫺49

5. ⫺8b ⫽ 96

6. ⫺3y ⫽ ⫺27

m 5

7. ᎏᎏ ⫽ 9

n 12 y 10. ᎏᎏ ⫽ 1 5 j 12. ᎏᎏ ⫽ ⫺3 ⫺8

8. ᎏᎏ ⫽ 5

x 7

9. 8 ⫽ ᎏᎏ c 2

11. ⫺60 ⫽ ᎏᎏ


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EXAMPLE

For use with pages 96 –101

3 Writing and Solving an Equation If you divide your camera film equally over your 7-day vacation, you will take 25 pictures a day. How many exposures do you have?

Lesson 2.6

Solution Let e represent the number of exposures you have. Write a verbal model. Then use the verbal model to write an equation. Total number of exposures ⫽ Number of pictures taken per day Number of days e ᎏᎏ ⫽ 25 7 e 7 p ᎏᎏ ⫽ 7 p 25 7

e ⫽ 175

Substitute values. Multiply each side by 7. Simplify.

Answer: You have 175 exposures.

Exercises for Example 3 13. Your parents divide money evenly among you and your three siblings. Each of you receives $75. Find the total amount your parents gave. 14. You purchase 9 yards of fabric. The total cost is $45. Find the cost per yard of the fabric.

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EXAMPLE

Solve equations involving decimals.

1 Adding and Subtracting Decimals a. Find the sum ⫺3.7 ⫹ 1.82. Use the rule for adding numbers with different signs. Subtract 1.82 from ⫺3.7. ⫺3.7 ⫹ 1.82 ⫽ ⫺1.88 ⫺3.7 > 1.82, so the sum has the same sign as ⫺3.7. b. Find the difference ⫺5.06 ⫺ 4.05. First rewrite the difference as a sum: ⫺5.06 ⫹ (⫺4.05). Then use the rule for adding numbers with the same sign. Add ⫺5.06 and ⫺4.05. ⫺5.06 ⫹ (⫺4.05) ⫽ ⫺9.11 Lesson 2.7

Both decimals are negative, so the sum is negative.

Exercises for Example 1 Find the sum or difference.

EXAMPLE

1. ⫺2.15 ⫹ (⫺7.5)

2. ⫺3.68 ⫹ 0.23

3. 5.27 ⫹ (⫺7.12)

4. ⫺8.25 ⫺ 1.28

5. ⫺2.65 ⫺ (⫺4.9)

6. ⫺11.43 ⫺ (5.28)

2 Multiplying and Dividing Decimals a. ⫺0.25(9.95) ⫽ ⫺2.4875

Different signs: Product is negative.

b. ⫺2.85(⫺4.8) ⫽ 13.68

Same sign: Product is positive.

c. ⫺45.92 ⫼ (⫺8.2) ⫽ 5.6

Same sign: Quotient is positive.

d. ⫺180.12 ⫼ 15.8 ⫽ ⫺11.4

Different signs: Quotient is negative.

Exercises for Example 2 Find the product or quotient.

7. 3.8(⫺8.2) 9. ⫺2.7(⫺0.3) 11. ⫺30.6 ⫼ 8.5


8. ⫺5.4(1.2) 10. 7.875 ⫼ ⫺6.3 12. ⫺21.46 ⫼ ⫺2.9

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EXAMPLE


3 Solving Addition and Subtraction Equations Solve the equation. a. z ⫹ 1.85 ⫽ 0.78

Solution a. z ⫹ 1.85 ⫽ 0.78 z ⫹ 1.85 ⫺ 1.85 ⫽ 0.78 ⫺ 1.85 z ⫽ ⫺1.07 b.

EXAMPLE

x ⫺ 2.59 ⫽ ⫺1.45 x ⫺ 2.59 ⫹ 2.59 ⫽ ⫺1.45 ⫹ 2.59 x ⫽ 1.14

b. x ⫺ 2.59 ⫽ ⫺1.45 Write original equation. Subtract 1.85 from each side. Simplify. Write original equation. Add 2.59 to each side. Simplify.

4 Solving Multiplication and Division Equations Solve the equation. y ⫺9.6

Lesson 2.7

a. ⫺0.4b ⫽ 1

b. ᎏᎏ ⫽ ⫺8.1

Solution a. ⫺0.4b ⫽ 1

Write original equation.

1 ⫺0.4b ᎏᎏ ⫽ ᎏ ᎏ ⫺0.4 ⫺0.4

b ⫽ ⫺2.5 y ᎏᎏ ⫽ ⫺8.1 b. ⫺9.6 y ⫺9.6 ᎏᎏ ⫽ ⫺9.6(⫺8.1) ⫺9.6

冢

Divide each side by ⫺0.4. Simplify. Write original equation.

冣

Multiply each side by ⫺9.6.

y ⫽ 77.76

Simplify.


13. a ⫹ 6.98 ⫽ ⫺3.54

14. t ⫹ 70.12 ⫽ 4.28

15. x ⫺ 4.79 ⫽ ⫺11.82

16. m ⫺ 13.56 ⫽ ⫺12.02

17. 12.4x ⫽ ⫺169.88

18. ⫺7.9y ⫽ ⫺40.29

c ⫺50.12

19. ᎏᎏ ⫽ ⫺0.04

66


x ⫺13.2

20. ᎏᎏ ⫽ 20.1
