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Solving Equations CHAPTER Solving Equations How far can an BEFORE athlete run in In the previous chapter you’ve . • Performed operations with integers 10 seconds? • Written and evaluated variable expressions Now In Chapter 2 you’ll study . • Using addition, multiplication, and distributive properties • Simplifying variable expressions • Solving equations using mental math • Solving equations using addition, subtraction, multiplication, or division • Solving equations with decimals WHY? So you can solve real-world problems about . • dinosaurs, p. 67 • giant pumpkins, p. 76 • fitness, p. 81 • aviation, p. 89 • mountain climbing, p. 94 • computers, p. 100 • baseball, p. 106 60 PAFL_229843_C02_CO.indd 60 12/17/08 10:58:48 AM Chapter 2 Resources Go to thinkcentral.com • Online Edition • eWorkbook • @HomeTutor • State Test Practice • More Examples • Video Tutor .BUI MATH In the Real World Track Race The world’s fastest athletes can run 100 meters in under 10 seconds. In this chapter, you will use equations to solve problems like finding the average rate at which an athlete runs a race. What do you think? Suppose an athlete runs at an average rate of 10.5 meters per second for 10 seconds. Use the formula distance ϭ rate ϫ time to find the distance the athlete runs. PAFL_229843_C02_CO.indd 61 12/17/08 10:58:54 AM CHAPTER Chapter Prerequisite Skills PREREQUISITE SKILLS QUIZ Review Vocabulary Preparing for Success To prepare for success in this chapter, test your numerical expression, p. 5 knowledge of these concepts and skills. You may want to look at the variable expression, p. 5 pages referred to in blue for additional review. order of operations, p. 16 absolute value, p. 23 1. Vocabulary Describe the difference between a numerical expression decimal, p. 746 and a variable expression. Perform the indicated operation. (pp. 750–752) 2. 3.8 ϩ 7.1 3. 8.23 Ϫ 4.97 4. 5.5 ϫ 9.4 5. 6.93 Ϭ 2.1 Write a variable expression to represent the phrase. (p. 5) 6. 7 more than a number 7. The quotient of a number and 4 Perform the indicated operation. (pp. 29, 34, 42) 8. Ϫ19 ϩ 12 9. 8 Ϫ (Ϫ20) 10. 6(Ϫ7) 11. Ϫ75 Ϭ (Ϫ5) NOTETAKING STRATEGIES USING DEFINITION MAPS When you study a new concept, you Note Worthy can use a definition map to define the concept, describe the concept’s attributes, and give specific examples. A definition You will find a notetaking map for power is shown below. strategy at the beginning of each chapter. Look for additional notetaking and What is it? What are its attributes? study strategies throughout A power is a product A power consists of a base the chapter. of repeated factors: and an exponent. 23 ϭ 2 p 2 p 2. The base is the number POWER that is used as a factor. The base for 23 is 2. What are examples? The exponent is how many 3 6 4 times the base repeats. 2 (0.5) n The exponent for 23 is 3. A definition map will be helpful in Lesson 2.3. 62 PANL_9780547587776_C02_SG.indd 62 11/4/10 6:28:25 PM 2 1 Properties and Operations LESSON . BEFORE Now WHY? You found sums and You’ll use properties of So you can compare the lengths Vocabulary products of numbers. addition and multiplication. of two fish, as in Ex. 48. additive identity, p. 64 multiplicative identity, p. 64 In English, commute means to change locations, and associate means to group together. These words have similar meanings in mathematics. Commutative properties let you change the positions of numbers in a sum or product. Associative properties let you group numbers in a sum or product together. Commutative and Associative Properties Commutative Property of Addition Commutative Property of Multiplication Words In a sum, you can add the Words In a product, you can multiply numbers in any order. the numbers in any order. Numbers 4 ϩ (Ϫ7) ϭ Ϫ7 ϩ 4 Numbers 8(Ϫ5) ϭ Ϫ5(8) Algebra a ϩ b ϭ b ϩ a Algebra ab ϭ ba Associative Property of Addition Associative Property of Multiplication Words Changing the grouping of Words Changing the grouping of the numbers in a sum does not the numbers in a product does change the sum. not change the product. Numbers (9 ϩ 6) ϩ 2 ϭ 9 ϩ (6 ϩ 2) Numbers (3 p 10) p 4 ϭ 3 p (10 p 4) Algebra (a ϩ b) ϩ c ϭ a ϩ (b ϩ c) Algebra (ab)c ϭ a(bc) Example 1 Using Properties of Addition Music You buy a portable CD player for $48, rechargeable batteries with charger for $25, and a CD case for $12. Find the total cost. Solution The total cost is the sum of the three prices. Use properties of addition to group together prices that are easy to add mentally. 48 ϩ 25 ϩ 12 ϭ (48 ϩ 25) ϩ 12 Use order of operations. ϭ (25 ϩ 48) ϩ 12 Commutative property of addition ϭ 25 ϩ (48 ϩ 12) Associative property of addition ϭ 25 ϩ 60 Add 48 and 12. ϭ 85 Add 25 and 60. Answer The total cost is $85. Video Tutor Go to thinkcentral.com Lesson 2.1 Properties and Operations 63 PAFL_229843_C0201.indd 63 12/17/08 12:22:30 PM Example 2 Using Properties of Multiplication .25 ؍ ؊7 and y ؍ Study Strategy Evaluate 4xy when x .4xy ϭ 4(؊7)(25) Substitute ؊7 for x and 25 for y Another Way You can also evaluate the expression [4(Ϫ7)](25) Use order of operations. Ϫ 4( 7)(25) in Example 2 [Ϫ7(4)](25) Commutative property of multiplication simply by multiplying from left to right: Ϫ7[(4)(25)] Associative property of multiplication 4(Ϫ7)(25) ϭ Ϫ28(25) Ϫ7(100) Multiply 4 and 25. .ϭ Ϫ700 Ϫ700 Multiply ؊7 and 100 However, this calculation is more difficult to do mentally. Checkpoint In Exercises 1–3, evaluate the expression. Justify each of your steps. 1. (17 ϩ 36) ϩ 13 2. 8(Ϫ3)(5) 3. 3.4 ϩ 9.7 ϩ 7.6 4. Evaluate 5 x2 y when x ϭ Ϫ6 and y ϭ 20. Example 3 Using Properties to Simplify Variable Expressions Simplify the expression. a. x ϩ 3 ϩ 6 ϭ (x ϩ 3) ϩ 6 Use order of operations. ϭ x ϩ (3 ϩ 6) Associative property of addition ϭ x ϩ 9 Add 3 and 6. b. 4(8y) ϭ (4 p 8)y Associative property of multiplication ϭ 32y Multiply 4 and 8. Checkpoint Simplify the expression. 5. m ϩ 5 ϩ 9 6. 6(3k) 7. 4 ϩ x ϩ (Ϫ1) 8. (2r)(Ϫ5) Identity Properties When 0 is added to any number, or when any number is multiplied by 1, the result is identical to the original number. These properties of 0 and 1 are called identity properties, and the numbers 0 and 1 are called identities. Note Worthy Identity Properties In your notebook, be sure to Identity Property of Addition Identity Property of Multiplication include important properties such as those from this lesson. Words The sum of a number and the Words The product of a number and the You may find it helpful to write additive identity , 0, is the number. multiplicative identity , 1, is the number. each property using words, numbers, and algebra, as Numbers Ϫ6 ϩ 0 ϭ Ϫ6 Numbers 4 p 1 ϭ 4 shown at the right. Algebra a ϩ 0 ϭ a Algebra a p 1 ϭ a 64 Chapter 2 Solving Equations PAFL_229843_C0201.indd 64 12/17/08 12:22:35 PM Example 4 Identifying Properties Statement Property Illustrated a. (Ϫ5)(1) ϭ Ϫ5 Identity property of multiplication b. 2 (Ϫ9) ϭ Ϫ9 ϩ 2 Commutative property of addition c. y 2 ϩ 0 ϭ y 2 Identity property of addition d. 2(pq) ϭ (2p)q Associative property of multiplication Review Help Unit Analysis You can use unit analysis to find a conversion factor that To write the conversion factor converts a given measurement to different units. A conversion factor, 1 foot __ , you need to know 1 foot 12 inches such as __ , is equal to 1: 12 inches that 1 foot ϭ 12 inches. To 1 foot 12 inches __ ϭ __ 1 help you convert measures, 12 inches 12 inches see the Table of Measures on p. 793. So, the identity property of multiplication tells you that multiplying a measurement by a conversion factor does not change the measurement. Example 5 Multiplying by a Conversion Factor Roller Coasters The Steel Dragon 2000 is the world’s longest roller coaster. Its length is 2711 yards. How long is the roller coaster in feet? Solution 1 Find a conversion factor that converts yards to feet. The statement 1 yard ϭ 3 feet gives you two conversion factors. 1 yard 3 feet Factor 1: __ Factor 2: __ 3 feet 1 yard Unit analysis shows that a conversion factor that converts yards to feet has feet in the numerator and yards in the denominator: feet yards p _ ϭ feet yards So, factor 2 is the desired conversion factor. 2 Multiply the roller coaster’s length by factor 2 from Step 1. 3 feet Use the conversion factor. 2711 yards ϭ 2711 yards p __ 1 yard Divide out common unit. 8133 feet Multiply. The Steel Dragon 2000, located Answer The roller coaster is 8133 feet long. in Nagashima, Japan Checkpoint 9. Identify the property illustrated by the statement z 4 p 1 ϭ z 4 .
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