Algebraic Expressions

Bowling is a popular sport among 8.1 The Parts of Cars You Don’t See! people all over the world. In Relationships between Quantities...... 561 the United States, 8.2 Tile Work the most popular form of Simplifying Algebraic Expressions...... 569 bowling is ten-pin bowling. 8.3 Blueprints to Floor Plans to Computers Other forms of bowling include Using the Distributive Property to Simplify five-pin bowling, nine-pin Algebraic Expressions...... 579 skittles, and duckpin bowling, 8.4 Are They Saying the Same Thing? to name a few. Multiple Representations of Equivalent Expressions...... 593 8.5 Like and Unlike Combining Like Terms...... 605 8.6 DVDs and Songs: Fun with Expressions Using Algebraic Expressions to Analyze © Carnegie Learning and Solve Problems...... 617

559

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Chapter 8 Algebraic Expressions 27/08/13 4:11PM

© Carnegie Learning they make? car manufacturers to know how many parts are neededfor each vehicle Can you thinkof some car parts thatyou cannot see? Doyou thinkitishelpful for is estimated thatthere are roughly 17,000 to 20,000 parts inanaverage car. about theother parts you cannot see like thebrakes orthefuelfilter. Actually, it some parts of acar like thetires, thesteeringwindshield. But wheel,andthe what H   In thislesson, you will: Learning Goals ave you ever wondered how many parts make upacar? Sure, you can see

Write numerical andalgebraic expressions. Predict thenext term inasequence. You Don’t See! Don’t You Cars of Parts The Relationships between Quantities 8.1 Relationships between Quantities   Key Terms

term sequence •

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Chapter 8 by buildingatrainwithonecar, thentwocars,andfinallythree carsasshown. Josh likesbuildingtrainsoutoftrainparts.Eachcarneedsfourwheels.begins sequences ofnumbers. letters, orotherobjects.Thenumberoftraincarsandthewheelsneededform A 1. 2. sequence

Number ofwheels: Number ofcars: Write twosequencesusingJosh’s traincars. total wheelswillheneedtobuildatrainwith5cars? Describe howmanytotalwheelsJoshwillneedtobuildatrainwith4cars.How

Building Trains Algebraic Expressions is a pattern involvinganordered isapattern arrangementofnumbers,geometricfigures, 27/08/13 4:11PM

© Carnegie Learning sequence. Thesecondtermistheobjectornumberinsequence,andsoon. a product ofnumbersandvariables.Thefirstterm isthefirstobjectornumberin sequence formthe Each numberofcarsinthefirstsequenceandeachwheelssecond 7. 6. 5. 4. 3. 8. a trainwithanunknownnumberofcars. Write analgebraicexpression torepresent thenumber ofwheelsneededtobuild has anynumberofcars. Describe inyourownwords howtocalculatethenumberofwheelsneedediftrain 6 cars,7and8cars. Explain howyouwoulddeterminethenumberofwheelsneededtomakeatrainwith What are thetwoquantitiesthatchangeinJosh’s train-buildingsituation? What isthefifth terminthefirstsequence? What isthefirstterminsecondsequence? terms, ormembers,ofthesequence.A 8.1

Relationships between Quantities term isanumber, avariable,or

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563 27/08/13 4:11PM 9. Josh would like to add more trains to his collection. The cost of each car is $8. a. Write a sequence that represents the cost of buying 1 car, 2 cars, 3 cars, 4 cars, and 5 cars.

b. Describe the mathematical process that is being repeated to determine the total cost for a set of cars.

c. What are the variable quantities, or the quantities that are changing, in this situation? Include the units that are used to measure these quantities.

d. What quantity remains constant? Include the units that are used to measure this quantity.

e. Which variable quantity depends on the other variable quantity?

f. Write an algebraic expression for the total cost, in dollars, for a train set with n cars. Rewrite the first five terms of the sequence. © Carnegie Learning

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variable, orproduct ofnumbersandvariables. sometimes numbersandoperations.Atermofanalgebraicexpression isanumber, previously,As youlearned analgebraicexpression containsatleastonevariableand 10. d. c. b. a. Write analgebraicexpression torepresent eachsequence. e.

2, 4,6,8,… 2, 4,8,16,… 3, 5,7,9,… 1, 3,5,7,… 101, 102,103,104,… by the variable The first term is 3 multiplied Parts of Algebraic Expressions is addition,andthesecondoperationsubtraction. The expression has three terms: 3 Consider theexpression 3 The firstoperationisaddition. x . multiplied by the variable The secondtermis4 x 3

x 1

1 4 4 y 8.1

y 2 x

2 , 4

7. 7 The secondoperationissubtraction. y Relationships between Quantities , and 7. The first operation y . constant termof7. The third termisa

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565 27/08/13 4:12PM 1. Let’s consider two algebraic expressions: 8 1 5x and 8 2 5x a. Identify the number of terms in each algebraic expression.

b. Identify the operation in each algebraic expression.

c. Identify the terms in each algebraic expression.

d. What is the same in both expressions?

e. What is different in the expressions?

2. Identify the number of terms, and then the terms themselves for each algebraic expression. a. 4 2 3x

b. 4a 2 9 1 3a © Carnegie Learning

c. 7b 2 9x 1 3a 2 12

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3. c. b. a. Write analgebraicexpression thatrepresents eachsituation. g. f. e. d.

in Diego drove ataconstantrateof60milesperhour. Howmanymilesdidhedrive many cookiesdoeseachfriendreceive ifsheshares with Jackie has12cookies.Shewantstoshare themequally amongherfriends.How bought lunch? The costofaschoollunchis$1.60.Howmuchmoneywascollectedif $8.75. Whatisthecostof Used paperbackbookscost$6.25eachwithashippingandhandlingof and Chairs cost$35andsofas$75.Howmuchwouldittopurchase person pay? number ofpeople, The costtorent astorageunitis$125.Thecostwillbeshared equallyamonga withdraws Donald has$145inhissavingsaccount.Howmuchisleftaccountifhe t hours? y sofas? d dollars? p , whoare storingtheirbelongings.Howmuchwilleach x books? 8.1

Relationships between Quantities f friends? n

students x • chairs

567 27/08/13 4:12PM 4. Write an algebraic expression that represents each word expression. a. the quotient of a number, n, b. three more than a number, n divided by 7

c. one-fourth of a number, n d. fourteen less than three times a number, n

e. six times a number, n, f. three more than a number, n, subtracted from 21 added to 21

g. one-fourth of a number, n, h. Ten times the square of a number, w, minus 6 divided by 12.

5. Construct an algebraic expression for each description. a. There are 2 terms. The first term is a constant added to the second term, which is a product of a number and a variable.

b. There are 4 terms. The first term is a quotient of a variable divided by 11. This is added to a second term, which is a constant. The third term is a second variable multiplied by three-fourths. The third term is subtracted from the first 2 terms. The last term is a different constant added to the other 3 terms.

c. The cube of a variable subtracted from a constant and then added to the square of the same variable.

d. A number multiplied by the square of a variable minus a number multiplied by the same variable minus a constant. © Carnegie Learning

Be prepared to share your solutions and methods.

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© Carnegie Learning Is theorder of thedirections important? Can you thinkof other directions thatpeople ormachines doto perform tasks? instruct computers to dotasks. program. This iswhy computer programmers must bevery careful how they confusion thatmight happenifscrolling onadrop-down menuclosed outa As withany directions, theorder of thedirections isimportant. Just imagine the the computer to perform thetask you want. mouse clickonanicon, even whenyou close outaprogram requires directions for can perform thetask. Just thinkaboutit: every drop-down menuyou select, every Computer programming istheability of writingdirections so thatthecomputer They alluse programming. a browser. What dothese andmany other computer programs have incommon? blog, you probably use some type of text program. Ifyou surfthenet, you use I f you type your papers, you probably use aword processing program. Ifyou    In thislesson, you will: Learning Goals

algebraic expressions. Use algebra tiles to simplify Use theOrder of Operations. simplify expressions. Addition andMultiplication to Commutative Properties of Use theAssociative and Expressions Simplifying Algebraic Work Tile    Key Terms

of Addition Associative Property Multiplication Property of Commutative Property of Addition Commutative 8.2 Simplifying Algebraic Expressions   

like terms simplify of Multiplication Associative Property •

569 27/08/13 4:12PM Problem 1 Going Bowling

1. You and your friends are going bowling. It costs $2.50 to rent shoes, and it costs $3.50 to bowl one game. a. How much will it cost you to rent shoes and bowl 1 game?

b. How much will it cost you to rent shoes and bowl 2 games?

c. How much will it cost you to rent shoes and bowl 3 games?

2. Describe how you calculated the total cost for each situation.

3. What are the variable quantities in this problem?

4. What are the constant quantities in this problem?

5. What variable quantity depends on the other variable quantity? © Carnegie Learning

6. Write an algebraic expression to determine the total cost to rent shoes and bowl games. Let g represent the unknown number of games bowled.

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© Carnegie Learning 7. are reversed. These expressions are equivalent,eventhoughthefactorsinsecondterms Expressions 1and2demonstratetheCommutativeProperty ofMultiplication. These expressions are equivalent,eventhoughtheorder ofthetermsisreversed. Expressions 1and3demonstratetheCommutativeProperty ofAddition. For anynumbers more factorsinamultiplicationproblem doesnotchangetheproduct. The For anynumbers terms inanadditionproblem doesnotchangethesum. The you addormultiplytwomore numbersdoesnot affect thesumorproduct. The Commutative Commutative Commutative b. a. Given thefourexpressions shown:

state oneotherexampleoftheCommutativeProperty ofAddition. state oneotherexampleoftheCommutativeProperty ofMultiplication. Let represent thetotalcosttorent shoesandbowlgames. There are severalwaystowriteanalgebraicexpression to same relationship. These expressions are equivalentbecausetheyexpress the 2.50 Expression g 1 represent the number of games bowled. 3.50 a a

Properties

and and Property Property

g 1

b b

, , 2.50 Expression a a

× 1

of of of

b 1 b

Addition Multiplication Addition

5 g

(3.50)

b b ×

a 2

a .

and . statesthatchangingtheorder oftwoormore Expression 3.50 8.2 Multiplication statesthatchangingtheorder oftwoor g

1 Simplifying Algebraic Expressions 2.50

g Expression statethattheorder inwhich (3.50) 1 2.50

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571 27/08/13 4:12PM When determining the total cost to rent shoes and bowl games, you used the Order of Operations. Recall that these rules ensure that the order in which numbers and operations are combined is the same for everyone. You should remember that when evaluating any expression, you must use the Order of Operations.

8. Evaluate each numerical expression using the Order of Operations. Show all your work. a. 22 2 4(3) b. 9(3) 1 4(5)

c. 60 2 32 d. (8 2 5)2 1 52

e. 7(12) 2 2(3 1 2) f. (10 2 6) 1 22 © Carnegie Learning

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© Carnegie Learning Problem 2 Problem and factors inamultiplicationproblem doesnotchangetheproduct. Foranynumbers The ( in anadditionproblem doesnotchangethesum.Foranynumbers The parentheses wheneveryouwant. involving the order ofaddendsandtoaddparentheses whenever youwant.Foranyexpression powerful properties. Foranyexpression involving The CommutativeandAssociativeProperties ofAdditionandMultiplicationare very 1. 2. a 3.

Associative Associative c How dotheexpressions inthefirstcolumncompare tothoseinthesecondcolumn? Evaluate eachnumericalexpression usingtheOrder ofOperations. Compare thetwoexpressions ineachrow. Whatdoyounotice? b , ( )

1 a

only c More Properties

b 5 ) × multiplication,youcanalsochangetheorder ofthefactorsandinclude

c 1

Property Property 5 ( b

× (

c b ).

(19 of of (4 c (12

(16 ). Multiplication Addition 1 1 3 42) 17) 3 25)4 5)2 1 1 3 statesthatchangingthegrouping oftheterms 8 8.2 statesthatchangingthegrouping ofthe

Simplifying Algebraic Expressions 19 only 4 12(25 16(5 1 1 addition,theyallowyoutochange (42 (17 3 3 1 2) 1 4) 3) 8) a , b , and c ,

• a

, b , 573 27/08/13 4:12PM Benjamin and Corinne are both trying to determine this sum.

22 1 17 1 3 1 8

Their methods are shown.

Benjamin 22 +17 39 Corinne +3 42 (22 + 8) + (17 + 3) +8 30 + 20 50 50

4. Describe the differences between the methods Benjamin and Corinne used to calculate the sum.

Use the 5. Determine each sum by writing an equivalent numerical properties to expression using the Commutative and Associative Properties. look for number a. 5 1 7 1 5 b. 5 1 13 1 2 combinations that make it simpler to add like Corinne did.

c. 2 1 21 1 8 1 9 d. 4 1 3 1 6 1 5 © Carnegie Learning

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© Carnegie Learning Problem 3 Problem raised tothesamepower. Thenumericalcoefficientsofliketermscanbedifferent. In analgebraicexpression, liketermsare twoormore termsthathavethesamevariable terms possible,andifalllikehavebeencombined. expression withfewerterms.Anexpression isinsimplestformifitcontainsthefewest To simplifyanexpression istousetherulesofarithmeticand algebra torewrite that The CommutativeandAssociativeProperties canhelp yousimplifyalgebraicexpressions. Let’s usealgebratilestoexplore simplifying algebraicexpressions. You can represent the algebraic expression 3

Simplifying Algebraic Expressions 1 coef Numerical 1 cient 1 x x x V 3x +2 1 1 y y ariable x x 8.2 Constant Simplifying Algebraic Expressions x

1 1 1 2 using algebra tiles. me see the the see me really help help really math! Models Models x x

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Chapter 8

a. below eachalgebraicexpression. Represent eachalgebraicexpression usingalgebratiles.Then,sketchthemodel

e. c.

3 4 4 x y Algebraic Expressions

1 1 2 3

b. f. d.

2 2 5 x x x

+3 1 1 y 27/08/13 4:12PM

2. g. e. a. Simplify eachalgebraicexpression. f. e. d. c. b. a. by combiningliketerms. Use yourmodelsfrom Question1towriteeachalgebraic expression insimplestform c.

(3 55 (3 2 (4 (3 (4 (2 (3 (16.2 x y x y x y x x

1 1 1 1 1 1 1 1

1 3 x 2) 3) 2) 3) 1) 2) 2)

65 1 x

1 1 1 1 1 1 1 1 4.1) y 4

(4 5 4 (4 (4 (2 5 1 x x x 75 y y y x

1 1 1 1 (10.4

2) 3) 3) 1)

2 2)

h. f. b. d.

( (2 (8 (

x __ 8.2 4 3

x x models I just regroup

1 x

like termsusingmy

1 1

1 1) all thesametiles When Icombine 5) 17 2

Simplifying Algebraic Expressions 1

together. ) 1 y 1 (3 ) (3 1

y __ 4 3

(11 y

x 1

1 2) 2) x

2 1 8 (4

)

1 terms means to add or or add to means terms subtract terms with with terms subtract the same variables. variables. same the 2) So, combining like like combining So, Like 3x + 5x. 5x. + 3x Like That's 8x.” That's

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577

27/08/13 4:12PM Talk the Talk

1. How do you know when an algebraic expression is in simplest form?

2. How do you know when a numerical expression is in simplest form?

Be prepared to share your solutions and methods. © Carnegie Learning

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© Carnegie Learning D blueprints—do you thinkpeoplewillnolongerplanbuildings aswell? as more work isbeingdone oncomputers, peopleare no longerusingpaper homes, apartment buildings—well, almost any buildingyou can thinkof. However, Architects anddraftspersons would create thefloorplan for buildings, malls, Not very longago, theblueprint was thebackbone of thebuildingprocess. remain standing. a buildingcan withstand damagefrom anearthquake orahurricane andstill buildings inthearea. Incertain cities, there are special requirements so that materials willbeneeded,andhow thenew structure willinteract withother construction takes place, architects planwhatthe buildingwilllooklike, what your house orapartment buildingwas constructed? Before theactual    In thislesson, you will: Learning Goals o you ever wonder how your school was built? Or have you wondered how

using algebra tiles. Model theDistributive Property the Distributive Property. Write algebraic expressions using using theDistributive Property. Simplify algebraic expressions 8.3 to Simplify Algebraic Expressions Using the Distributive Property Plans toComputers Blueprints toFloor Using theDistributive Property to Simplify Algebraic Expressions     Key Terms

over Subtraction Distributive Property of Division Distributive Property of Divisionover Addition over Subtraction Distributive Property of Multiplication over Addition Distributive Property of Multiplication •

579 27/08/13 4:12PM Problem 1 Installing Carpet

The Lewis family just bought a new house. They will install new carpet in two adjacent rooms before they move in.

13 ft 7 ft

11 ft Family Room Office 11 ft

1. How would you calculate the total area of the rooms?

Brian I calculated the area of each room. Then, I added the two areas together to get the area of both rooms. (11 x 13) + (11 x 7)

2. Calculate the area using Brian’s expression. © Carnegie Learning

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© Carnegie Learning that if There isalsothe numbers The Distributive Property ofMultiplicationoverAddition. The twoexpressions, (11 4. 3. Distributive you aboutthetwoexpressions BrianandSarawrote? What doyounoticeaboutyourresults usingeachexpression? Whatdoesthattell Calculate thearea usingSara’s expression. a , b a , and , b , and c 8.3

are anyreal numbers,then Distributive Property c

, a Using theDistributive Property to Simplify Algebraic Expressions

• ( b

3 Sara I multiplied the width by the total length of both rooms to find the total area of both rooms. 1 1 (13 + 7) 1 (13 1

of 1 13)

c Multiplication Property ) 1 5 (11

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b 3

of 7)and11(13 1

Multiplication a

• a over

• . ( b

Addition 2

c 1 ) 5 7),are equalbecauseofthe

over a statesthatforanyreal

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Subtraction 2

•

c . , whichstates

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Chapter 8 1 expressions withtheDistributiveProperty. Let’s usealgebratilestoexplore rewriting algebraic

model usingalgebratilesisshown. multiply the5byeachtermofquantity( Distributive Property torewrite thisexpression. Inthiscase, two factors:5andthequantity( Consider theexpression 5( Algebraic Expressions Algebra Tiles 5 x x just likeaddingthe appears tobe quantity x+1 1 1 1 1 1

This model five times! 5( x 11)5

x 11).Thisexpression has 1

x x x x x x 1 1 x 11).You canusethe x +1 x 11).The speaking about an algebraic algebraic an about speaking "the quantity." For example example For quantity." "the expression use the words words the use expression 1 1 1 1 1 1 (

x + 3) in words would would words in 3) + x be "the quantity x quantity "the be When you are are you When plus three."” plus

27/08/13 4:12PM

1. a. the expression usingtheDistributiveProperty. Create andsketchamodelofeachexpression usingyouralgebratiles.Then,rewrite c. b.

3( 2( 4(2 x x

x 1 1

1 2) 3) 1) 8.3

Using theDistributive Property to Simplify Algebraic Expressions

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583 27/08/13 4:12PM d. (3x 1 1)2 © Carnegie Learning

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2. b. a. Rewrite eachexpression usingtheDistributiveProperty. Then,simplifyifpossible. d. c. f. e.

2( __ 5 2( __ 2( 2 2 3 1 x

(4 (6 x x

x x 1 1 1

2(3 1 1 4) 5) 5) 2) 12) 8.3 1 x 1

1 2 4(

2( 8 7)

x Using theDistributive Property to Simplify Algebraic Expressions x

1 1 7) 5) addition or subtraction, subtraction, or addition multiplication "over" the the "over" multiplication I “"distribute" the the “"distribute" I like this: like 2(x + 2) 2) + 2(x 2x + 4 + 2x Ah, so so Ah, ↓ ↓ ↓

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585 27/08/13 4:12PM Problem 3 Splitting an Expression Equally

So far in this lesson, you multiplied expressions together using the Distributive Property of Multiplication over Addition and the Distributive Property of Multiplication over Subtraction. Now let’s think about how to divide expressions. How do you think the Distributive Property will play a part in dividing expressions? Let’s find out.

1. Represent 4x 1 8 using your algebra tiles. a. Sketch the model you create. I know that multiplication and division are inverse operations. So, I should start thinking in reverse.

b. Divide your algebra tile model into as many equal groups as possible. Then, sketch the model you created with your algebra tiles. © Carnegie Learning

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number of equal groups from part (c). The product you calculateVerify should you equal created 4 equal groups by multiplying your expression from part (d) by the Write anexpression torepresent eachgroup from yoursketchesinpart(b). How manyequalgroups didyoucreate? Division overAddition. using theDistributiveProperty of from Question1.You canrewrite it Let’s considerthedivisionexpression 8.3 ______ 4 x

4 1

Using theDistributive Property to Simplify Algebraic Expressions

5 5 5

x 1 ___ 4 4

x x 1

1 1 2 2

__ 8 4

rewrite the expression, expression, the rewrite the denominator into into denominator the you have to divide divide to have you both terms in the the in terms both numerator. When you you When

• x

1 587 8. 27/08/13 4:12PM The model you created in Question 1 is an example that shows that the Distributive Property holds true for division over addition. 2. Represent 2x 1 6y 1 4 using your algebra tiles. a. Sketch the model you created. Be careful to not combine unlike terms using the same type of algebra tiles.

b. Divide your algebra tile model into equal groups. Then, sketch the model you created with your algebra tiles.

c. How many equal groups did you create?

d. Write an expression to represent each group from part (b). © Carnegie Learning e. Verify that you created equal groups by multiplying your expression from part (d) by the number of equal groups from part (c). The product should equal 2x 1 6y 1 4.

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numbers and 4. Distributive a. Represent 6 e. d. c. b. ______Question 2usingtheDistributive Rewrite thedivisionexpression from Then, simplifytheexpression. Property ofDivisionoverAddition. 2 x

1 Sketch themodelyoucreated. Write anexpression torepresent eachgroup from part(b). How manyequalgroups didyoucreate? created withyouralgebratiles. Divide youralgebratilemodelintoequalgroups. Then,sketchthemodelyou equal 6 by thenumberofequalgroups from part(c).Theproduct youcalculateshould Verify thatyoucreated equalgroups bymultiplying yourexpression from part(d)

2 6

1 c

8.3 ﬁ 4 1

Property 0, then 1 3(

3( x

Using theDistributive Property to Simplify Algebraic Expressions x 1

1 1). ______ a 1)usingyouralgebratiles.

1 of c

b Division

__ c a

b c over

Addition you must divide the the divide must you denominator into all all into denominator the terms in the the in terms the Remember that that Remember states that if numerator. a ,

b , and c are real

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589 27/08/13 4:12PM The expression in Question 4 can be simplified using the Distributive Property of Division over Addition in two ways.

5. Analyze each correct method. Explain the reasoning used in each.

Method 1 3(x + 1) 6 + 3(x + 1) __6 ______ = + 3 3 3 = 2 + (x + 1) Method 2 x + 1) ______6 + 3x + 3 ______6 + 3( = = x + 3 3 3 ______ 3x + 9 = 3 __3x __9 + = 3 3 = x + 3

Reasoning:

The Distributive Property also holds true for division over subtraction.

The Distributive Property of Division over Subtraction states that if a, b, and c are real

______a 2 b __a __b © Carnegie Learning numbers and c ﬁ 0, then c 5 c 2 c .

For example: ______2x 2 5 5 ___ 2x 2 __5 2 2 2 5 x 2 __ 5 2

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Be prepared toshare yoursolutionsandmethods. 6. a. Property youused. Simplify eachexpression usingaDistributiveProperty. Then,statewhichDistributive c. e. b. d.

______ ______3 32 3( 6 2 x x x 1

2 3

2 1 1 4 1 2(

7 7 4

2 1) 3 x

8.3 1 4)

Using theDistributive Property to Simplify Algebraic Expressions

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591 27/08/13 4:12PM © Carnegie Learning

592 • Chapter 8 Algebraic Expressions

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© Carnegie Learning S conservation haschangedsince thetimeof Roosevelt andMuir? conservation. Doyou thinklandconservation isstill taking place? Doyou think John MuirandPresident Theodore Roosevelt. Both menwere pioneers inland yo -SEMihtee) NationalPark. The park exists because of two conservationists: The firs use thelandfor recreation. owned plot of natural land.This landisset asidefor animalsafety, andpeoplecan national park is, butmany experts agree thataNationalPark isagovernment-     In thislesson, you will: Learning Goals o, whatmakes a park anationalpark? There are different ideas onwhata

to problem situations. Write thecorresponding expressions Determine iftwo expressions are equivalent. Graph expressions onacalculator. and graphs. Compare expressions usingproperties, tables, t nationalpark established was California’s Yosemite (pronounced of Equivalent Expressions Multiple Representations Thing? Same the Saying They Are 8.4 Multiple Representations of Equivalent Expressions  Key Term

equivalent expressions

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593 27/08/13 4:12PM Problem 1 Off to Camp!

Harley and Jerome belong to a Youth Outdoors Club. Each member needs to raise money to pay for camp by selling tins of popcorn. Each popcorn tin sold earns the member $4 toward the trip. Harley and Jerome have each sold 5 popcorn tins. Club chairperson Ms. Diani, asked all members to think about how much money they could earn toward their trip if they continue to sell more popcorn tins.

Harley and Jerome decide to let p represent the number of additional tins of popcorn to be sold. Harley expressed his potential earnings by the expression 4( p 1 5). Jerome expressed his potential earnings by the expression 4p 1 20.

1. Describe the meaning of each person’s expression in terms of the problem situation. Include a description for the units of each number or variable in the expression.

There are many ways to determine if Harley and Jerome’s expressions both accurately describe this situation. Two algebraic expressions are equivalent expressions if, when any values are substituted for variables, the results are equal. You can verify that two expressions are equivalent by using properties to simplify the expressions to the same expression, or by graphing them to show that the graphs are the same. Tables can be useful to show that expressions are not equivalent or whether they may be equivalent.

Let’s explore each possibility.

2. State the property that verifies that Harley and Jerome’s expressions are equivalent. © Carnegie Learning

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What wouldyouneedtodoverifythetwoexpressions are equivalent? What canyoudeterminebasedonthevaluesintable? Complete thetableshownforeachadditionalnumberoftinssold. Number of Additional Popcorn Tins Sold 25 15 3 7 8.4

Multiple Representations of Equivalent Expressions Harley’s Expression 4( p

 5) Jerome’s Expression 4 p

 20

•

595 27/08/13 4:12PM You can use a graphing calculator to determine if two expressions are equivalent. The graphing calculator will create a graph for each expression. Follow the steps.

Step 1: Press Y5 and enter what is shown.

y1 5 4(x 1 5)

y2 5 4x 1 20

To distinguish between the graphs of y1 and y2, move your ENTER cursor to the left of y2 until the \ flashes. Press one time to select \.

Step 2: Press WINDOW to set the bounds and intervals for the graph.

Xmin 5 0, Xmax 5 50, and Xscl 5 10,

Ymin 5 0, Ymax 5 100, and Yscl 5 10.

Step 3: Press GRAPH .

4. Sketch both graphs on the coordinate plane shown.

30 Money Earned (dollars)

x 10 20 30 40 © Carnegie Learning Tins Sold

5. How does the graph verify that the two expressions are equivalent?

596 • Chapter 8 Algebraic Expressions

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equivalent.” 1. were equivalentusinga But you you But table, Iwouldhaveto a. Choose three different valuesfor b. substitute every know forsureif two expressions

known value! Do thetwoexpressions appeartobeequivalent?Explain yourreasoning. Do the two expressions appear to be equivalent? Explain your reasoning. Wow. .to Equal orNot?

x 8.4

Multiple Representations of Equivalent Expressions x x andcompleteeachtable. __ 2 3

(

 7) __ 2 1

(

 9) __ 2 3



__ 3 7

__ 2 x



__ 2 9

•

597 27/08/13 4:12PM 2. Determine if the two expressions are equivalent. Graph each expression using a graphing calculator and sketch the graph in the coordinate plane shown. Press WINDOW to set the bounds: Xmin 5 0, Xmax 5 10, Xscl 5 1, Ymin 5 0, Ymax 5 10, and Yscl 5 1. a. Are the two expressions equivalent? Explain your reasoning. 2(x 1 2) 1 3x 5x 1 4 y Don't forget to put arrows on 9 each end of your line. 8 The arrows show that 7

6 the line goes on

5 forever.”

x 1 2 3 4 5 6 7 8 9

b. Are the two expressions equivalent? Explain your reasoning. __1 x 1 5 2 __1 (x 1 5) 2

2 © Carnegie Learning 1

x 1 2 3 4 5 6 7 8 9

598 • Chapter 8 Algebraic Expressions

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2. 1. a. have prepared 21mealstotal. The cooksdividetheincomingorders amongthemselves. Sofartoday, thecooks A localrestaurant isbusiestoverlunchandhasthree cookswhoworkatthistime. c. b. b. a. 25 oatmealenergy bars. The teamreceives $1.25foreachoatmealenergy barsold.Hehasalready sold Christopher issellingoatmealenergy barstoraisemoneyforhisbaseballteam. c.

Write anexpression, andshowyourcalculations. If 18additionalorders comein,howmanymealswill eachcookprepare? prepares. Let Write anexpression torepresent theunknownnumberofmealseachcook Write anexpression, andshowyourcalculations. If 42additionalorders comein,howmanymealswill eachcookprepare? for his baseball team. Let Write an expression to represent the unknown amount of money Christopher will raise baseball team?Write anexpression, andshowyour calculations. If Christophersells45more energy bars,howmuchmoneywillheraiseforthe baseball team?Write anexpression, andshowyourcalculations. If Christophersells10more energy bars,howmuchmoneywillheraisefor the Writing Expressions m represent thenumberofadditionalorders. 8.4

c represent the number of additional energy bars sold. Multiple Representations of Equivalent Expressions

•

599 27/08/13 4:12PM 3. The Music-For-All Club is offering a special deal to all new members. You can join the club at no cost and receive 10 free CDs. However, you must also agree to purchase at least 12 more CDs within the first year at the club’s special price of $11.99 per CD. a. What will 20 CDs from the club cost? Show your calculations.

b. What will 25 CDs from the club cost? Show your calculations.

c. Write an expression to represent the total cost of an unknown number of CDs. Let m represent the number of CDs you will get from the club.

4. Four friends decide to start a summer business doing yard work in their neighborhood. They will split all their earnings evenly. They have two lawnmowers, but they need to buy gas, rakes, trash bags, and a pair of pruners. They spend $100 buying supplies to get started. a. How much profit will each friend receive if they earn $350 the first week? Show your calculations.

b. How much profit will each friend receive if they earn $475 the first week? Show your calculations.

c. Write an expression that represents the unknown profit for each friend. © Carnegie Learning Let d represent the amount of money earned.

600 • Chapter 8 Algebraic Expressions

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1. Let What quantityischanginginthisriddle? Why doyouthinkwillalwaysendwiththesamenumber? you notice? What isyouranswer?Compare youranswerto partner’s answer. Whatdo Step 6:Subtractyouroriginalnumber. Step 5:Dividethedifference by3. Step 4:Subtract6from theproduct. Step 3:Multiplythesumby3. Step 2:Add9toyournumber. Step 1:Pickanumberbetween1and30. Try thisnumberriddlewithapartner. Followthesteps.

n represent theoriginalnumber. Write andsimplifyanexpression foreachstep. Number Magic

8.4

Multiple Representations of Equivalent Expressions

•

601 27/08/13 4:12PM Talk the Talk

Complete the graphic organizers by stating each rule using variables and providing an example.

Commutative Property of Commutative Property of Addition Multiplication For any number For any number a and b a and b

Example: Example:

Number Properties

For any number For any number a, b, and c a, b, and c

Example: Example:

Associative Property Associative Property © Carnegie Learning of Addition of Multiplication

602 • Chapter 8 Algebraic Expressions

462340_C1_CH08_pp559-632.indd 602 27/08/13 4:12 PM multiplication over addition multiplication over subtraction For any number For any number a, b, and c a, b, and c

Example: Example:

distributive properties

For any number For any number a, b, and c a, b, and c

Example: Example:

division over addition division over subtraction © Carnegie Learning

Be prepared to share your solutions and methods.

8.4 Multiple Representations of ChapterEquivalent 8 Expressions Summary • 603

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604 • Chapter 8 Algebraic Expressions

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© Carnegie Learning U population intheUnited States? they useful? What are some other categories thatare used to describe the rural, urban, andsuburban. What does each of thecategories mean andwhy are useful. For example, thepopulationof theUnited States issometimes dividedinto    In thislesson, you will: Learning Goals sing categories to classify people, places, andother objects isoften very

Combine like terms inalgebraic expressions. Use modelsto combine like terms inalgebraic expressions. Develop modelsfor algebraic terms. Combining Like Terms Unlike and Like 8.5 Combining Like Terms •

605 27/08/13 4:12PM Problem 1 Like and Unlike

You will use the algebra tiles shown to model algebraic expressions.

x y 1 1 x 1 y 1 1 We have used some of x y these tiles before.

y y 2 x x 2

1. What are the dimensions of each of the tiles? a.

e. © Carnegie Learning

606 • Chapter 8 Algebraic Expressions

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2. a. Then sketchyourmodel. Represent eachalgebraicexpression usingalgebratilesandtheoperationssymbols. b.

3 2 x 1 2 1

2 x

y 2 2

1 1 3 terms and 2 operations operations 2 and terms 3 so I will have 3 groups of of groups 3 have will I so The first expression has has expression first The algebra tiles separated separated tiles algebra by 2 addition signs. addition 2 by 8.5

Combining Like Terms

•

607 27/08/13 4:12PM c. 3x 1 x 1 2y

d. 5x 1 1 1 y2 1 3

e. 2x 1 3 1 3x 1 x2

3. Which of the expressions in parts (a)–(e) have different terms with identical tiles? © Carnegie Learning

608 • Chapter 8 Algebraic Expressions

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4. a. Rearrange thetilesinQuestion2,part(c)sothatthere are onlytwoterms. c. b. a. fewest possible. Rearrange thetilesinQuestion2part(b)toreduce thenumberoftermsto d. c. b.

Sketch yournewmodel.Includetheoperationsymbol. Could yourearrange thetilessothere wasonlyone term?Explainyourreasoning. Sketch yournewmodel.Includetheoperationsymbol. What are thetermsthatcanbecombinedcalled? Could yourearrange thetilessothere wasonlyoneterm?Explainyourreasoning. Write thealgebraicexpression represented. Write thealgebraicexpression represented. 8.5

Combining Like Terms

•

609 27/08/13 4:12PM 6. Rearrange the tiles in Question 2 part (d) to reduce the number of terms to the fewest possible. a. Sketch your new model. Include the operation symbol.

b. Write the algebraic expression represented.

c. Could you rearrange the tiles so there are fewer terms? Explain your reasoning.

7. Rearrange the tiles in Question 2 part (e) to reduce the number of terms to the fewest possible. a. Sketch your new model. Include the operation symbol.

b. Write the algebraic expression represented. © Carnegie Learning

c. Could you rearrange the tiles so there are fewer terms? Explain your reasoning.

610 • Chapter 8 Algebraic Expressions

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8. operations symbols. The algebraicexpression 2(3 b. c. a.

Model yournewexpression usingalgebratilesand operationssymbols. Then, writethealgebraicexpression thatisrepresented. Rearrange thetilestoreduce thenumberoftermstofewestpossibleterms. 2(3x Use thedistributiveproperty torewrite theexpression withoutparentheses. 1

2) 2 2 3 5 x 1 x x x 2) 2 3 isrepresented withalgebratilesandthe 2 1 8.5 1 1

Combining Like Terms 1 1 1

•

611 27/08/13 4:12PM 9. Now, you will create your own expression and model. a. Write your own algebraic expression that has 4 terms with 2 pairs of like terms.

b. Model your expression using algebra tiles and operations symbols.

c. Rearrange the tiles to reduce the number of terms to the fewest possible. What is the same about everyone,s expression in the end?

e. Could you rearrange the tiles so there are fewer terms? Explain your reasoning.

612 • Chapter 8 Algebraic Expressions

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1. a. Represent 4 d. c. b.

Sketch themodelofexpression. What property tellsyouthat4 Represent thealgebraicexpression from part(b)using algebratilesandoperators. Write anequivalentalgebraicexpression withoneterm. What abouttheExpressions? x 1 2 x usingalgebratilesandtheoperationssymbols. x

1 2 x

5 (4 1 2) x 8.5 ?

Combining Like Terms

•

613 27/08/13 4:12PM 2. Represent 2x 1 5 1 3x using algebra tiles and the operations symbols. a. Sketch the model of the expression.

b. Write an equivalent algebraic expression by combining like terms.

c. Represent the algebraic expression from part (b) using algebra tiles and operations symbols.

d. What property tells you that 2x 1 5 1 3x 5 2x 1 3x 1 5?

614 • Chapter 8 Algebraic Expressions

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operations symbols. Represent thealgebraicexpression from part(b)using algebratilesand Write anequivalentalgebraicexpression bycombiningliketerms. Sketch themodelofexpression. x 1 2 x 2 2 3 x

2 x 2 usingalgebratilesandoperationssymbols. 8.5

Combining Like Terms help me see what what see me help is like and unlike! and like is The tiles really really tiles The

•

615 27/08/13 4:12PM Problem 3 Without the Tiles

Simplify each algebraic expression by combining like terms. So, to combine like 1. 3x 1 5y 2 3x 1 2y 5 terms I will look for the same variable with the same 2. 4x2 1 4y 1 3x 1 2y2 5 exponent.

3. 7x 1 5 2 6x 1 2 5

4. x2 1 5y 1 4x2 2 3y 5

5. 4x3 2 5y 1 3x2 1 2x2 5

6. x 1 5y 1 6x 2 2y 2 3x 1 2y2 5

Simplify each algebraic expression using the distributive properties first, and then combining like terms.

7. 4(x 1 5y) 2 3x 5 8. 2(2x 1 5y) 1 3(x 1 3y) 5

4x + 6y 9. 3x 1 5(2x 1 7) 5 10. ______ 2 3y 5 2

3______x 2 9y 11. 3(x 1 2y) 1 12. 2(x 1 3y) 1 4(x 1 5y) 2 3x 3 5 5

Be prepared to share your solutions and methods.

616 • Chapter 8 Algebraic Expressions

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© Carnegie Learning P or adirector’s chair. Doyou thinkmusicfans fans orart collect souvenirs too? Another collectible it—and sometimes tryto get theautographs of the director, crew, andactors. ones like you see atthemovie theater. Collectors willoften get acopy andframe One of themost popularthingsmovie fans collect are themovie posters—the hands onsouvenirs from afilm. Film souvenirs are very popularamongmovie fans. Collectors willtryto get their parks to concert halls, souvenirs are astaple of most publicevents andlocations.   In thislesson, you will: Learning Goals eople just love collecting souvenirs. From theaters to zoos, from amusement

Analyze andsolve problems withalgebraic expressions. Represent problem situations withalgebraic expressions. might Analyze andSolve Problems Using Algebraic Expressions to Fun with Expressions DVDs and Songs: be a copy of the screenplay—the actual script of a movie— 8.6 Using Algebraic Expressions to Analyze andSolve Problems

•

617 27/08/13 4:13PM Problem 1 DVDs

Jack, Jenny, John, and Jeannie each collect DVDs. Jack likes western movies, Jenny likes comedies, John likes action movies, and Jeannie likes sports movies.

Jenny says: “I have twice as many DVDs as Jack.”

John says: “I have four more DVDs than Jenny.”

Jeannie says: “I have three times as many as John.”

1. If Jack has 10 DVDs, determine the number of DVDs for each friend. Explain your reasoning. a. Jenny

b. John

c. Jeannie

d. all of the four friends together

2. If Jeannie has 24 DVDs, determine the number of DVDs for each friend. Explain your reasoning. a. John

b. Jenny

d. all of the four friends together

618 • Chapter 8 Algebraic Expressions

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3. 4. 5. a. Write analgebraicexpression torepresent thenumber ofDVDsfor: c. a. represents thenumberofDVDsforeachfriend. Let a. altogether ifJackhas: Use yourexpression from part(e)todeterminethenumberofDVDstheyhave e. d. c. b.

the males. 25 DVDs. 10 DVDs. Simplify theexpression youwrote inpart(d). all fourfriendstogether Jeannie John Jenny x represent thenumberofDVDsthatJackhas.Write analgebraicexpression that

8.6

Using Algebraic Expressions to Analyze andSolve Problems b. d. b.

the females. 101 DVDs. 2 DVDs.

•

619 27/08/13 4:13PM 6. Let y represent the number of DVDs Jeannie has. Write an algebraic expression that represents the number of DVDs for each friend. a. John The number of DVDs that John has is less than Jeannie. So, the expression I write in part (a) has to be b. Jenny less than y.

c. Jack

d. all four friends together

e. Simplify the expression you wrote in part (d).

7. Use your expression from part (e) to determine how many DVDs they have altogether if Jeannie has: a. 72 DVDs. b. 24 DVDs.

8. Write an algebraic expression to represent the number of DVDs for: a. the males. b. the females.

620 • Chapter 8 Algebraic Expressions

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10. 11. 9. a. Write analgebraicexpression torepresent thenumber ofDVDsfor: c. a. represents thenumberofDVDsforeachfriend. Let e. d. c. b. a. altogether ifJennyhas: Use yourexpression from part(e)todeterminethenumberofDVDstheyhave

the males. 50 DVDs. Jack Simplify theexpression youwrote inpart(d). all fourfriendstogether Jeannie John 20 DVDs. z represent thenumberofDVDsJennyhas.Write analgebraicexpression that

8.6

Using Algebraic Expressions to Analyze andSolve Problems b. d. b.

the females. 34 DVDs. 24 DVDs.

•

621 27/08/13 4:13PM 12. Let j represent the number of DVDs John has. Write an algebraic expression that represents the number of DVDs for each friend. a. Jenny

b. Jack

c. Jeannie

d. all four friends together

e. Simplify the expression you wrote in part (d).

13. Use your expression from part (e) to determine the number of DVDs they have altogether if John has: a. 24 DVDs. b. 8 DVDs.

14. Write an algebraic expression to represent the number of DVDs for: a. the males. b. the females.

622 • Chapter 8 Algebraic Expressions

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Marty has3timesasmanysongsonhisMP3Marilynhers. Maryanne has3more thantwice the number of songs on her MP3 as Melvin has on his. Marilyn hashalfasmanysongsonherMP3Marvinhis. Marvin has5more songsonhisMP3thanMelvinhas onhis. a. Write analgebraicexpression torepresent thenumber ofsongsfor: e. a. altogether ifMelvinhas: Use yourexpression from part( f. c. a. expression thatrepresents thenumberofsongsoneachfriend’s MP3player. Let

the males. all fivefriendstogether 15 songs. Simplify theexpression youwrote inpart(e). Maryanne Marvin x represent thenumberofsongsonMelvin’s MP3player. Write analgebraic Songs onanMP3Player

8.6

Using Algebraic Expressions to Analyze andSolve Problems f ) tocalculatethenumberofsongstheyhave b. b. d. b.

the females. 47 songs. Marty Marilyn

•

623 27/08/13 4:13PM 4. Let y represent the number of songs on Marilyn’s MP3 player. Write an algebraic expression that represents the number of songs on each friend’s MP3 player. a. Marvin

b. Melvin

c. Maryanne

d. Marty

e. all five friends together

f. Simplify the expression you wrote in part (e).

5. Use your expression from part (f) to calculate the number of songs they have altogether if Marilyn has: a. 15 songs. b. 20 songs.

Be prepared to share your solutions and methods.

624 • Chapter 8 Algebraic Expressions

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expressions (8.4) equivalent like terms (8.2) simplify (8.2) term (8.1) sequence (8.1) The missingtermsofthesequenceare: 24,30,36. number oflegs. Each anthas6legs.So,eachtimeanisadded, add6tothe Number oflegs:6,12,18,____,____ Number ofants:1,2,3,4,5,6 Write themissingtermsinsequence. Example to determinethenextterms. between adjacentknowntermsandapplythatsamerelationship the sequence,andsoon.Lookatrelationship second termistheobjectornumberin first objectornumberinthesequence, numbers andvariables.Thefirsttermisthe A termisanumber, variable,orproduct of geometric figures, letters,orotherobjects. ordered arrangementofnumbers, involvingan A sequenceisapattern in aSequence Predicting theNext Term

Summary         Properties

Distributive Property of Divisionover Subtraction (8.3) Distributive Property of Divisionover Addition (8.3) Distributive Property of Multiplication over Subtraction (8.3) Distributive Property of Multiplication over Addition (8.3) Associative Property of Multiplication (8.2) Associative Property of Addition (8.2) Commutative Property of Multiplication (8.2) Commutative Property of Addition (8.2) store and process information like like information process and store contains synapses that transfer, transfer, that synapses contains contains hundreds of trillions of of trillions of hundreds contains these little computers which is is which computers little these more than all the computers computers the all than more a computer? Only your brain brain your Only computer? a Did you know your brain brain your know you Did Chapter 8 on Earth! on Summary

•

625 27/08/13 4:13PM Writing an Algebraic Expression to Represent a Sequence or Situation Whenever you perform the same mathematical process over and over, you can write a mathematical phrase, called an algebraic expression, to represent the situation. An algebraic expression is a mathematical phrase involving at least one variable and sometimes numbers and operation symbols.

Examples

The algebraic expression 100 1 3n, where n is the term number, represents the sequence shown. 103, 106, 109, 112,…

The algebraic expression 2.49p represents the cost of p pounds of apples. One pound of apples costs $2.49.

Using the Associative and Commutative Properties of Addition and Multiplication to Simplify Expressions The Commutative Properties of Addition and Multiplication state that the order in which you add or multiply two or more numbers does not affect the sum or the product. The Associative Properties of Addition and Multiplication state that changing the grouping of the terms in an addition or multiplication problem does not change the sum or the product. Use these rules to group addends or factors into expressions that are easy to compute in your head.

Example

7 1 9 1 23 1 11 5 (7 1 23) 1 (9 1 11)

5 30 1 20

626 • Chapter 8 Algebraic Expressions

462340_C1_CH08_pp559-632.indd 626 27/08/13 4:13 PM Using Algebra Tiles to Simplify Algebraic Expressions To simplify an expression is to use the rules of arithmetic and algebra to rewrite the expression as simply as possible. An expression is in simplest form if all like terms have been combined. Like terms in an algebraic expression are two or more terms that have the same variable raised to the same power. Only the numerical coefficients of like terms are different. In an algebraic expression, a numerical coefficient is the number multiplied by a variable.

Example

You can use algebra tiles to represent each term in the algebraic expression. Combine like tiles and describe the result. (3x 1 2) 1 (5y 1 3) 1 ( x 1 2y)

1 1 1 1 1 1 1 1 x x x + y y y y y +x y y = x x x x y y y y y y y 1 1

4x 1 7y 1 5

Modeling the Distributive Property Using Algebra Tiles The Distributive Property of Multiplication can be modeled by placing the tiles that match the first factor along the left side of a grid and the tiles that match the second factor along the top of a grid. The Distributive Property of Division can be modeled by separating a model of an expression into parts with an equal number of each type of tile.

Examples

a. 3(2 x 1 1) 5 6x 1 3 b. 6y 1 12 5 3(2y 1 4)

y 1 1 x x 1 y 1 1 1 x x 1 1 x x 1 y 1 1 1 x x 1 y 1 1

Chapter 8 Summary • 627

462340_C1_CH08_pp559-632.indd 627 27/08/13 4:13 PM Simplifying Algebraic Expressions Using the Distributive Property of Multiplication The Distributive Property of Multiplication over Addition states that for any numbers a, b, and c, a 3 (b 1 c) 5 a 3 b 1 a 3 c. The Distributive Property of Multiplication over Subtraction states that if a, b, and c are real numbers, then a 3 (b 2 c) 5 a 3 b 2 a 3 c.

Examples

9(4x 1 2) 4(2x 2 6) 36x 1 18 8x 2 24

Simplifying Algebraic Expressions using the Distributive Property of Division The Distributive Property of Division over Addition states that if a, b, and c are real ______a 1 b __a __b numbers and c  0, then c 5 c 1 c . The Distributive Property of Division over ______a 2 b __a __b Subtraction states that if a, b, and c are real numbers and c  0, then c 5 c 2 c .

Examples

______2x 1 8 5 ___2x 1 __8 ______3x 2 7 5 ___3x 2 __7 2 2 2 3 3 3

628 • Chapter 8 Algebraic Expressions

462340_C1_CH08_pp559-632.indd 628 27/08/13 4:13 PM Determining If Two Expressions Are Equivalent Using a Table or a Graph You can determine if two expressions are equivalent in one of two ways: (1) by calculating values for each or (2) by graphing each. If the output values of each expression are the same for each input, then the expressions are equivalent. If the graph of each expression is the same, then the expressions are equivalent.

Example

The table shows that the expression ( x 1 12) 1 (4x 2 9) and the expression 5x 1 3 are equivalent.

x ( x 1 12) 1 (4x 2 9) 5x 1 3

0 3 3

1 8 8

2 13 13

The outputs for each expression are the same, so ( x 1 12) 1 (4x 2 9) 5 5x 1 3.

The graph shows that the expression 3( x 1 3) 2 x and the expression 3x 1 3 are not equivalent.

14 3(xϩ3)Ϫx

12 3xϩ3 10

The graphs shown are not the same line, so 3( x 1 3) 2 x  3x 1 3.

Chapter 8 Summary • 629

462340_C1_CH08_pp559-632.indd 629 27/08/13 4:13 PM Writing and Solving an Expression Using the Distributive Property You can use the Distributive Property to solve a problem for any input.

Example

Two friends split the cost of a rental car for three days. Each day the car costs $26 for insurance and fees plus 18¢ per mile driven. If the two friends plan to drive the same number of miles each day, how much will each one pay? Let m represent the number of miles driven each day.

3(26 1 0.18m) 78 1 0.54m ______ 5 ______2 2 78 0.54m 5 ___ 1 ______2 2

5 39 1 0.27m

If each friend drives 250 miles each day, how much will each friend pay? 39 1 0.27(250) 5 106.5, so each friend will pay $106.50.

Combining Like Terms Using Algebra Tiles Algebra tiles can be used to model algebraic expressions. The tiles can then be rearranged so that multiple terms with identical tiles, called like terms, can be combined into one term. After all of the like terms are combined, the original algebraic expression can be rewritten with the fewest possible terms.

Example

Represent the algebraic expression 3x 1 5 1 x 2 1 using algebra tiles and the operation symbols. Sketch your model.

x 11 1 x 2 1

x 1

630 • Chapter 8 Algebraic Expressions

462340_C1_CH08_pp559-632.indd 630 27/08/13 4:13 PM Rearrange the tiles to combine like terms and to reduce the number of terms to the fewest possible terms.

x 1 1

x 1

Write the algebraic expression represented by the model. 4x 1 4

Using Algebraic Expressions to Analyze and Solve Problems Many real-life situations can be represented using algebraic expressions. The algebraic expressions can then be used to answer questions about the situation.

Example

Sophia, Hector, Gavin and Jenna are comparing their video game collection. Sophia has 5 more games than Hector. Gavin has twice as many games as Sophia. Jenna has 12 fewer games than Gavin.

a. Let x represent the number of video games that Hector has. Write an algebraic expression that represents the number of video games that each person has. Hector: x Sophia: x 1 5 Gavin: 2(x 1 5) Jenna: 2(x 1 5) 2 12 All 4 friends together: x 1 x 1 5 1 2(x 1 5) 1 2(x 1 5) 2 12 5 6x 1 13 © Carnegie Learning

Chapter 8 Summary • 631

462340_C1_CH08_pp559-632.indd 631 27/08/13 4:13 PM b. Use your expression to determine the number of video games they have altogether if Hector has: 18 video games 6(18) 1 13 5 121 They have 121 video games. 46 video games 6(46) 1 13 5 289 They have 289 video games. © Carnegie Learning

632 • Chapter 8 Algebraic Expressions

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