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1 6. Epressions and Equations In this chapter ou will learn about: using letters in algebraic epressions and equations simplifing algebraic epressions using multiplication collecting like terms in algebraic epressions substituting numbers into algebraic epressions solving simple linear equations writing and solving linear equations for word problems writing word problems for linear equations 6. Simplifing Algebraic Epressions In this section, ou will simplif algebraic epressions using multiplication, addition, and subtraction. Get Read You will need: colored straws inde cards 274

2 Eercise 6. Warm Up Work out these multiplications: a 3 2 b 5 3 c 2 4 d 4 6 e 5 7 f 6 3 g 4 9 h Work out these multiplications: a b c d e f g h a What do ou notice about our answers to questions 2a d? b What do ou notice about our answers to questions 2e h? c Making Decisions Does it matter in what order ou multipl numbers? 275

3 4 Work out these additions: a b c 4 9 d e 8 + f 8 7 g 6 2 h Work out these additions: a b c d e f g h a What do ou notice about our answers to questions 5a d? b What do ou notice about our answers to questions 5e h? c Making Decisions Does it matter in what order ou add integers? 276

4 Main Eercise 7 Faisal has three straws. He does not know their lengths, so he calls the: length of the red straw length of the blue straw length of the ellow straw z When Faisal joins two red straws together, the total length is 2: 2 = 2 Work out these lengths: a 2 =? b z 2 =? c 3 =? d 4 =? z z? 2?? z Math fact: We alwas write the number before the letter, so 2 not 2. e 5 =?? 277

5 THINK ABOUT IT TALK ABOUT IT 8 a Hind sas z + z + z + z + z is the same as 5z. Is she correct? Draw a picture to help ou. b Fatima sas + 5 is the same as 5. Is she correct? Hint + 5 = c Eman sas that 5 is the same as 5. Is she correct? 9 Cop and complete in our notebook: a 2 a = a 2 =? b 3? = 3 = 3? c 4 =? =? d 6 q =? 6 =?q 0 Work with a partner. a Find matching epressions among these: A 7 b B 7d C d 7 D 7b E c 7 F b + 7 G 7c H b 7 I 7 d J 7 c b Which card does not match an of the others? Draw a picture to help ou

6 Eample Mansour joins two straws of length a to make a longer straw of length 2a: 2 a = 2a. He makes 3 of these longer straws: a 2a Then he joins them all together to make an even longer straw: a The length of this straw is 6a: 3 2a = 6a. In our notebooks, work out these lengths. Use colored straws to help ou. a 4 2a =? b 2 3a =? c 5 3 =? d 2 3 =? e 5 2 =? a a a a 2a 2a 6a a a a 2a a a a a a 279

7 TALK ABOUT IT 2 In our notebook, simplif: a 7 2m b 5 4q c 3 6p d 4 3 e 7t 6 f 5z 9 3 a Write three different multiplications that give the answer 24. b Compare our answers with two other classmates. Did ou write the same multiplications? Are ou all correct? 4 In our notebook, cop and complete these multiplication blocks. To find the number in each block, multipl the numbers in the two blocks below. a b? ??? e 2 6 Word fact: Simplif means to write in a shorter or easier wa. Hint Start with = 24. Choose two numbers to put in the boes. Then decide where to put the. 280

8 c d e Eample Latifa has red straws of length and blue straws of length. She arranges two red straws and two blue straws as the sides of a rectangle: What is the area of Latifa s rectangle???? 7 3 q??? 3 4 z??? d 2 5 Area =?? 28

9 Answer The area of a rectangle is length width. This rectangle has length = and width =, so Area =. In shorter form this is written as. 5 Here are five straws of different colors and different lengths: The sides of these rectangles are formed b the straws. What is the area of each rectangle? a b a Math fact: We do not need to write a multiplication sign between a number and a letter or between two letters. For eample, 2 = 2 = When there are two or more different letters in a multiplication, we write them in alphabetical order. b z 282

10 c d Eample Latifa makes a rectangle with red straws of length and blue straws of length. She makes another rectangle of the same size on top: 283

11 Latifa sas that this bigger rectangle has length and width 2, so its total area = length width = 2. But the bigger rectangle is also 2 times the size of the smaller rectangles. Each smaller rectangle has area, so the bigger rectangle has total area = 2, or 2 in short. Now Latifa arranges her small rectangles side b side: This longer rectangle has length 2 and width, so its total area = length width = 2. It is also two times the size of the smaller rectangles, so its total area = 2 also. Latifa finds that: 2 = 2 = 2 All are the same as 2. 6 In our notebook, simplif: a a 6b b 8b a c q 5p d p 3q e 4m n f n 7m 284

12 Eample Latifa continues to make bigger rectangles b joining together small rectangles of length and width. She makes a big rectangle si times the size of the small rectangles: Area of big rectangle = length width = 3 2. She also knows that: Area of big rectangle = 6 area of small rectangle = 6. So 3 2 =

13 Net, Latifa rearranges the straws to make another big rectangle si times the size of the small rectangles: This time, Area of big rectangle = length width = 2 3. She also knows that: Area of big rectangle = 6 area of small rectangle = 6. So 2 3 = 6. Latifa has shown that: 3 2 = 2 3 = 6 all are the same as 6. 7 In our notebook, simplif: a 2a 6b b 3q 4p c 4m 5n d 7 3 Math fact: When there are two or more numbers in a multiplication, multipl them all together and write this new number in front of the letters. It does not matter in what order ou multipl the numbers. 286

14 e 2n 8p f 4 4b Eample Jumaa needs to simplif 5 2. Answer First he makes 5 using red straws of length : Then he takes awa 2 (two straws): He can see there are three straws left: So 5 2 = Take awa 2 287

15 TALK ABOUT IT 8 In our notebook, simplif: a 5 b 5b 3b c 6z 3z d 7 2 e 3a 2a 9 In our notebook, simplif: a 5a + 2a 3a Hint First make 5a. Then add 2 more a. Then take awa 3a. How man a are ou left with? b 3z + 6z 4z c 2q + q 2q d 2n + 8n 3n e 3b + 2b 4b 20 Work with a partner. Take four inde cards. Write these four epressions on them: Math fact: a is written as just a. We don t need to write the. Word fact: An algebraic epression is made up of numbers and letters but has no = sign

16 THINK ABOUT IT a Choose two cards. Ask our partner to add the epressions on the two cards. Check their answer. b Now have our partner choose two cards (not both the same as the ones ou chose). Add the epressions on the two cards. Ask our partner to check our answer. c Repeat parts a and b. d What is the biggest total ou can get b adding the epressions on two cards? 2 In our notebook, simplif: a Hint You have 3 red straws and 2 blue straws: You add another red straw: What is the simplest wa to describe the straws ou have now? b

17 THINK ABOUT IT c 5b + 3 2b Hint d a + 6 7a e f 8n + 5m n 3m 22 Rashid sas that 5 + simplifies to 5. Is he correct? 23 In our notebook, simplif: a Hint Think of as the area of a rectangle with length and width : 4 is the area of a rectangle formed b joining four of these rectangles. 2 is the area of a rectangle formed b joining two of these rectangles. What does mean? 5b b b b b 2b b + Word fact: In Q2a, 3 and are like terms, but 3 and 2 are not like terms. We can simplif b putting together like terms. This is called collecting like terms. We cannot put together unlike terms. Area = + 290

18 It is the area of a rectangle formed b joining the 4 rectangle and the 2 rectangle: If ou then take awa an area of 3, what are ou left with? b c 2ab + 4ab 5ab Hint Think of ab as the area of a rectangle with length a and width b. d 5pq 3pq + 0pq Take awa 3 29

19 6.2 Substituting Numbers into Epressions In this section ou will substitute numbers into algebraic epressions. Get Read You will need: a number line fraction tiles Eercise 6.2 Warm Up What is the missing number? Cop and complete in our notebook: a = 3 +? b? + 8 = c 3 +? = + 4 d 8 3 = +? e 2? = fraction circles

20 f = 20? g = 8? h 7 5 =? 2 Work out in our notebook: a b 5 3 c d 5 3 e f 5 3 g h 5 3 Hint To find 5 + 3, start at 5 on the number line, then move right three units. To find 5 3, start at 5 on the number line, then move left three units. Hint 5 3 is the difference (or distance) between 5 and 3 on the number line. Math fact: Integers are whole numbers. The can be positive or negative numbers, but the are alwas whole numbers. Integers can be shown on a number line:

21 THINK ABOUT IT 3 Amna sas that = 2. Hessa sas the answer is 8. Mona sas the answer is 2. Who is correct? 4 a Hind and Latifa go to the Dubai indoor ski slope. Hind has a thermometer app on her cellphone. It shows the outside temperature as 32 C. When Hind puts the phone on the snow, the app shows 6 C. What is the difference between these two temperatures? b Organizing Information When Hind and Latifa come out, it is evening and the temperature is 8 C. Draw a simple diagram to show the temperature changes felt b Hind and Latifa. 5 Work these out in our notebook. Simplif our answers if possible. a b c C Hint You can use fraction tiles or fraction circles to help ou. 294

22 THINK ABOUT IT d e f Salem sas: Before ou can add fractions the must have the same denominator. Is he correct? 7 Work these out in our notebook. Simplif our answers if possible. a b c d e f g Hint = =

23 8 Work these out in our notebook. Use methods such as tid numbers or partitioning. Then check our answers with a calculator. a b c d e f Main Eercise 9 Jumaa has a collection of plastic bricks. He has red bricks labeled that are 3 cm long, and white bricks that are cm long. 3 cm cm When Jumaa joins one cm brick to one brick, what will the total length be? 3 cm cm + = 3 + = 4 The total length is 4 cm. + = 3 cm? cm 296

24 Cop and complete in our notebook. a + 2 =? + 2 =? b + 3 =? + 3 =? c + 4 =? + 4 =? d =? =? Hint e 2 =? 2 =? 0 Jumaa also has a ellow brick labeled that is 6 cm long and a green brick labeled z that is 8 cm long. Cop and complete in our notebook: a + 3 =? + 3 =? b + 5 =? + 5 =? c z + 4 =? + 4 =? d z + 6 =? + 6 =? 3 cm? cm 6 cm cm z 8 cm Hint Look at the bricks. What is the length of? What is the length of z? Draw a diagram to help ou. 297

25 2 Jumaa puts brick net to brick. What is the total length? + = = 9 The total length is 9 cm. Cop and complete in our notebook: a + z =? +? =? b + z =? +? =? c + + z =? +? +? =? 3 cm 3 cm 3 cm 6 cm 6 cm z 8 cm z 8 cm 6 cm cm Jumaa now puts together a brick, a brick, and two cm bricks. What is the total length? 298

26 TALK ABOUT IT = = The total length is cm. Cop and complete in our notebook: a =? +? + 3 =? b + z + 4 =? +? + 4 =? c + + z + 3 =? +? +? + 3 =? d =? + 4 +? =? e z = 5 +? +? +? =? 3 Saif sas that is the same as Is he correct? 4 Cop and complete in our notebook: a =?? =? Hint 3 cm? 6 cm cm cm What is the difference between the length of and the length of? 6 cm 3 cm b z =?? =? 299

27 c z =?? =? d + z =? +?? =? e + z 2 =? +? 2 =? f + 4 z =? + 4? =? 5 Jumaa puts three bricks together: He sas that the total length of the three bricks is 8 cm. He writes: + + = = 8. Jumaa can also show his working out like this: + + = 3 = 3 = 3 6 = 8. Work out these lengths for Jumaa s bricks: a + 2 b 2 + z c 2z + 6 cm 3 cm 6 cm 6 cm 8 cm z 8 cm Hint Look at Q9 parts d and e. 6 cm 300

28 d e 2z f 3 2 Hint What does 2 mean? What does 2 mean? What does it mean to add them? Hint 2 6 Cop and complete in our notebook: a 2z = 2?? =? 6 =? b 5 z = 5?? =? 8 =? c 4 2 = 4? 2? =?? =? d 3z 7 = 3? 7? =?? =? + 3 cm 3 cm 6 cm 6 cm?

29 7 Ibrahim is in the Burj Khalifa. He rides five floors up in the elevator. He calls this trip. Then he rides ten floors down in the elevator. He calls this trip. Ibrahim sas that the number of floors from where he started is + = = 5. So he finished five floors below where he started. Here are some other elevator rides Ibrahim could take: a = 7 b = 3 c = 4 d = 5 Work out: a a + c b a + b c c + d d b + c e a b f b d 0 down 5 up Hint Use a vertical number line to help ou:

30 g Suggesting Conclusions Ibrahim and his friend Mubarak take the same three elevator rides but in different order. Ibrahim does a then b then c. Mubarak does b then a then c. Will the end on the same floor? 8 Khadija and Moza are clearing up after a part. There are five bottles of juice that have not been finished. Khadija and Moza wonder what the will get if the add together the juice in some of the bottles. b + a =? +? b 8 b + a = = 7 8 Cop and complete in our notebook: a + =? +? =? b b + =? +? =? c + b =? +? =? d z + a =? +? =? e a =?? =? z 7 8 b 8 a a =?

31 f Making Decisions Which of the answers would be too much for one bottle? 9 Hind and Jameela are making girls dresses from pieces of cloth. The have four kinds of cloth, called p, q, r, and s: Each piece of p is 0.48 m long. Each piece of q is.37 m long. Each piece of r is 2.5 m long. Each piece of s is 0.73 m long. One tpe of garment needs 2.99 m of cloth. Hind sas: p + r = = 2.99 So the can use piece of p and piece of r to make this garment. Jameela sas: r + s = = 3.24 So the can also use one piece of r and one piece of s to make this garment, but there will be = 0.25 m of cloth left over. Hind and Jameela find more was to make this garment. For each wa, work out the total length of cloth used. Also work out how much cloth will be left over after making the garment. a 7p b 5s c p + 2q d 4s + p e 2p + 3s f Generating Solutions Another tpe of garment needs.69 m of cloth. 304

32 Find four different was of using p, q, r and s to make this garment. For each wa, work out how much cloth will be left over. 20 A tai driver charges 6 AED for callout and then 2 AED for each kilometer traveled. He works out that the cost of a journe is: Cost = 2d + 6 where d is the number of kilometers traveled. Find the cost of these journes: a 5 kilometers b 0 kilometers c 2 kilometers d 4 kilometers 2 Faisal has just started working in a bank. Part of his job is to bu lunch sandwiches for the bank staff. The costs of the sandwiches are: Tuna Grilled chicken Beef burger 30 AED 25 AED 40 AED The number of people wanting sandwiches is different each da. Faisal writes that the total cost of buing the sandwiches is: Cost = 30T + 25G + 40B Where: T is the number wanting tuna G is the number wanting grilled chicken B is the number wanting beef burger tuna sandwhich 30 AED chicken sandwhich 25 AED beef burger 40 AED 305

33 Work out the total cost of buing sandwiches for each da. a Sunda: 0 tuna, 8 grilled chicken, 5 beef burgers. b Monda: 8 tuna, 2 grilled chicken, 7 beef burgers. c Tuesda: 5 tuna, 2 grilled chicken, 0 beef burgers. d Wednesda: 0 tuna, 8 grilled chicken, 5 beef burgers. 6.3 Solving Equations In this section ou will solve simple equations. Eercise 6.3 Warm Up The scales are balanced. If = 2 Find the value of. 2 The scales are balanced. If = 2 Find the value of and. 306

34 3 The scales are balanced. If = 4 Find the value of and. 4 In our notebook draw strip diagrams for these number sentences. Find the value of? in each number sentence. a 0 +? = 7 b? 5 = 6 c 4? = 28 d? 3 = 7 Hint This is a strip diagram for 0 +? = 7: 5 Find the value of? in these number sentences. Show our working in our notebook. a 2? +3 = b =? 0 What number should ou add to 0 to make 7? 7? c d? 6 = 2 2? =

35 Main Eercise In algebra, we write letters for the unknowns, instead of shapes like,, or Δ. We use letters instead of shapes because: Letters like or a are easier to write and to sa than shapes like. If there are several unknowns, we can use a different letter for each one. = is called an equal sign. An equation must have an equal sign. It means that the value of the left side equals the value of the right side. 6 Cop these into our notebook. Net to each one write equation or not equation. a a + 3 = 6 b 5 b + 2 c 3d + 4 = 0 d t 5 e = ab 308

36 Eample Solve 2a = 3 Answer Taking awa from 2a gives 3, so 2a is 4. This means that a must be half of 4. So a = 7. 7 Eplaining Solve these equations b inspection. Eplain how ou got our answers. a n + 6 = 3 b w = 22 c 63 = 9p d 5 = 2n + 3 e 7r = 20 Word fact: The method used in this eample is called solving b inspection. Hint To solve an equation b inspection, look carefull at each part of the equation and think about what it does. Hint Remember 9p means p multiplied b 9. Word fact: To solve an equation means to find numbers for the letters that make the two sides of the = sign balance. This means that when ou substitute these numbers into the equation, the two sides are equal. 309

37 Eample What number can we substitute for to make this equation balance? + 3 = 7 Answer Step I guess 3. Step 2 Word fact: A number that makes an equation balance is called a solution. I substitute 3 into the equation = 6 (smaller than 7). I need a bigger number. Step 3 I guess 4. Step 4 I substitute 4 into the equation = 7 7 is eactl correct. Step 5 So the solution is = 4. This method is called guess-and-check. Word fact: In guess-and-check we first guess a solution. Then we check if the solution works b substituting it into the equation. If the equation does not balance, choose either a bigger or a smaller number and check again. 8 Eplaining Find a value for that balances each equation. Write our first guess. Did ou have to change it? If es, wh? a + 5 = 2 b 4 + = c 5 = 3 30

38 d 5 = + 2 e 7 = 5 f 7 = 3 Eample Salama and Wafaa want to solve this equation: z 3 = 6 Answer Salama guesses z = 2. But Wafaa sas: 2 is too big because 2 3 is 2 divided b 3 which equals 7. Then Salama suggests a smaller number: z = 8. Wafaa sas: 8 3 = 6 because 8 divided b 3 is 6. So the solution is z = 8. This is also a form of guess-and-check. 9 Eplaining Find a value for that balances each equation. Write our first guess. Did ou have to change it? If es, wh? a 6 + h = 3 b 7 = 84 c z 9 = 6 d = 4 e 4 3 = 5 Hint z means z divided b

39 Eample We can use strip diagrams to find the value of an unknown. Solve 3 5 = 3. Answer Step Step 2 Draw a strip diagram. Look at what the strip diagram tells ou. The strip diagram tells us that 3 = 8 So = 6. 0 In our notebook draw a strip diagram for each equation. Then find a solution. a 4 = 20 b = 5 c = 7 d 4 = 5 e 4 = 32 Eample 5 a Solve + 4 = 0 Answer 3 Step Write the equation. 32

40 Step 2 Step 3 b Solve 4 = 28 Answer Step + 4 = 0 Subtract 4 from both sides: = 0 4 Answer the question. = 6 Write the equation. 4 = 28 Step 2 Divide both sides b 4: 4 4 = 28 4 Step 3 c Solve 5 = 6 Answer Step Answer the question. = 7 Write the equation. 5 = 6 Step 2 Multipl both sides b 5: 5 5 = 6 5 Step 3 Answer the question. = 30 Solve each equation b changing both sides in the same wa. Write our working in our notebook. a + 5 = 2 b 6 = 9 c 4 = 2 d 5 = 3 Math fact: If ou do the same thing to both sides of an equation, it is still balanced. 33

41 e 3 = 8 f 5 = 30 g 4 = 6 h 3 = 2 Eample Solve = 4. Answer Step Subtract 2 from both sides: = = 2 Step 2 Divide both sides b 4: 4 4 = 2 4 = 3 2 Solve each equation using two steps. Then check our answer. Write our working in our notebook. a = 0 b 5 2 = 8 c = d 4 3 = 2 e = 23 f 6 2 = 4 Math fact: You can check if our solution is correct b substituting it into the equation. See if the two sides balance. Math fact: To solve some equations, ou need two steps. g = 8 h 9 =

42 THINK ABOUT IT 3 How can ou decide which method is best for solving an equation? Eample Solve = 4 in two different was. Answer Strip diagram B inspection 4 is 2 more than 4 so 4 equals 2 so equals 3. 4 In our notebook, solve these equations using strip diagrams and b inspection. a = 3 b 4 5 = 3 c = 0 d 2 3 = 2 e 2q + 7 = 9 f z =

43 THINK ABOUT ITNot Eample Solve d 4 Answer 3 = in two different was. I guess d = = 0. This is too small. Guess-and-check I know the solution must be a multiple of 4. So m net (bigger) guess is d = =. So, d = 6. d 4 3 =. Balancing Add 3 to both sides: d = + 3. d 4 = 4. Multipl both sides b 4: d 4 4 = 4 4. d = 6. 5 In our notebook, solve these equations using guess-and-check and b balancing. a = 3 b 4 5 = 3 c = 0 d 2 3 = 2 e 2q + 7 = 9 f z 5 3 = for sale or 6 Khalfan sas that each method of solving equations gives the same answer. Is he correct? 36

44 6.4 Using Equations to Represent Word Problems In this section ou will write and solve equations that describe word problems. Eercise 6.4 Warm Up a What is 3 more than 5? b What is 6 more than? c What is 9 less than 8? d What is 5 less than 2? 2 a Double means.? b What is double 7? c What is double 50? d Half means.? e What is half of 6? f What is half of 54? 3 Ahmed has three more candies than his brother. His brother has seven candies. How man sweets does Ahmed have? Draw a diagram or picture to help ou. 37

45 4 Amna has a bag of dates. She eats five and has 3 left. a In our notebook write a number sentence to describe the problem. Use d as the unknown. b Solve the number sentence. c How man dates were in the bag at the beginning? 5 Abdulla has a bo of pencils. He bus si more. He now has 5 pencils. a In our notebook write a number sentence to describe the problem. Use p as the unknown. b Solve the number sentence. c How man pencils did Abdulla have at the beginning? 6 Mira bus four boes of eggs. Each bo has the same number of eggs. She finds she has 32 eggs in total. a In our notebook write a number sentence to describe the problem. Use as the unknown. b Solve the number sentence. c How man eggs are in each bo? 7 Humaid divides a bag of grapes between himself and five friends. Each bo gets seven grapes. a In our notebook write a number sentence to describe the problem. Use q as the unknown. 38

46 b Solve the number sentence. c How man grapes were in the bag at the beginning? 8 Khawla withdraws 50 AED from her bank account. She has 240 AED left in her account. a You want to find out how much mone Khawla had in her account before the withdrawal. Which of these number sentences could ou use? A 50 + a = 240 B 50 a = 240 C a 50 = 240 D 240 a = 50 b Solve the number sentence ou chose in part a. 9 Maram bus three pencils and a ruler. Each pencil costs 2 AED. Maram spends AED altogether. a You want to find out how much the ruler cost. Which of these number sentences could ou use? A = B = C = D = b Solve the number sentence ou chose in part a. 39

47 0 Eplaining Noura bought three packets of buttons. Each packet has 0 buttons. Noura lost some buttons and now has 26 left. Afra sas to find out how man buttons Noura lost, ou solve this number sentence: 3 b 0 = 26 So Noura lost 2 buttons. Is Afra correct? Wh or wh not? Eample Salama has a bag of candies. She eats 4 candies and has 0 left. How man candies were in the bag at the beginning? Answer Step Step 2 Step 3 Step 4 Choose a letter for the unknown. Use for the number of candies in the bag at the beginning. Write an equation to represent the word problem. 4 = 0 Solve the equation. We can do this b balancing: = = 4 Answer the question. Salama had 4 candies in the bag at the beginning. 320

48 Main Eercise Khalfan collects model cars. Mansour gives him four more cars. Khalfan now has 2 model cars. How man model cars did Khalfan have at the start? a Write an equation to represent the word problem. b Solve the equation. c Write a word sentence to answer the question. 2 Huda bought three apples and an orange. Apples cost 4 AED each. The total cost was 6 AED. How much did the orange cost? a Write an equation to represent the word problem. b Solve the equation. c Write a word sentence to answer the question. 3 Jasem has five bags of olives. Each bag has 2 olives. Jasem eats some olives and has 52 olives left. How man olives did Jasem eat? a Write an equation to represent the word problem. b Solve the equation. c Write a word sentence to answer the question. 4 At Eid al Fitr, Rashid s uncle shares some mone equall between Rashid and his four brothers. Each bo got 60 AED. How much mone did Rashid s uncle share between the bos? a Write an equation to represent the word problem. 32

49 b Solve the equation. c Write a word sentence to answer the question. 5 When ou solve an equation that describes a word problem, wh is it important to write a word sentence with our solution? Eample Abdullah bus four pens and a notebook. The notebook costs 32 AED He spends 60 AED in total. How much does each pen cost? Answer Step Step 2 Step 3 Step 4 Choose a letter for the unknown. Use p for the cost of each pen. Write an equation to represent the word problem. 4p + 32 = 60 Solve the equation. We can do this in two steps: Subtract 32 from both sides: 4p = p = 28 Then divide both sides b 4: 4p 4 = 28 4 p = 7 Answer the question. Each pen costs 7 AED. 6 Nasser hires a tai to take him to town. The tai charges 4 AED for callout and 2 AED for each kilometer traveled. Nasser pas 24 AED. How man kilometers did he travel? 322

50 a Which letter will ou use for the distance traveled? b Write an equation to represent the problem. c Solve the equation. d Write a word sentence to answer the question. 7 Shaikha s mother shares a bowl of nuts equall between Shaikha and her three sisters. Shaikha gives si of the nuts to her pet bird. She finds she has nine left for herself. How man nuts did Shaikha s mother share out? a Which letter will ou use for the number of nuts Shaikha s mother shared out? b Write an equation to represent the problem. c Solve the equation. d Write a word sentence to answer the question. 8 Sultan fills five large water containers and one small container. The small container holds two liters. The total amount of water is 27 liters. a You want to find out how much each large container holds. Which of these equations could ou use? A = 27 B = 27 C 2 5 = 27 D = 27 b Solve the equation ou chose in part a. c Write a word sentence to answer the question.. 323

51 9 Mona s friend Laila divided a bag of candies equall between Mona and three other friends. Mona alread had two candies and now has 6. How man candies did Laila share out? Miriam writes this equation to answer the question: 4 6 c = 2 Khadija writes this equation to answer the question: c = 6 Which equation will give the correct answer? Wh? 20 At the souk, Fatima and two of her friends each bu the same number of bracelets for themselves. Fatima also bus two etra bracelets for her mother. In total Fatima and her friends bought 4 bracelets. How man bracelets did each girl bu for herself? a Write an equation to represent the problem. b Solve the equation in two different was. Choose from these methods: inspection guess-and-check strip diagram balancing c Write a word sentence to answer the question. 2 As a reward for good work, Reem s math teacher divides a bo of chocolates equall between the eight students in the class. Reem eats three of her chocolates and has four left. How man chocolates were in the bo? a Write an equation to represent the problem. 324

52 b Solve the equation in two different was. Choose from these methods: inspection guess-and-check strip diagram balancing c Write a word sentence to answer the question. 6.5 Creating Word Problems from Equations In this section ou will create word problems from equations. Eercise 6.5 Warm Up On the left is a list of number sentences with as the unknown. On the right is a list of word problems. Read both lists. Number sentence A 4 + = 7 B 4 = 24 Word problem Jameela shares a bag of candies between four friends. Each friend receives five candies. How man sweets were in the bag? 2 Faisal catches 6 fish. He gives some of the fish to his uncle. He has nine fish left. How man did he give to his uncle? 325

53 C 4 = 5 D 6 = 9 E 5 = 0 F 3 = 6 G 8 = 32 3 Hamda has some mone in her purse. She spends 5 AED and has 0 AED left. How much mone did she have at the beginning? 4 Jasem has four model aircraft. His uncle gives him more model aircraft. He now has seven. How man was he given? 5 Mansour bus gifts of equal value for his three brothers. Each gift cost 6 AED. How much did Mansour spend in total? 6 Latifa has eight friends who pla in her football team. She bus each of them an ice cream. She spends 32 AED. What is the cost of an ice cream? 7 Salma bus four boes of eggs. She has 24 eggs altogether. How man eggs are in each bo? Match each word problem to a number sentence b writing the number for the word problem net to the letter for the appropriate number sentence. Make a table like this. One of the matches has been done for ou. Number sentence Word problem A? B 7 C? D? E? F? G? 326

54 2 In our notebook, write a word problem for each number sentence. a 5 + a = 7 b 3 a = 5 c a 5 = 4 d 9 a = 6 e a 5 = 8 3 In our notebook, write a word problem for each number sentence: a 3 b + 4 = 3 b 5 h 2 = 3 c = 6 d 8 2 z = 2 Eample This is an equation that uses a letter for the unknown: + 6 = 3 It can be used to represent man different word problems. For eample: Salma has a bo of pens. She puts si more pens in the bo. There are now 3 pens in the bo. How man pens were in the bo at the beginning? 327

55 THINK ABOUT IT TALK ABOUT IT Main Eercise 4 In the eample, would it affect the solution if ou changed the in the equation to another letter? 5 In our notebook, write a word problem for each equation: a + 8 = 4 b 4 = 28 c 5 + = 9 d 5 = 7 e 3 = 5 f 2 = 28 g Show a partner one of our word problems. Can the write the equation for the problem? This is a good wa to check if our word problem makes sense. Eample This is an equation that can be solved in two steps: 4 2 = 4 It can be used to represent man different word problems. For eample: Hamdan has four packets of cookies. He eats two cookies and has 4 left. How man cookies are in each packet? 328

56 6 In our notebook, write a word problem for each equation: a 5 +3 = 3 b 6 4 = 4 c 2 3 = 6 d 6 2 = 3 e = 5 Eample This is an equation that can be solved in two steps: = 22 A word problem it can represent is: I bu three boes of pencils. I alread had four pencils. I now have 22 pencils. How man pencils were in each bo? Math fact: An equation can describe man different word problems. The same equation can also represent this word problem: I have three equal bunches of flowers. I also have four separate flowers. I have 22 flowers in total. How man flowers are in each bunch? Two or more word problems can be solved with the same equation. 329

57 TALK ABOUT IT 7 In our notebook, write three different word problems for each equation: a 4 3 = 7 b = 20 c 0 2 = 4 d 4 3= e = 7 8 Show our answers for Q7 to a partner. Discuss which letters are better to use for the unknowns in the equations. 330

58 Summar In algebra, we use letters for unknowns. This makes number sentences easier to write. 3 means 3 times, or multiplied b 3. We do not need to write a multiplication sign between a number and a letter. When we write a number and a letter multiplied together, we alwas put the number first. 3b not b3 3 means divided b 3. In an epression like 5z, we write the letters in alphabetical order. Simplifing an epression means: writing ever part of the epression in its shortest form writing fractions in their lowest form collecting like terms Substituting into an epression means putting numbers in place of the letters in the epression. An equation must have an = sign in it. Solving an equation means finding a number for the unknown that will make the two sides of the = sign balance. This number is called a solution. Equations can be solved b inspection, guess-and-check, drawing a strip diagram, or a balancing method, which means doing the same things to both sides of the equation to keep it balanced. To solve word problems, first read the problem carefull. Then write an equation to describe the problem. Then solve the equation. Finall, write our answer in a word sentence. The same equation can be used to describe and solve man different word problems. 33

59 Review Simplif these epressions: a 8 b 4 q 2 p c 4z + 3p z + 5p d 4ab + 2cd + 3ab cd e You are given a = 5, b = 6, and c = 3. Find the value of these epressions: a a + b + c b a + b c c ab + c d a + bc e ab + bc 3 You are given p = 2, q = 5, and r = 6. Find the value of these epressions: a 3p + 5 b 2q 4 c 4p r d qr + 2pq e r 3 332

60 4 Sultan needs to bu eggs to make cakes. Each cake needs three eggs. Sultan also needs si eggs for the famil breakfast. He writes: Number of eggs = 3c + 6 where c is the number of cakes he will make. Work out how man eggs Sultan needs to bu if he makes: a 2 cakes b 3 cakes c 5 cakes 5 A tai driver charges 6.50 AED for callout and.80 AED for each kilometer traveled. He writes this for the cost of a journe: Cost =.8d where d is the number of kilometers traveled. Work out the cost of these journes: a 5 kilometers b 8 kilometers c 0 kilometers d 5 kilometers 6 Moza s factor makes furniture. The wood is supplied in three different lengths:.2 m, 2. m and.8 m. Moza writes: Total length of wood needed =.2a + 2.b +.8c where a is the number of.2 m lengths needed b is the number of 2. m lengths needed c is the number of.8 m lengths needed 333

61 THINK ABOUT IT Work out the total length of wood needed for these different tpes of furniture: a a = 3, b = 2, c = 4 b a = 4, b = 3, c = 5 c a =, b = 5, c = 3 7 What are the differences between epressions and equations? 8 Solve these equations b inspection. Write down our working in our notebook. a + 8 = 2 b 5 = 7 c 6 = 36 d 8 = 4 e = 25 f = 0 9 In our notebook solve these equations using strip diagrams. a 3 = 2 b = 30 c 25 = d 5 = 7 e = 8 334

62 TALK ABOUT IT 0 Solve these equations b changing both sides in the same wa. Write down our working in our notebook. a + 5 = 8 b 5 = 45 c 3 = 5 d = 8 e 6 5 = 25 Solve each of these equations in two different was. In our notebook write down our working for both was. a 6 = 9 b 3 = 8 c = 28 d 3 5 = 7 e = Discuss with a partner how ou decided which methods to use for solving the equations in Q. 3 Jasem needs 250 AED to bu a computer game for his brother s birthda. He has 70 AED in his bank account. The birthda is in five weeks time. How much mone must Jasem save each week? a Write an equation to represent the word problem. b Solve the equation. c Write a word sentence to answer the question. 335

63 TALK ABOUT IT THINK ABOUT IT 4 For each of these equations, write two different word problems that it can describe: a 3 = 5 b = 9 c 5 4 = 2 d 22 3 = 4 5 Show our answers for Q4 to a partner. Talk about different was to solve the equations. 6 For equations that need two steps to solve, which method do ou think is the quickest? I can statements What can ou do? I can write algebraic epressions using letters for the unknowns. I can rewrite algebraic epressions in shorter form. I can decide which are like terms in an algebraic epression. I can simplif algebraic epressions b collecting like terms. I can substitute numbers into an algebraic epression and calculate the value. I can decide if an algebraic epression is an equation or not. I can solve equations b inspection. I can solve equations b guess-and-check. I can solve equations using strip diagrams. 336

64 I can solve equations b balancing. I can use equations to solve word problems. I can write word problems from equations. 337