Algebraic Techniques 1 PDF

360 Chapter 8 Algebraic techniques 1 Chapter 8 8A Algebraic techniques 1 What you will learn...

360 Chapter 8 Algebraic techniques 1 Chapter 8 8A Algebraic techniques 1 What you will learn Introduction to formal algebra 8B Substituting positive numbers into algebraic expressions 8C Equivalent algebraic expressions 8D Like terms 8E Multiplying, dividing and mixed operations 8F Expanding brackets 8G Applying algebra EXTENSION 8H Substitution involving negative numbers and mixed operations 8I Number patterns EXTENSION 8J Spatial patterns EXTENSION 8K Tables and rules EXTENSION 8L The Cartesian plane and graphs EXTENSION ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  number and algebra 361 nSW Syllabus for the australian Curriculum Strand: number and algebra Substrand: alGEBRaiC TECHniQuES Outcome A student generalises number properties to operate with algebraic expressions. (MA4–8NA) Designing robots Algebra provides a way to describe everyday activities using mathematics alone. By allowing letters like x or y to stand for unknown numbers, different concepts and relationships can be described easily. Engineers apply their knowledge of algebra and geometry to design buildings, roads, bridges, robots, cars, satellites, planes, ships and hundreds of other structures and devices that we take for granted in our world today. To design a robot, engineers use algebraic rules to express the relationship between the position of the robot’s ‘elbow’ and the possible positions of a robot’s ‘hand’. Although they cannot think for themselves, electronically programmed robots can perform tasks cheaply, accurately and consistently, without ever getting tired or sick or injured, or the need for sleep or food! Robots can have multiple arms, reach much farther than a human arm and can safely lift heavy, awkward objects. Robots are used extensively in car manufacturing. Using a combination of robots and humans, Holden’s car manufacturing plant in Elizabeth, South Australia fully assembles each car in 76 seconds! Understanding and applying mathematics has made car manufacturing safer and also extremely efficient. ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party 362 Chapter 8 Algebraic techniques 1 pre-test 1 If = 7, write the value of each of the following. a +4 b –2 c 12 – d 3× 2 Write the value of × 4 if: a =2 b =9 c = 10 d = 2.5 3 Write the answer to each of the following computations. a 4 and 9 are added b 3 is multiplied by 7 c 12 is divided by 3 d 10 is halved 4 Write down the following, using numbers and the symbols +, ÷, × and –. a 6 is tripled b 10 is halved c 12 is added to 3 d 9 is subtracted from 10 5 For each of the tables, describe the rule relating the input and output numbers. For example: Output = 2 × input. a Input 1 2 3 5 9 Output 3 6 9 15 27 b Input 1 2 3 4 5 Output 6 7 8 9 10 c Input 1 5 7 10 21 Output 7 11 13 16 27 d Input 3 4 5 6 7 Output 5 7 9 11 13 6 If the value of x is 8, find the value of: a x+3 b x–2 c x×5 d x÷4 7 Find the value of each of the following. a 4×3+8 b 4 × (3 + 8) c 4×3+2×5 d 4 × (3 + 2) × 5 8 Find the value of each of the following. a 50 – (3 × 7 + 9) b 24 ÷ 2 – 6 c 24 ÷ 6 – 2 d 24 ÷ (6 – 2) 9 If = 5, write the value of each of the following. a –4 b ×2–1 c ÷5+2 d × 7 + 10 e × f × ÷ g 3× – 15 h 2 ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  number and algebra 363 8A introduction to formal algebra A pronumeral is a letter that can represent any number. The choice of letter used is not significant mathematically, but can be used as an aide to memory. For instance, h might stand for someone’s height and w might stand for someone’s weight. The table shows the salary Petra earns for various hours of work if she is paid $12 an hour. number of hours Salary earned ($) 1 12 × 1 = 12 2 12 × 2 = 24 3 12 × 3 = 36 n 12 × n = 12n Rather than writing 12 × n, we write 12n because multiplying a pronumeral by a number is common and this notation saves space. We can also write 18 ÷ n as 18. n Using pronumerals, we can work out a total salary for any number of hours of work. let’s start: Pronumeral stories Ahmed has a jar with b biscuits in it that he is taking to a birthday party. He eats 3 biscuits and then shares the rest equally among 8 friends. Each friend receives b − 3 biscuits. This is a short story for 8 the expression b − 3. 8 Try to create another story for b − 3 , and share it with others in the class. 8 Can you construct a story for 2t + 12? What about 4(k + 6)? x + y + 3 is an example of an algebraic expression. Key ideas x and y are pronumerals, which are letters that stand for numbers. In the example x + y + 3, x and y could represent any numbers, so they could be called variables. a a × b is written as ab and a ÷ b is written as. b A term consists of numbers and pronumerals combined with multiplication or division. For example, 5 is a term, x is a term, 9a is a term, abc is a term, 4 xyz is a term. 3 A term that does not contain any pronumerals is called a constant term. All numbers by themselves are constant terms. An (algebraic) expression consists of numbers and pronumerals combined with any mathematical operations. For example, 3x + 2yz is an expression and 8 ÷ (3a – 2b) + 41 is also an expression. Any term is also an expression. A coefficient is the number in front of a pronumeral. For example, the coefficient of y in the expression 8x + 2y + z is 2. If there is no number in front, then the coefficient is 1, since 1z and z are equal. ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party 364   Chapter 8 Algebraic techniques 1 Example 1 The terminology of algebra a List the individual terms in the expression 3a + b + 13c. b State the coefficient of each pronumeral in the expression 3a + b + 13c. c Give an example of an expression with exactly two terms, one of which is a constant term. Solut ion Explanatio n a There are three terms: 3a, b and 13c. Each part of an expression is a term. Terms get added (or subtracted) to make an expression. b The coefficient of a is 3, the coefficient The coefficient is the number in front of a pronumeral. of b is 1 and the coefficient of c is 13. For b the coefficient is 1 because b is the same as 1 × b. c 27a + 19 This expression has two terms, 27a and 19, and 19 is (There are many other expressions.) a constant term because it is a number without any pronumerals. Example 2 Writing expressions from word descriptions Write an expression for each of the following. a 5 more than k b 3 less than m c the sum of a and b d double the value of x e the product of c and d Solut ion Explanatio n a k+5 5 must be added to k to get 5 more than k. b m–3 3 is subtracted from m. c a+b a and b are added to obtain their sum. d 2 × x or just 2x x is multiplied by 2. The multiplication sign is optional. e c × d or just cd c and d are multiplied to obtain their product. Example 3 Expressions involving more than one operation Write an expression for each of the following without using the × or ÷ symbols. a p is halved, then 4 is added b the sum of x and y is taken and then divided by 7 c the sum of x and one-seventh of y d 5 is subtracted from k and the result is tripled ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  number and algebra 365 SoluTion ExplanaTion p p is divided by 2, then 4 is added. a +4 2 x+y b ( x + y) ÷ 7 = x and y are added. This whole expression is divided 7 by 7. By writing the result as a fraction, the brackets are no longer needed. y 1 y c x+ or x + y x is added to one-seventh of y, which is. 7 7 7 d (k – 5) × 3 = 3(k – 5) 5 subtracted from k gives the expression k – 5. Brackets must be used to multiply the whole expression by 3. Exercise 8A R K I NG WO U F Example 1 1 The expression 4x + 3y + 24z + 7 has four terms. C R PS MA Y a List the terms. b What is the constant term? LL HE M AT I C A T c What is the coefficient of x? d Which letter has a coefficient of 24? Example 2 2 Match each of the word descriptions on the left with the correct mathematical expression on the right. a the sum of x and 4 a x-4 x b 4 less than x B 4 c the product of 4 and x C 4-x d one-quarter of x D 4x 4 e the result from subtracting x from 4 E x f 4 divided by x F x+4 R K I NG WO U F 3 For each of the following expressions, state: C R PS MA Y i the number of terms; and LL HE M AT I C A T ii the coefficient of n. a 17n + 24 b 31 – 27a + 15n 4 c 15nw + 21n + 15 d 15a – 32b + xy + 2n 3 d e n + 51 f 5bn – 12 + + 12n 5 ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party 366 Chapter 8 Algebraic techniques 1 8A R K I NG WO U F 4 Write an expression for each of the following without using the × or ÷ symbols. C R PS MA Y a 1 more than x b the sum of k and 5 LL HE M AT I C A T c double the value of u d 4 lots of y e half of p f one-third of q g 12 less than r h the product of n and 9 i t is subtracted from 10 j y is divided by 8 Example 3 5 Write an expression for each of the following without using the × or ÷ symbols. a 5 is added to x, then the result is doubled. b a is tripled, then 4 is added. c k is multiplied by 8, then 3 is subtracted. d 3 is subtracted from k, then the result is multiplied by 8. e The sum of x and y is multiplied by 6. f x is multiplied by 7 and the result is halved. g p is halved and then 2 is added. h The product of x and y is subtracted from 12. 6 Describe each of these expressions in words. a 7x b a+b c (x + 4) × 2 d 5 – 3a R K I NG WO U F 7 Nicholas buys 10 lolly bags from a supermarket. C R PS MA Y LL HE a If there are 7 lollies in each bag, how many lollies does he buy in total? M AT I C A T b If there are n lollies in each bag, how many lollies does he buy in total? Hint: Write an expression involving n. 8 Mikayla is paid $x per hour at her job. Write an expression for each of the following. a How much does Mikayla earn if she works 8 hours? b If Mikayla gets a pay rise of $3 per hour, what is her new hourly wage? c If Mikayla works for 8 hours at the increased hourly rate, how much does she earn? 9 Recall that there are 100 centimetres in 1 metre and 1000 metres in 1 kilometre. Write expressions for each of the following. a How many metres are there in x km? b How many centimetres are there in x metres? c How many centimetres are there in x km? ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  Number and Algebra 367   R K I NG WO U F 10 A group of people go out to a restaurant, and the total amount they must pay is $A. They C R PS MA Y decide to split the bill equally. Write expressions to answer the following questions. LL HE M AT I C A T a If there are 4 people in the group, how much do they each pay? b If there are n people in the group, how much do they each pay? c One of the n people has a voucher that reduces the total bill by $20. How much does each person pay now? a+b 11 There are many different ways of describing the expression in words. One way is ‘The sum of 4 a and b is divided by 4.’ What is another way? R K I NG WO U F 12 If x is a whole number between 10 and 99, classify each of these statements as true or false. C R PS MA Y a x must be smaller than 2 × x. LL HE M AT I C A T b x must be smaller than x + 2. c x – 3 must be greater than 10. d 4 × x must be an even number. e 3 × x must be an odd number. 13 If b is an even number greater than 3, classify each of these statements as true or false. a b + 1 must be even. b b + 2 could be odd. c 5 + b could be greater than 10. d 5b must be greater than b. 14 If c is a number between 10 and 99, sort the following in ascending order (i.e. smallest to largest). 3c, 2c, c – 4, c ÷ 2, 3c + 5, 4c – 2, c + 1, c × c. Enrichment: Many words compressed 15 One advantage of writing expressions in symbols rather than words is that it takes up less space. For instance, ‘twice the value of the sum of x and 5’ uses eight words and can be written as 2(x + 5). Give an example of a worded expression that uses more than 10 words and then write it as a mathematical expression. ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party 368 Chapter 8 Algebraic techniques 1 8B Substituting positive numbers into algebraic expressions Substitution involves replacing pronumerals (like x and y) with numbers and obtaining a single number as a result. For example, we can evaluate 4 + x when x is 11, to get 15. let’s start: Sum to 10 The pronumerals x and y could stand for any number. What numbers could x and y stand for if you know that x + y must equal 10? Try to list as many pairs as possible. If x + y must equal 10, what values could 3x + y equal? Find the largest and smallest values. To evaluate an expression or to substitute values means to replace each pronumeral in an Key ideas expression with a number to obtain a final value. For example, if x = 3 and y = 8, then x + 2y evaluated gives 3 + 2 × 8 =19. A term like 4a means 4 × a. When substituting a number we must include the multiplication sign, since two numbers written as 42 is very different from the product 4 × 2. Replace all the pronumerals with numbers, then evaluate using the normal order of operations seen in Chapter 1: – brackets – multiplication and division from left to right – addition and subtraction from left to right. For example: (4 + 3) × 2 − 20 ÷ 4 + 2 = 7 × 2 − 20 ÷ 4 + 2 = 14 − 5 + 2 = 9+2 = 11 ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  Number and Algebra 369   Example 4 Substituting a pronumeral Given that t = 5, evaluate: 10 a t+7 b 8t c +4−t t Solut ion Explanatio n a t + 7 = 5+ 7 Replace t with 5 and then evaluate the expression, = 12 which now contains no pronumerals. b 8t = 8 × t Insert × where it was previously implied, then =8×5 substitute in 5. If the multiplication sign is not = 40 included, we might get a completely incorrect answer of 85. 10 10 Replace all occurrences of t with 5 before evaluating. c +4−t = +4−5 t 5 Note that the division (10 ÷ 5) is calculated before the = 2+4−5 addition and subtraction. =1 Example 5 Substituting multiple pronumerals Substitute x = 4 and y = 7 to evaluate these expressions. a 5x + y + 8 b 80 – (2xy + y) Solut ion Explanatio n a 5x + y + 8 = 5 × x + y + 8 Insert the implied multiplication sign between 5 and x = 5×4+7+8 before substituting the values for x and y. = 20 + 7 + 8 = 35 b 80 − (2 xy + y ) = 80 − (2 × x × y + y ) Insert the multiplication signs, and remember the order = 80 − (2 × 4 × 7 + 7) in which to evaluate. = 80 − (56 + 7) = 80 − 63 = 17 ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  370 Chapter 8 Algebraic techniques 1 Example 6 Substituting with powers and roots If p = 4 and t = 5, find the value of: a 3p2 b t 2 + p3 c p 2 + 32 SoluTion ExplanaTion a 3 p2 = 3 × p × p Note that 3p2 means 3 × p × p, not (3 × p)2. = 3×4×4 = 48 b t 2 + p 3 = 52 + 4 3 t is replaced with 5, and p is replaced with 4. = 5× 5+ 4 × 4 × 4 Remember that 43 means 4 × 4 × 4. = 25 + 64 = 89 c p 2 + 32 = 4 2 + 32 Recall that the square root of 25 must be 5 because 5 × 5 = 25. = 25 =5 Exercise 8B R K I NG WO U F 1 Use the correct order of operations to evaluate the following. C R PS MA Y a 4+2×5 b 7–3×2 c 3×6–2×4 d (7 – 3) × 2 LL HE M AT I C A T Example 4a 2 What number would you get if you replaced b with 5 in the expression 12 + b? 3 What number is obtained when x = 3 is substituted into the expression 5 × x ? 4 What is the result of evaluating 10 – u if u is 7? 5 Calculate the value of 12 + b if: a b=5 b b=8 c b = 60 d b=0 R K I NG WO U F Example 4b,c 6 If x = 5, evaluate each of the following. Set out your solution in a manner similar to that C R PS MA Y LL HE shown in Example 4. M AT I C A T a x+3 b x×2 c 14 – x d 2x + 4 e 3x + 2 – x f 13 – 2x 20 g 2(x + 2) + x h 30 – (4x + 1) i +3 x 10 x+7 10 − x j ( x + 5) × k l x 4 x ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  Number and Algebra 371   R K I NG m 7x + 3(x – 1) n 40 – 3x – x o x + x( x + 1) WO U F 30 6(3 x − 8) C p + 2 x ( x + 3) q 100 – 4(3 + 4x) r R PS MA Y LL HE x x+2 M AT I C A T Example 5 7 Substitute a = 2 and b = 3 into each of these expressions and evaluate. a 2a + 4 b 3a – 2 c a+b d 3a + b e 5a – 2b f 7ab + b g ab – 4 + b h 2 × (3a + 2b) i 100 – (10a + 10b) 12 6 ab 100 j + k +b l a b 3 a +b 8 Evaluate the expression 5x + 2y when: a x = 3 and y = 6 b x = 4 and y = 1 c x = 7 and y = 3 d x = 0 and y = 4 e x = 2 and y = 0 f x = 10 and y = 10 9 Copy and complete each of these tables. a n 1 2 3 4 5 6 n+4 5 8 b x 1 2 3 4 5 6 12 – x 9 c b 1 2 3 4 5 6 2(b – 1) d q 1 2 3 4 5 6 10q – q Example 6 10 Evaluate each of the following, given that a = 9, b = 3 and c = 5. a a 3c 2 b 5b 2 c a2 – 33 d 2b 2 + – 2c 3 2b 3 e a + 3ab f 2 b +4 2 g 24 + h (2c) – a2 2 6 R K I NG WO U F 11 A number is substituted for b in the expression 7 + b and gives the result 12. What is the C R PS MA Y value of b? LL HE M AT I C A T 12 A number is substituted for x in the expression 3x – 1. If the result is a two-digit number, what value might x have? Try to describe all the possible answers. 13 Copy and complete the table. x 5 9 12 x+6 11 7 4x 20 24 28 ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party 372 Chapter 8 Algebraic techniques 1 8B R K I NG WO U F 14 Assume x and y are two numbers, where xy = 24. C R PS MA Y a What values could x and y equal if they are whole numbers? Try to list as many as possible. LL HE M AT I C A T b What values could x and y equal if they can be decimals, fractions or whole numbers? 15 Dugald substitutes different whole numbers into the expression 5 × (a + a). He notices that the result always ends in the digit 0. Try a few values and explain why this pattern occurs. Enrichment: Missing numbers 16 a Copy and complete the following table, in which x and y are whole numbers. x 5 10 7 y 3 4 5 x+y 9 14 7 x–y 2 3 8 xy 40 10 0 b If x and y are two numbers where x + y and x × y are equal, what values might x and y have? Try to find at least three (they do not have to be whole numbers). ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  number and algebra 373 8C Equivalent algebraic expressions In algebra, as when using words, there are often many ways to express the same thing. For example, we can write ‘the sum of x and 4’ as x + 4 or 4 + x, or even x + 1 + 1 + 1 + 1. No matter what number x is, x + 4 and 4 + x will always be equal. We say that the expressions x + 4 and 4 + x are equivalent because of this. By substituting different numbers for the pronumerals it is possible to see whether two expressions are equivalent. Consider the four expressions in this table. 3a + 5 2a + 6 7a + 5 – 4a a+a+6 a=0 5 6 5 6 a=1 8 8 8 8 a=2 11 10 11 10 a=3 14 12 14 12 a=4 17 14 17 14 From this table it becomes apparent that 3a + 5 and 7a + 5 – 4a are equivalent, and that 2a + 6 and a + a + 6 are equivalent. let’s start: Equivalent expressions Consider the expression 2a + 4. Write as many different expressions as possible that are equivalent to 2a + 4. How many equivalent expressions are there? Try to give a logical explanation for why 2a + 4 is equivalent to 4 + a × 2. Two expressions are called equivalent when they are equal, regardless of what numbers are Key ideas substituted for the pronumerals. For example, 5x + 2 is equivalent to 2 + 5x and to 1 + 5x + 1 and to x + 4x + 2. This collection of pronumerals and numbers can be arranged into many different equivalent expressions. ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party 374 Chapter 8 Algebraic techniques 1 Example 7 Equivalent expressions Which two of these expressions are equivalent: 3x + 4, 8 – x, 2x + 4 + x ? SoluTion ExplanaTion 3x + 4 and 2x + 4 + x are equivalent. By drawing a table of values, we can see straight away that 3x + 4 and 8 – x are not equivalent, since they differ for x = 2. x =1 x =2 x=3 3x + 4 7 10 13 8–x 7 6 5 2x + 4 + x 7 10 13 3x + 4 and 2x + 4 + x are equal for all values, so they are equivalent. Exercise 8C R K I NG WO U F 1 a Copy the following table into your workbook and complete. C R PS MA Y LL HE M AT I C A T x=0 x=1 x=2 x=3 2x + 2 ( x + 1) × 2 b Fill in the gap: 2x + 2 and (x + 1) × 2 are __________ expressions. 2 a Copy the following table into your workbook and complete. x=0 x=1 x=2 x=3 5x + 3 6x + 3 b Are 5x + 3 and 6x + 3 equivalent expressions? R K I NG WO U F 3 Show that 6x + 5 and 4x + 5 + 2x are equivalent by completing the table. C R PS MA Y LL HE M AT I C A T 6x + 5 4 x + 5 + 2x x=1 x=2 x=3 x=4 ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  Number and Algebra 375   R K I NG Example 7 4 For each of the following, choose a pair of equivalent expressions. WO U F a 4x, 2x + 4, x + 4 + x C R PS MA Y LL HE b 5a, 4a + a, 3 + a M AT I C A T c 2k + 2, 3 + 2k, 2(k + 1) d b + b, 3b, 4b – 2b 5 Match up the equivalent expressions below. a 3x + 2x A 6 – 3x b 4 – 3x + 2 B 2x + 4x + x c 2x + 5 + x C 5x d x+x–5+x D 4–x e 7x E 3x + 5 f 4 – 3x + 2x F 3x – 5 R K I NG WO U F 6 Write two different expressions that are equivalent C R PS MA Y LL  HE to 4x + 2. M AT I C A T 7 The rectangle shown opposite has a perimeter given b b by b + l + b + l. Write an equivalent expression for the perimeter.  8 There are many expressions that are equivalent to 3a + 5b + 2a – b + 4a. Write an equivalent expression with as few terms as possible. R K I NG WO U F 9 The expressions a + b and b + a are equivalent and only contain two terms. How many C R PS MA Y LL HE expressions are equivalent to a + b + c and contain only three terms? Hint: Rearrange the M AT I C A T pronumerals. 10 Prove that no two of these four expressions are equivalent: 4 + x, 4x, x – 4, x ÷ 4. 11 Generalise each of the following patterns in numbers to give two equivalent expressions. The first one has been done for you. a Observation: 3 + 5 = 5 + 3 and 2 + 7 = 7 + 2 and 4 + 11 = 11 + 4. Generalised: The two expressions x + y and y + x are equivalent. b Observation: 2 × 5 = 5 × 2 and 11 × 5 = 5 × 11 and 3 × 12 = 12 × 3. c Observation: 4 × (10 + 3) = 4 × 10 + 4 × 3 and 8 × (100 + 5) = 8 × 100 + 8 × 5. d Observation: 100 – (4 + 6) = 100 – 4 – 6 and 70 – (10 + 5) = 70 – 10 – 5. e Observation: 20 – (4 – 2) = 20 – 4 + 2 and 15 – (10 – 3) = 15 – 10 + 3. f Observation: 100 ÷ 5 ÷ 10 = 100 ÷ (5 × 10) and 30 ÷ 2 ÷ 3 = 30 ÷ (2 × 3). ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party 376 Chapter 8 Algebraic techniques 1 8C R K I NG WO 12 a Show that the expression 4 × (a + 2) is equivalent to 8 + 4a using a table of values for a U F between 1 and 4. C R PS MA Y LL HE b Write an expression using brackets that is equivalent to 10 + 5a. M AT I C A T c Write an expression without brackets that is equivalent to 6 × (4 + a). Enrichment: Thinking about equivalence 13 3a + 5b is an expression containing two terms. List two expressions containing three terms that are equivalent to 3a + 5b. 14 Three expressions are given: expression A, expression B and expression C. a If expressions A and B are equivalent, and expressions B and C are equivalent, does this mean that expressions A and C are equivalent? Try to prove your answer. b If expressions A and B are not equivalent, and expressions B and C are not equivalent, does this mean that expressions A and C are not equivalent? Try to prove your answer. Each shape above is made from three identically-sized tiles of length l and breadth b. Which of the shapes have the same perimeter? ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  number and algebra 377 8D like terms Whenever we have terms with exactly the same pronumerals, they are called ‘like terms’ and can be collected and combined. For example, 3x + 5x can be simplified to 8x. If the two terms do not have exactly the same pronumerals, they must be kept separate; for example, 3x + 5y cannot be simplified – it must be left as it is. let’s start: Simplifying expressions Try to find a simpler expression that is equivalent to 1a + 2b + 3a + 4b + 5a + 6b + … + 19a + 20b What is the longest possible expression that is equivalent to 10a + 20b + 30c? Assume that all coefficients must be whole numbers greater than zero. Compare your expressions to see who has the longest one. Like terms are terms containing exactly the same pronumerals, although not necessarily in Key ideas the same order. – 5ab and 3ab are like terms. – 4a and 7b are not like terms. – 2acb and 4bac are like terms. Like terms can be combined within an expression to create a simpler expression that is equivalent. For example, 5ab + 3ab can be simplified to 8ab. If two terms are not like terms (such as 4x and 5y), they can still be added to get an expression like 4x + 5y, but this expression cannot be simplified further. Example 8 identifying like terms Which of the following pairs are like terms? a 3x and 2x b 3a and 3b c 2ab and 5ba d 4k and k e 2a and 4ab f 7ab and 9aba SoluTion ExplanaTion a 3x and 2x are like terms. The pronumerals are the same. b 3a and 3b are not like terms. The pronumerals are different. c 2ab and 5ba are like terms. The pronumerals are the same, even though they are written in a different order (one a and one b). ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party 378   Chapter 8 Algebraic techniques 1 d 4k and k are like terms. The pronumerals are the same. e 2a and 4ab are not like terms. The pronumerals are not exactly the same (the first term contains only a and the second term has a and b). f 7ab and 9aba are not like terms. The pronumerals are not exactly the same (the first term contains one a and one b, but the second term contains two a terms and one b). Example 9 Simplifying using like terms Simplify the following by collecting like terms. a 7b + 2 + 3b b 12d – 4d + d c 5 + 12a + 4b – 2 – 3a d 13a + 8b + 2a – 5b – 4a e 12uv + 7v – 3vu + 3v Solut ion Explanatio n a 7b + 2 + 3b = 10b + 2 7b and 3b are like terms, so they are combined. They cannot be combined with 2 because it contains no pronumerals. b 12d – 4d + d = 9d All the terms here are like terms. Remember that d means 1d when combining them. c 5 + 12a + 4b - 2 - 3a 12a and 3a are like terms. We subtract 3a because it = 12a - 3a + 4b + 5 - 2 has a minus sign in front of it. We can also combine the = 9a + 4b + 3 5 and the 2 because they are like terms. d 13a + 8b + 2a - 5b - 4a Combine like terms, remembering to subtract any term = 13a + 2a - 4a + 8b - 5b that has a minus sign in front of it. = 11a + 3b e 12uv + 7v - 3vu + 3v Combine like terms. Remember that 12uv and 3vu are = 12uv + 3vu + 7v + 3v like terms (i.e. they have the same pronumerals). = 9uv + 10v ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  number and algebra 379 Exercise 8D R K I NG WO U F 1 For each of the following terms, state all the pronumerals that occur in it. C R PS MA Y LL a 4xy b 3abc c 2k d pq HE M AT I C A T 2 Copy the following sentences into your workbook and fill in the gaps to make the sentences true. More than one answer might be possible. a 3x and 5x are ____________ terms. b 4x and 3y are not ____________ ____________. c 4xy and 4yx are like ____________. d 4a and ____________ are like terms. e x + x + 7 and 2x + 7 are ____________ expressions. f 3x + 2x + 4 can be written in an equivalent way as ____________. R K I NG WO U F Example 8 3 Classify the following pairs as like terms (L) or not like terms (N). C R PS MA Y LL HE a 7a and 4b b 3a and 10a c 18x and 32x M AT I C A T d 4a and 4b e 7 and 10b f x and 4x g 5x and 5 h 12ab and 4ab i 7cd and 12cd j 3abc and 12abc k 3ab and 2ba l 4cd and 3dce 4 Simplify the following by collecting like terms. a a+a b 3x + 2x c 4b + 3b d 12d – 4d e 15u – 3u f 14ab – 2ab g 8ab + 3ab h 4xy – 3xy Example 9 5 Simplify the following by collecting like terms. a 2a + a + 4b + b b 5a + 2a + b + 8b c 3x – 2x + 2y + 4y d 4a + 2 + 3a e 7 + 2b + 5b f 3k – 2 + 3k g 7f + 4 – 2f + 8 h 4a – 4 + 5b + b i 3x + 7x + 3y – 4x + y j 10a + 3 + 4b – 2a k 4 + 10h – 3h l 10x + 4x + 31y – y m 10 + 7y – 3x + 5x + 2y n 11a + 4 – 3a + 9 o 3b + 4b + c + 5b – c p 7ab + 4 + 2ab q 9xy + 2x – 3xy + 3x r 2cd + 5dc – 3d + 2c s 5uv + 12v + 4uv – 5v t 7pq + 2p + 4qp – q u 7ab + 32 – ab + 4 R K I NG WO U F 6 Ravi and Marissa each work for n hours per week. Ravi earns $27 per hour and Marissa earns C R PS MA Y $31 per hour. LL HE M AT I C A T a Write an expression for the amount Ravi earns in one week. b Write an expression for the amount Marissa earns in one week. c Write a simplified expression for the total amount Ravi and Marissa earn in one week. ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party 380 Chapter 8 Algebraic techniques 1 8D R K I NG WO 7 The length of the line segment shown could be expressed as a + a + 3 + a + 1. U F C R PS MA Y LL HE M AT I C A T a a 3 a 1 a Write the length in the simplest form. b What is the length of the segment if a is equal to 5? 8 Let x represent the number of marbles in a standard-sized bag. Xavier bought 4 bags and Cameron bought 7 bags. Write simplified expressions for: a the number of marbles Xavier has b the number of marbles Cameron has c the total number of marbles that Xavier and Cameron have d the number of extra marbles that Cameron has compared to Xavier 9 Simplify the following by collecting like terms. a 3xy + 4xy + 5xy b 4ab + 5 + 2ab c 5ab + 3ba + 2ab d 10xy – 4yx + 3 e 10 – 3xy + 8xy + 4 f 3cde + 5ecd + 2ced g 4 + x + 4xy + 2xy + 5x h 12ab + 7 – 3ab + 2 i 3xy – 2y + 4yx R K I NG WO U F 10 a Test, using a table of values, that 3x + 2x is equivalent to 5x. C R PS MA Y LL b Prove that 3x + 2y is not equivalent to 5xy. HE M AT I C A T 11 a Test that 5x + 4 – 2x is equivalent to 3x + 4. b Prove that 5x + 4 – 2x is not equivalent to 7x + 4. c Prove that 5x + 4 – 2x is not equivalent to 7x – 4. Enrichment: How many rearrangements? 12 The expression a + 3b + 2a is equivalent to 3a + 3b. a List two other expressions with three terms that are equivalent to 3a + 3b. b How many expressions, consisting of exactly three terms added together, are equivalent to 3a + 3b? All coefficients must be whole numbers greater than 0. ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  number and algebra 381 8E Multiplying, dividing and mixed operations To multiply a number by a pronumeral, we have already seen we can write them next to each other. For example, 7a means 7 × a, and 5abc means 5 × a × b × c. The order in which numbers or pronumerals are multiplied is unimportant, so 5 × a × b × c = a × 5 × c × b = c × a × 5 × b. When writing a product without × signs, the numbers are written first. 7 xy We write as shorthand for (7xy) ÷ (3xz). 3 xz 10 10 5 × 2 2 We can simplify fractions like by dividing by common factors, such as = =. 15 15 5 × 3 3 7 xy 7 xy 7 y Similarly, common variables can be cancelled in a division like , giving =. 3 xz 3 xz 3z let’s start: Rearranging terms 5abc is equivalent to 5bac because the order of multiplication does 5 ×a×b×c=? not matter. In what other ways could 5abc be written? a × b is written ab. Key ideas a a ÷ b is written. b a × a is written a2. Because of the commutative property of multiplication (e.g. 2 × 7 = 7 × 2), the order in which values are multiplied is not important. So 3 × a and a × 3 are equivalent. Because of the associative property of multiplication (e.g. 3 × (5 × 2) and (3 × 5) × 2 are equal), brackets are not required when only multiplication is used. So 3 × (a × b) and (3 × a) × b are both written as 3ab. Numbers should be written first in a term and pronumerals are generally written in alphabetical order. For example, b × 2 × a is written as 2ab. When dividing, any common factor in the numerator and denominator can be cancelled. 2 For example: 4 a1b 2a 1 1 = 2 bc c Example 10 Simplifying expressions with multiplication a Write 4 × a × b × c without multiplication signs. b Simplify 4a × 2b × 3c, giving your final answer without multiplication signs. c Simplify 3w × 4w. SoluTion ExplanaTion a 4 × a × b × c = 4abc When pronumerals are written next to each other they are being multiplied. ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party 382 Chapter 8 Algebraic techniques 1 b 4a × 2b × 3c = 4 × a × 2 × b × 3 × c First, insert the missing multiplication signs. =4×2×3×a×b×c Now we can rearrange to bring the numbers to the front. = 24abc 4 × 2 × 3 = 24 and a × b × c = abc, giving the final answer. c 3w × 4w = 3 × w × 4 × w First, insert the missing multiplication signs. =3×4×w×w Rearrange to bring numbers to the front. = 12w2 3 × 4 = 12 and w × w is written as w2. Example 11 Simplifying expressions with division a Write (3x + 1) ÷ 5 without a division sign. 8ab b Simplify the expression. 12b SoluTion ExplanaTion 3x + 1 The brackets are no longer required as it becomes a (3x + 1) ÷ 5 = 5 clear that all of 3x + 1 is being divided by 5. 8ab 8 × a × b Insert multiplication signs to help spot common b = 12b 12 × b factors. 2× 4 ×a× b = 8 and 12 have a common factor of 4. 3× 4 × b 2a Cancel out the common factors of 4 and b. = 3 Exercise 8E R K I NG WO U F 1 Chen claims that 7 × d is equivalent to d × 7. C R PS MA Y a If d = 3, find the values of 7 × d and d × 7. b If d = 5, find the values of 7 × d and d × 7. LL HE M AT I C A T c If d = 8, find the values of 7 × d and d × 7. d Is Chen correct in his claim? 2 Classify each of the following statements as true or false. a 4 × n can be written as 4n. b n × 3 can be written as 3n. c 4 × b can be written as b + 4. d a × b can be written as ab. e a × 5 can be written as 50a. 12 2×6 3 a Simplify the fraction. (Note: This is the same as.) 18 3×6 2000 2 × 1000 b Simplify the fraction. (Note: This is the same as.) 3000 3 × 1000 2a 2×a c Simplify. (Note: This is the same as.) 3a 3×a ISBN: 9781107626973 © David Greenwood et al. 2013 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party  Number and Algebra 383   R K I NG 4 Match up these expressions with the correct way to write them. WO U F a 2×u A 3u C R PS MA Y LL 5 HE M AT I C A T b 7×u B u c 5÷u C 2u u d u×3