Chapter 12

12 NCM7 2nd ed SB TXT.fm Page 416 Saturday, June 7, 2008 6:58 PM

PATTERNS AND ALGEBRA

Algebra has been a powerful tool for mathematicians for over 4000 years and was used in ancient Babylon and Egypt. It was brought by the Arabs from India to Europe. The word ‘algebra’ comes from ‘al-jabr’, an Arabic word for one of the steps used to solve equations. Algebra today has a wide range of uses—from putting a satellite into space to calculating the quantities of ingredients in a biscuit recipe!

12

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12 NCM7 2nd ed SB TXT.fm Page 417 Saturday, June 7, 2008 6:58 PM

0123456789

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0123456789 567890123 3 8901234 456789 012345 0123456789 678901234

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In this chapter you will: 56789012345678 5678

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12 NCM7 2nd ed SB TXT.fm Page 418 Saturday, June 7, 2008 6:58 PM

Start up Worksheet 12-01 Brainstarters 12

1 If = 8, find the value of: a +3 b 4+ e 3× f ×7 i 9 + − 12 j 2× +5 2 If a e i

= 20, find the value of: −5 b 3× × f 2× + 11 − 9 j 6+2×

3 Find the value of: a 3 × (−4) b −2 × 5 f 3 − (−9) g −4 × (−5) 4 If a f

= −2, find the value of: +2 b −1 − g 2× −1

c −6 g ÷2 k 11 − + 1

d + h 32 ÷ l ÷8−3

c ÷4 g 60 − k 100 ÷

d +9 h 25 − + 6 l − 10 × 2

−5

c 8 × (−2) h −1 × (−6)

d −7 + 5 i 2 + (−3)

e −4 − 10

c h

d 3+ i 3×

e 5−

×

+1

5 If p = 3 and q = 2, find the value of: a p+q b p−q c q−p d p×q e p÷q p+q f 5×q−p g 4+p×q h ----------i 10 × q − 5 × p 5 6 Complete each table for the given rule. a y=x+2 b m=5×n x

−1

0

1

2

n

y

0

1

2

6

2

−1

m

c d=3×c−1 c

−1

1

2

d l=m×m 3

4

d

m

5

l

12-01 Variables and expressions In Chapter 6, we used letters of the alphabet to stand for numbers when describing number patterns. In this chapter, we will explore variables further.

Example 1 An envelope contains an unknown number, y, of paperclips. Find an expression to represent the number of paperclips in this diagram. Solution 1 envelope of paperclips + 3 paperclips = y + 3 paperclips

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Example 2 Each cup holds the same unknown quantity of marbles. Let k stand for the number of marbles in the cup. Find an expression to represent the number of marbles in the diagram. k marbles

k marbles

k marbles

Solution 3 cups of marbles + 5 marbles = 3 lots of k + 5 =3×k+5 = 3k + 5 The expression 3k + 5 is called an algebraic expression. Unlike an algebraic rule or formula, such as y = 2x − 9, it does not contain an equals sign. An algebraic expression uses variables and numerals to describe a quantity. The algebraic expression 3k + 5 describes the total number of marbles in the diagram above.

Example 3 If m stands for the number of coins in each envelope and p stands for the number of coins in each cup, write the algebraic expression for the number of coins in this diagram.

p coins

m coins

p coins

p coins

m coins

m coins

p coins

Solution p + m + 2p + 2m + p

Exercise 12-01 1 Select the correct alternative from A, B, C and D to complete this statement. If y stands for the number of paperclips in each envelope, then 3y + 2 is shown by: A B

C

C

2 If y stands for the number of paperclips in an envelope, draw what is represented by each of the following algebraic expressions. a y+1 b 2 lots of y c 4+y d 1+y+2 e y+5 f y+y g y+2+y+1 h 2y + 3 i 4 + 2y j y + 1 + 2y + 4 k y+2+y+2 l 3 + 2y + 1 CHAPTER 12 ALGEBRA

Ex 1

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12 NCM7 2nd ed SB TXT.fm Page 420 Saturday, June 7, 2008 6:58 PM

Ex 2

3 Write an algebraic expression for what is shown by each of the following diagrams. Let k stand for the number of marbles in each cup. ( stands for one marble.) a

b

c

d

e

f

g

h

i

j

4 Write an algebraic expression for the number of paperclips in each of the following diagrams, using y to stand for the number of paperclips in each envelope.

Ex 3

a

b

c

d

e

f

g

h

i

j

5 Write an algebraic expression for what is shown by each of the following diagrams, using m to stand for the number of coins in each envelope and p to stand for the number of coins in each cup.

420

a

b

c

d

e

f

g

h

NEW CENTURY MATHS 7

12 NCM7 2nd ed SB TXT.fm Page 421 Saturday, June 7, 2008 6:58 PM

i

j

k

l

m

n

6 a Which of the expressions from Question 5 means the same amount as 3m + 2p?

12-02 Algebraic abbreviations Mathematicians prefer to write expressions as simply as they can. For instance: 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 9 × 5 and 2 × 2 × 2 × 2 × 2 = 25 In algebra, we try to write expressions as simply as possible too. Already we know that 2m a is a simpler way of writing 2 × m, and --- is a simpler version of a ÷ 2. 2 In algebraic expressions such as 2m + 3w − 4k, we call each part of the expression a term. algebraic expression

2m

+

3w

-

4k

three terms

Example 4 1 Simplify each of the following expressions. a 3+3+3+3+3 b m+m+m+m+m

c 13K + 5K − 8K

Solution a 3 + 3 + 3 + 3 + 3 can be written as 5 × 3. b m + m + m + m + m is the same as 1m + 1m + 1m + 1m + 1m and can be rewritten as 5m. c 13K + 5K − 8K can be rewritten as 10K (because 13 + 5 − 8 = 10) 2 Simplify each of the following expressions. a 3×m b m×w c 5×B×A e 3×m×2×w f k×k g m×m×m

d 8×a×n×b×m h 2×y×y

Solution Simplifying an algebraic expression means writing the expression in the shortest way. a 3 × m = 3m b m × w = mw c 5 × B × A = 5AB Note that we always write the number at the start of the term, followed by the variables in alphabetical order. d 8 × a × n × b × m = 8abmn e 3 × m × 2 × w = 6mw Note that we multiply the numbers first, then write the variables in alphabetical order. f k × k = k2

g m × m × m = m3

h 2 × y × y = 2y2

CHAPTER 12 ALGEBRA

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12 NCM7 2nd ed SB TXT.fm Page 422 Saturday, June 7, 2008 6:58 PM

Example 5 Write each of these expressions in expanded form. a 3AB b 47abcd Solution a 3AB = 3 × A × B b 47abcd = 47 × a × b × c × d Note that expanding is the opposite of simplifying.

Exercise 12-02 1 How many terms are there in each of the following expressions? a 2k + 1 b m + 3p + m c 4x d 2w + 7y + w e 4 + 7n f 4m − 2k − 7 g 9 h 3p + 2q i x + y2 + z j 4pq + 6pq + 7p k 6k − 3k + 2m − m l 6ef + 4e 2 2 m 7m + 3n − 2m + 6n n 12cde − 5def o j2 p 12f − 4g + 3f + 7g + 10f − 9g Ex 4

2 Simplify each of the following expressions. a 5+5+5+5 b 2+2+2+2+2+2+2+2+2 c a+a+a+a+a+a d e+e+e+e e w+w f 5m + 2m + 6m g 8h + 3h + 2h + h h 5m − 2m i 23k + 17k j 12w − 4w − 5w k 7p + 5p − 2p l 6n + 3n − 10n ma + a + a + b + b n 3d − 2d + 5d o m + 2m − 3m p 12q − 4q + q

Ex 4

3 Simplify the following expressions. a 4×w b 8×a×b c 3×c×a×d d a×c×b×5 e 2×w×3×h f g×2×f×3 g 2×w×c×3×m h 2×m×s×4×h×b i the product of m and w j 2 × y × 3 × r × q × 5 × p × d k a×a l d×d×d m4 × f × f n g×2×g o 4×h×h×h p l×4×l×2 q 2×m×4×m r 6×r×r×6×r s c×c×e×e×e t 5×j×7×j×k u y×z×y×z×y v 4×n×3×p×n×2×p×n 4 What confusion would be caused by removing the multiplication sign from 5 × 3? 5 What does the expression AB mean?

Ex 5

6 Write each of these expressions in expanded form. a AB b 2mns c 188ABC e 4m f m g x2 2 3 i 7d j 5s k x2y3

d 3mnabcdef h q3 l 9m3n2

7 Write 4m3 in expanded form. Select A, B, C or D. A 4m × 4m × 4m B 4m + 4m + 4m C 4×m+m+m

D 4×m×m×m

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12-03 From words to algebraic expressions

Worksheet 12-02

This section will improve our ability to change information written in words into algebraic expressions. For example, an algebraic expression for the sum of A and B is A + B.

What’s the expression?

Example 6 Skillsheet 12-01

If A represents any number, write an expression for: a three times that number b three less than that number c the next consecutive number d that number multiplied by itself e the square root of that number f one-third of that number Solution a 3A c A+1 e

A

Algebraic expressions

b A−3 d A2 1 A f --- A or ---- or A ÷ 3 3 3

Example 7 Write an expression for: a the sum of m and 5 c m increased by 10 e the product of m and 2

b the double of m d the difference between N and Q f the quotient of m and 2

Solution a m+5 c m + 10

b 2m d N−Q

e 2m

m 1 f m ÷ 2 or ---- or --- m 2 2

Exercise 12-03 1 Write an expression for each of the following. Use N to represent any number. a double the number b half the number c triple the number d one-quarter the number e one-tenth the number f the next consecutive number g 5 times the number h the sum of the number and 21 i the difference between the number and 10 j 2 more than the number k the number increased by 3 l the number times itself m the square root of the number

Ex 6

2 Imagine that you must repeat Question 1 using the pronumeral A, instead of N. What difference would this make to your answers? Does it matter which letter of the alphabet you choose to use? CHAPTER 12 ALGEBRA

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12 NCM7 2nd ed SB TXT.fm Page 424 Saturday, June 7, 2008 6:58 PM

Ex 7

3 If A, B and C represent any three numbers, write an expression for: a the sum of A and B b the sum of all three numbers A, B and C c the difference between B and C, where B is greater than C d the product of A and C e the product of all three numbers A, B and C f the quotient of A and B g the sum of A and B, divided by C h the quotient of C and B. 4 Write an expression for: a the sum of 3 and A c 5 added to C e 3 taken away from E g the sum of A, B and W i R to the power of 2

b d f h j

3 less than B 8 increased by D X decreased by F m divided by m the sum of A and B, divided by 2.

5 Write an expression for: a the number of students in a class if there are B boys and G girls b the number of pies needed at a party if there are N children and each child can eat two pies c the number of children remaining in class if X leave for the library out of a total group of T d the amount of money earned by selling N cakes at the school fete, where each cake is priced at $2 e the cost of each film ticket where the total cost is $M and there are three people going to the film f the total cost of buying A cans of lemonade and B ice-creams, where each can costs $1 and each ice-cream costs $2.

6 Write an expression for the cost of buying c children’s tickets at $21 each and a adult tickets at $27 each. Select A, B, C or D. A 21a + 27c B a+c C 27a + 21c D a + c + 21 + 27

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12-04 Like terms Earlier, we used p to represent the number of coins in each cup number of coins in each envelope

and m to represent the

.

We found that these algebraic expressions were the same: a and p

+

+ p

m

+

=

m

2p

b

+

2m

and m + p + m + p + m = 3m + 2p These are examples of collecting like terms. We can add together things that are the same. For subtraction:

−

=

3m + 2p − 2m = m + 2p This is also collecting like terms. We can subtract things that are the same. These algebraic expressions represent the total number of coins in the envelopes and cups.

Exercise 12-04 1 Match each expression in the left column with the correct expression on the right. a

A + p +

m

+

2m

m

b

2p

B p +

m

+

m

+ p

+

2p

c

3

C p +

m

+ p +

m

+ p

d

+

3m

3

D p +

2

+ p + 1

p

e

+

2m

E 2m

+

2

+ m

+ 1

f

3p

+

3p

+

4

F 2

+ p +

2

+

2p

2m

CHAPTER 12 ALGEBRA

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12 NCM7 2nd ed SB TXT.fm Page 426 Saturday, June 7, 2008 6:58 PM

2 Match each expression in the left column with the correct expression on the right. a A

−

+ p

m

−

2m

m

−

b m +

B

−

4

2

c + p

−

2p

p

−

d +

2p

D

−

6

4

e

p E

− +

2m

−

2p

2m

f +

2m

F

−

3m

+

2p

− 2p

2

C

− 2m

+

2p

m

m

+

2

3 If p represents the number of coins in a cup and m represents the number of coins in an envelope, write the simplest algebraic expression you can for each of these.

−

b

a +

p

+ p

m

m

c

+

− + p

3m

−

+

m

2p

d +

2p

+

2

p +

e

− +

2m

p +

m

+

−

−

4

f

+ +

2

g

−

m

p +

2

− m

−

1

− m

+

+

2p

3

h

−

2p

− 3p

426

3

+

NEW CENTURY MATHS 7

2m

−

− m

−

p

+

2p

−

p

12 NCM7 2nd ed SB TXT.fm Page 427 Saturday, June 7, 2008 6:58 PM

−

i p +

m

+ p +

m

+

4

−

−

2m

j

−

3m + 4p − 4 Simplify these expressions. a 3m + 2m b 6p − p d 2m + 3p + 2p e 3m + 4p + m + 2p g 6m + 4p − 2m h 7p + 5m − 3m

−

3

− m

−

2p

c m+m−m f 2m + p + m + 3p i 12m + 6p − 3m − 2p

5 Simplify 9m + 4n − 5m + 2n. Select A, B, C or D. A 14m + 6n B 10mn C 4m + 6n

D 14m + 11n

Mental skills 12A Maths without calculators

Multiplying by 5, 15, 25 or 50 It is easier to multiply a number by 10 than by 5, so to multiply by 5, we can halve the number, then multiply by 10. This is because 1--- × 10 = 5. We can use a similar strategy 2 to multiply by 15, 25 and 50. 1 Examine these examples. a To multiply by 5, halve the number, then multiply by 10. 18 × 5 = 18 × 1--- × 10

(or factorise 18 into 9 × 2)

2

= 9 × 10 = 90 b To multiply by 50, halve the number, then multiply by 100. 26 × 50 = 26 × 1--- × 100 2

(or factorise 26 into 13 × 2)

= 13 × 100 = 1300 c To multiply by 25, quarter the number, then multiply by 100. 44 × 25 = 44 × 1--- × 100 4

(or factorise 44 into 11 × 4)

= 11 × 100 = 1100 d To multiply by 15, halve the number, then multiply by 30. 8 × 15 = 8 × 1--- × 30

(or factorise 8 into 4 × 2)

2

= 4 × 30 = 120 2 Now simplify these. a 32 × 5 e 52 × 50 i 12 × 15 m 28 × 25

b f j n

14 × 5 36 × 25 16 × 25 36 × 50

c g k o

48 × 5 44 × 50 22 × 50 26 × 15

d h l p

18 × 50 12 × 25 14 × 15 22 × 35

CHAPTER 12 ALGEBRA

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Just for the record

Ancient algebra The Rhind Papyrus is one of the earliest mathematical documents we have, dating back to 1700 BC. It contains the following algebraic problem, written here in hierolglyphics. The algebraic translation is: x + 2--- x − 1--- (x + 2--- x) = 10 3

3

3

Note that: legs walking left mean ‘minus’ legs walking right mean ‘add’ Find out in whose honour the Rhind Papyrus is named.

Worksheet 12-03 Collecting like terms

Skillsheet 12-02 Algebra using diagrams

!

12-05 Adding and subtracting like terms The following pairs of terms are called ‘like terms’ because each set of terms has exactly the same variable(s). 3x and 5x 3mw and 2mw 12m and 32m 3m and m 5ab and 2ab xyz and 2yzx The following are not like terms because each set of terms has different variables. 3x and 5m 8wm and 2wq 5ab and 2abc 2p and 3p2

We can only add or subtract like terms (terms which have exactly the same variables).

Example 8 Simplify these expressions. a 3m + 5m b 3ab + 2ba

c 5x + x

Solutions a 3m + 5m = 8m (3m and 5m are like terms) c 5x + x = 6x (x is really 1x)

b 3ab + 2ba = 5ab (ab and ba are like terms) d 42mw − 17wm = 25mw (mw and wm are like terms)

428

NEW CENTURY MATHS 7

d 42mw − 17wm

12 NCM7 2nd ed SB TXT.fm Page 429 Saturday, June 7, 2008 6:58 PM

Example 9 Simplify these expressions by adding or subtracting like terms. a 3m + 5w + 2m + 3w b 24 + 5ab − 14 − 2ba c 4a2 + 2a − 3a2 + 6a Solution a 3m + 5w + 2m + 3w

like terms 5w + 3w

3m

+ 5w

+ 2m

+ 3w

= 5m + 8w

− 14

− 2ba

= 10 + 3ab

+ 6a

= a 2 + 8a

like terms 3m + 2m

b 24 + 5ab − 14 − 2ba

like terms 24 − 14

24

+ 5ab

like terms 5ab − 2ba

c 4a2 + 2a − 3a2 + 6a

like terms 4a2 − 3a2

4a2

+ 2a

− 3a2 like terms 2a + 6a

Exercise 12-05 1 Find the like terms in each of these sets. a 2my, 3x, 6am, 16x, 4mb c 8k, 3x, 2w, 12g, 23w e 2b, 2k, 2m, 2g, 3k g x, 3x2, y, x2 i x2y, 2x, 3y, 4x2y k p2, q2, 3p, 2q2, pq, 7q

b d f h j l

2mw, 3km, 4w, 5mw, 6m, 7aw 2p, 5mq, 5p, 7q, 7m 2ab, 2a, 2b, 2m, 3ba 4mn, 3m, 4, 2nm, mn, 2n 7, 2a, 4b, 5p, 9, d, 2 c2d, cd2, 3c, 2d, 5cd2

2 Find the like terms in this set. Select A, B, C or D. 3ab, 2a2, 2b2a, 2p2, 2ba A 2a2, 2p2 B 2a2, 2b2a, 2p2, 2ba C 3ab, 2ba

D 2b2a, 2ba

3 Simplify each of these expressions by adding like terms. a 2m + 5m b 4k + 7k c 2ab + 5ab d 5mn + 2nm e xy + xy f 3abc + 4abc + 2bac g 4ab + 3ab + 2ab h 12mn + 6mn + mn + nm i 6cde + 5dec + 3edc j 6mp + mp + 9mp k 3x2 + 2x2 + x2 l ef 2 + 4ef 2 CHAPTER 12 ALGEBRA

Ex 8

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Ex 9

4 Simplify each of these expressions by subtracting like terms. a 5m − 3m b 8d − 3d c d 45abc − 12abc e 45fg − 13fg f g 30cd − 12cd h 12xy − 7yx − xy i 2 2 2 2 j 6k − 3k k 4w − w l

12mk − 7mk 48mn − 29mn 8de − 12de 5xy2 − 2xy2 − xy2

5 Simplify each of these by adding or subtracting like terms. a 5m + 2m − 3m b 12a − 4a − 5a c d 12f + 15f − 18f e 8s2 + 7s2 − 6s2 f g 3pq − 2pq + pq h 11abc + 3abc − 5abc i 2 2 2 j 3d − 8d + 2d k − 4x + 7x − 3x l

3x + 5x + 7x 5mn + 6mn − 4nm 8p − 3p − 5p 8de + 12de − 5de + de

6 Simplify the following. a 3x + 4 + 5x + 6 d 10k − 4 − 6k + 12 g 2p + 3 − 3p + 8 j 5ab + 2p − 7ab − 2p m 4y2 + 2y + y2 + 5y o 9g2 − 12 + 2g2 + 7

b e h k n p

4m − 6 + 4m + 10 c 23xy + 23ab + 17xy − 17ab f 5k − 11 − 8k + 6 i 3p + 12m − 6p − 4m l 2 2 2 2 7m + 2n − 4m + 3n x2 − 3x + 4x2 − 7x

2mn + 3f + 5mn + 7f 15r + 15 − 15r − r 13 + 2ab − 6ab − 1 −10de + 3d + 5de − 2d

7 Simplify the following. a 6m + 3m d 2d + 9 + 3d g 11p − p j 4k + 5 + 17 + 11k m 2y + 1 + 2y + 2 + y + 3 p 12a − a − 2a − 6a s 20x + 3y2 − 7x − 2x

b e h k n q t

k + 2 + 3k 2f + 3g + 4f + 7g 7 + 5q − 3 3a + 2b + 6a + 7b 7r + 6s − 2r − 4s 2m2 + 4m2 − 6 12ab + 3cd − 4ab + 7cd

12mx + 4xm 4y + 3x + 2x + 8y 8abc − 4abc − abc x2 + 3x2 + 9x2 12 + 6x − 4x − 6 12pq − pq − 7qp

8 Simplify a5 + a5. Select A, B, C or D. A a10 B a25

c f i l o r

C 2a5

Q 2a10

12-06 Multiplying algebraic terms Example 10 Simplify each of the following. a 2 × 3a b 4x × (−2) Solution a 2 × 3a = 2 × 3 × a = 6a c 7k × 3k × 2 = 7 × 3 × 2 × k × k = 42k2

430

NEW CENTURY MATHS 7

c 7k × 3k × 2

d 3m2 × 4p2

b 4x × (−2) = 4 × x × (−2) = 4 × (−2) × x = −8x d 3m2 × 4p2 = 3 × 4 × m2 × 4p2 = 12m2p2

12 NCM7 2nd ed SB TXT.fm Page 431 Saturday, June 7, 2008 6:58 PM

When multiplying algebraic terms, multiply the numbers first, then multiply the variables. Variables are usually written in alphabetical order.

Exercise 12-06 1 Simplify each of the following. a 3 × 2y b 7 × 8q e 4×2×d f 4a × 3b i 3m × 2m j 5k × 11k m 10 × 2m × 6n n 4a × 3 × 7a

c g k o

2 Simplify each of the following. a 2 × 4a × 9b b 2mn × 4 e 5rst × 2rs f 3ab × 6cd i hjk × hj × jk j 4q2 × 6p2

c x × xy × y g p×9 k 2xy × 6xy × 9yx

d 2j × 3k × 7 h m × n × 2m l 2m × 3n × mn

3 Simplify each of the following. a −6 × 3x b −2 × 3k e 6 × (−m) f 4 × (−3y) i −1 × (−r) j −8x × (−7)

c −1 × p g −2 × (−2n) k −s × (−s)

d −j × 4 h −k × (−5) l −3d × (−d)

Ex 10

4t × 3 12c × 4d 12f × 9f 4x × 7x × 3

d h l p

3k × 9 6p × 9q 6g × 3 × 4h 5m × 2 × 3m

4 Which of the following will simplify to 14a2n2? Select A, B, C or D. A 2a × an × 7 B 2a × 7n × an C 7a2 × 2n D 7a2n2 × 2a2n2

12-07 Expanding expressions Let m represent the number of paperclips in an envelope and

represent one paperclip.

The number of paperclips in each envelope is unknown but it is the same number for each envelope. So: a 2 lots of (m + 3) = 2(m + 3) = 2m + 6

=

Skillsheet 12-02 Algebra using diagrams

m

3 lots of (2m + 1) = 3(2m + 1)

b

Expandominoes

m

m

m

=

m

m

m

m

m

m

+ 3

6m

=

m

m

m

m

m

m

What is happening? What do the parentheses (brackets) mean? CHAPTER 12 ALGEBRA

Worksheet 12-04

431

12 NCM7 2nd ed SB TXT.fm Page 432 Saturday, June 7, 2008 6:58 PM

The number outside is multiplying each term inside the parentheses: 2(m + 3) = 2 × m + 2 × 3 = 2m + 6 3(2m + 1) = 3 × 2m + 3 × 1 = 6m + 3 This is called expanding the expression, that is, removing the grouping symbols by multiplying each term separately, and simplifying. Y An example of expanding being used is when finding the perimeter of a rectangle of length Y and height X. It can be written in two ways. X Method 1: The perimeter can be found by adding together all the sides: P =X+X+Y+Y Y P = 2X + 2Y Method 2: The perimeter can be found by doubling the sum of the length and height: P = 2(X + Y) Both answers must be equal, so: 2(X + Y) = 2X + 2Y. expanding

2(X + Y) = 2X + 2Y These are two ways of expressing the same thing.

Example 11 Expand each of the following expressions. a 2(x + 3) b 3(x − 4) Solution a 2(x + 3) b 3(x − 4) c 5(2x − m)

c 5(2x − m)

2(x + 3) = 2 × x + 2 × 3 = 2x + 6 3(x − 4) = 3 × x + 3 × (−4) = 3x − 12 5(2x − m) = 5 × 2x + 5 × (−m) = 10x − 5m

Example 12 Expand each of the following expressions. a −2(x − 3) Solution a −2(x − 3) b −(5 − 4x)

432

b −(5 − 4x)

−2(x − 3) = −2 × x − 2 × (−3) = −2x + 6 −1(5 − 4x) = −1 × 5 − 1 × (−4x) −(5 − 4x) is the same as −1(5 − 4x) = −5 + 4x

NEW CENTURY MATHS 7

X

12 NCM7 2nd ed SB TXT.fm Page 433 Saturday, June 7, 2008 6:58 PM

Example 13 Expand: a m(c + 1) Solution a m(c + 1) b f(2m − 3r)

b f(2m − 3r) m(c + 1) = m × c + m × 1 = cm + m f(2m − 3r) = f × 2m + f × (−3r) = 2fm − 3fr

Exercise 12-07 1 Expand each of the following expressions. a 3(a + 2) b 2(h + 2) c 2(m + 3) e 4(2m + 3) f 7(a + b) g 12(2p + 5) i 6(3x + 4) j 12(2m + n) k 10(4p + 2q)

d 4(x + 6) h 5(a + 2) l 3(2a + 4b)

2 Expand each of the following expressions. a 4(x − 2) b 3(m − 7) c 8(k − 3) e 2(m − p) f 7(3f − 2g) g 4(3m − 5) i 6(3x − 4) j 3(1 − k) k 6(2 − 3p)

d 5(y − 5) h 5(3m − 3) l 10(2 − 2m)

3 Expand each of the following expressions. a −3(x + 2) b −7(p + 1) c −2(m − 3) e − 4(6 − y) f −9(a + 4) g −(k + 3) i −(6 − 2x) j −3(4m + 5) k −5(3y − 6)

d −5(k − 4) h −(m − 2) l −(4 − 7x)

4 Expand each of the following expressions. a m(n + s) b x(y − 2) e y(2x − 3) f f(4k − 7) i 2m(3k + 1) j 3k(5x − 2) m 5w(7x − 2m) n 2c(5b − 3a)

c g k o

k(m + 12) b(2a − 6) 4m(1 − m) 3p(6q + 2r)

d h l p

p(q − 4) n(2p + 7) xy(x + 2) 4j(5k + 7l)

5 Expand each of the following expressions. a x(x + 4) b a(2a + 5) e −r(2r − 1) f p(p + q) i −3v(v + 7) j pq(p + q) m mn(2m + 3n) n −(2k + 4j)

c g k o

−n(n + 2) −d(d + e) −(m − n) ab(a + b + c)

d h l p

2y(y − 4) w(8 − w) −3k(k − 1) 4m(m + 2n + 3p)

6 Expand −4(3 − x2). Select A, B, C or D. A −12 + 4x2 B −12 − 4x2 1 7 Expand a( --- + 1). Select A, B, C or D. a 1 1 A a + --B 1 + --a a

Ex 11

Ex 12

Ex 13

C −12 − x2

D −12 + x2

C2

D a+1 CHAPTER 12 ALGEBRA

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Working mathematically

Applying strategies and reasoning

Numerical expansions We can use the expansion process to multiply numbers without using a calculator. 1 We can use the diagram below to multiply 59 × 4. 59 can be written as (50 + 9) and so 59 × 4 = 4 × (50 + 9). Copy the diagram and complete the following. Area of A = Area of B =

4

Total =

50

9

A

B

What is 59 × 4?

59

2 We can use a similar process to find 73 × 9. 73 can be written as (70 + 3) and so 73 × 9 = 9 × (70 + 3). Copy the diagram below and complete the following. Area of A = Area of B =

9

Total = What is 73 × 9?

70

3

A

B

73

3 Draw diagrams to calculate each of the following. a 51 × 6

b 98 × 5

c 43 × 7

d 18 × 7

e 99 × 4

f 27 × 8

g 88 × 8

h 101 × 9

12-08 Expanding and simplifying expressions Example 14 Expand and simplify: a 2x + 3(x + y) Solution a 2x + 3(x + y) = 2x + 3 × x + 3 × y = 2x + 3x + 3y = 5x + 3y

434

NEW CENTURY MATHS 7

b 2a − (4 + a) b 2a − (4 + a) = 2a − 1(4 + a) = 2a − 1 × 4 − 1 × a = 2a − 4 − a =a−4

12 NCM7 2nd ed SB TXT.fm Page 435 Saturday, June 7, 2008 6:58 PM

Example 15 Expand and simplify the following. a 2(m + 1) + 3(m − 4)

b 6(p + 2) − 3(p − 4)

Solution a 2(m + 1) + 3(m − 4) = 2 × m + 2 × 1 + 3 × m + 3 × (− 4) = 2m + 2 + 3m − 12 = 5m − 10

b 6(p + 2) − 3(p − 4) = 6 × p + 6 × 2 − 3 × p − 3 × (− 4) = 6p + 12 − 3p + 12 = 3p + 24

Exercise 12-08 1 Expand each of the following and simplify by collecting like terms. a 3p + 2(p + q) b 7(a + 2) + 12 c 3(k − 5) + 7 d 12 + 2(n − 2) e 6k + 11(k + 2) f 12(v + 4) + 4v g 16(t − 2) − 8t h 5a − 2(a + 1) i 7x − 3(x + 2) j 13x − 3(x + 1) k 2(x + 1) − 5 l a − (2 + a) m b − (b − 4) n 3(x + 5) − 2x + 7 o 3(4 − 2x) − 10 p 5(3 − 2x) + 5x − 3 2 Simplify 3(x + 5) − 4. Select A, B, C or D. A 14x B 3x + 1 C 3x − 9 3 Expand and simplify the following. a 2(x + 2) + 4(x + 1) c m(m + 1) + 3(m + 1) e 4(d + e) + 3(d − e) g 3(x − 2) − 3(x − 4) i 7(y − 2) − 3(y + 4) k 3(d + 2) − (d + 8) m 6(m − 4) + 12(2m + 3) o 9(m + n) + 2(m − 3n)

b d f h j l n p

Ex 14

D 3x + 11 Ex 15

3(r + 3) + 2(r − 4) 2(q − 7) + 5(q + 9) 4(x + 1) − 2(x − 2) 12(x − 3) − 4(2x − 4) 2(2m + 1) + 4(2m − 2) 5(2t + 3) + 8(3t + 1) 5(p − q) + 7(p + q) 4(k + 7) − (k − 3)

4 Correct this student’s homework by checking her working out for each problem. Which questions did she get wrong? a 5x − (2 − 3x) = 5x − 2 − 3x = 2x − 2 d 4m − (3 + 2m) = 4m − 3 − 2m = 2m − 3 g 2p − 3 − (3 − 4p) = 2p − 3 − 3 − 4p = −2p − 6

b 6x + (4 − 5x) = 6x + 4 − 5x =x+4 e 5x − (4 − 2x) + 3x = 5x − 4 + 2x + 3x = 10x − 4 h 9x − 1 + (3x − 5) = 9x − 1 − 3x + 5 = 6x + 4

c 2x − (8 − x) = 2x − 8 + x = 3x − 8 f 8 + (4x + 5) − 2x = 8 + 4x + 5 − 2x = 2x + 13 i 10 − (3x − 2) + (5x − 3) = 10 − 3x − 2 + 5x − 3 = 5 + 2x CHAPTER 12 ALGEBRA

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Mental skills 12B Maths without calculators

Dividing by 5, 15, 25 or 50 It is easier to divide a number by 10 than by 5, so to divide by 5, we can divide by 10, then double the answer. We can use a similar strategy to divide by 15, 25 and 50. 1 Examine these examples. a To divide by 5, divide by 10 and double the answer. We do this because there are two 5s in every 10. 140 ÷ 5 = 140 ÷ 10 × 2 = 14 × 2 = 28 b To divide by 50, divide by 100 and double the answer. This is because there are two 50s in every 100. 400 ÷ 50 = 400 ÷ 100 × 2 =4×2 =8 c To divide by 25, divide by 100 and multiply the answer by 4. This is because there are four 25s in every 100. 600 ÷ 25 = 600 ÷ 100 × 4 =6×4 = 24 d To divide by 15, divide by 30 and double the answer. This is because there are two 15s in every 30. 240 ÷ 15 = 240 ÷ 30 × 2 = 80 × 2 = 160 2 Now simplify these. a 90 ÷ 5 e 1300 ÷ 50 i 300 ÷ 25 m 270 ÷ 15

Worksheet 12-05 Substitution Worksheet 12-06

b f j n

170 ÷ 5 800 ÷ 50 1000 ÷ 25 360 ÷ 45

c g k o

230 ÷ 5 400 ÷ 50 700 ÷ 25 900 ÷ 25

d h l p

330 ÷ 5 2200 ÷ 50 600 ÷ 15 7000 ÷ 50

12-09 Substitution Substitution occurs in many sports, when one player replaces another during a game. Substitution in mathematics involves replacing a variable with a value. Substituting 5 for k in k + 7 gives 5 + 7.

Formula 1 game

!

Evaluating an expression means substituting a number into the expression and working out the answer.

Evaluate means to find the value of.

436

NEW CENTURY MATHS 7

12 NCM7 2nd ed SB TXT.fm Page 437 Saturday, June 7, 2008 6:58 PM

Example 16 Evaluate k − 9 when k = 15. Solution k − 9 = 15 − 9 =6

Example 17 1 Evaluate 2k + 1 when k = 3. Solution 2k + 1 = 2 × 3 + 1 =6+1 =7

(always do multiplication before addition)

2 Evaluate 3x − 2y if x = 2 and y = −1. Solution 3x − 2y = 3 × 2 − 2 × (−1) = 6 − (−2) = 6 + 2 =8 3 Evaluate m(n − 3) if m = 5 and n = 7. Solution m(n − 3) = 5 × (7 − 3) = 5 × (4) = 20

(always do brackets first)

Exercise 12-09 1 Evaluate k + 3 when: a k=2 b k = 18

c k =119

d k = −21

2 Evaluate 45 − k when: a k=5 b k = −13

c k = 28

d k = 45

3 Evaluate 4k when: a k=2

b k = 11

c k = −4

d k=8

4 Find the value of 5k + 1 when: a k = −3 b k = 10

c k = 21

d k = 38

c k = 12

d k = 13

c k = −9

d k = 57

5 Find the value of 14k − 8 if: a k=2 b k = −5 k 6 Find the value of --- if: 3 a k = 15 b k = 39

CHAPTER 12 ALGEBRA

Ex 16

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12 NCM7 2nd ed SB TXT.fm Page 438 Saturday, June 7, 2008 6:58 PM

Ex 17

7 Evaluate: a 5n − 2, if n = 2 c 23 + 5t, if t = 9 e 5(p + 6), if p = 5 g n(3n + 1), if n = 0.5 i (d + 1) ÷ 4, if d = −9 k 10(4 − p), if p = 3 m (4d − 1) ÷ 3, if d = 7

b d f h j l n

3k + 8, if k = 11 100 − 2a, if a = 23 4 (2m − 3), if m = −8 2w + 19, if w = 0 15 − 3m, if m = 4 (3 + 2m) × m, if m = 5 y2 + y, if y = 6

8 If m = −2, what is the value of 2m3? Select A, B, C or D. A −12 B 16 C −64

D −16

9 Evaluate these expressions if a = 0, b = 2, c = 5, d = 10, e = 16. a 2a + b b 3d + c c 4d − e e 4b − a − c f 3e + 2c − 3d g 4b − 3d + 8c i 3e − 2d + 6c − 400a j b(2c + 3d) k 2d(19a + 3c − d)

d c+d+e h 5a + 2b − 3c + 4d l bc + de

10 Copy and complete this table by substituting the different substitution values given. Values

a

x = 3 and y = 4

b

x = 4 and y = 0

c

x = −2 and y = −1

d

x = 5 and y = 6

e

x = 1--- and y = 1---

f

x = 8 and y = 9

g

x = 21 and y = 5

h

x = 0 and y = 8

i

x = 2.5 and y = 1.5

j

x = 3--- and y = 2---

k

x = 3 and y = −4

l

x = 0.7 and y = 0.2

2

2x + y

xy

8x

3x − 2y

37 − 4y

10

12

24

1

21

2

4

5

11 Use these values to evaluate the following expressions.

a e i m

438

a

b

c

d

x

y

z

K

P

M

7

3

−2

8

7

0

−20

1

100

0

x+y z + 3y ab + Kz Ka + Pab

b f j n

K+P+b 2K + 3K + x 7Pxy y2

NEW CENTURY MATHS 7

c g k o

4d + 3c 5 × (x + c) zd − P z2

d h l p

P − 5a 2 × (a − x) db − xy a2 − c2

12 NCM7 2nd ed SB TXT.fm Page 439 Saturday, June 7, 2008 6:58 PM

Using technology

Substitution 1 The spreadsheet below shows values for two variables, a and b, in columns A and B. Cells C1 to G1 show different algebraic expressions. Enter the data into an spreadsheet.

2 C1 shows the label ‘ab’, which means a × b. In cell C2, enter the formula shown above to complete this calculation when a = 2 and b = 1. Fill Down to copy this formula into cells C3 to C7. 3 Using your knowledge of operations (+, −, × and ÷) in spreadsheet formulas, create formulas in cells D2 to G2. For each column, D to G, Fill Down to complete each calculation. Hint: Use brackets where necessary. 4 The spreadsheet below shows values for variables; for example, m = 5, n = 2, and so on. Enter the data into a spreadsheet.

5 Complete these calculations by writing a formula for each in the given cell. a m+p (answer in cell A4) b t÷p (answer in cell B4) c n + 2p (answer in cell C4) d B−A (answer in cell D4) (answer in cell B5) e (p − n) ÷ t (answer in cell A5) f np + tu mn g -------p

(answer in cell C5)

h u − At

(answer in cell D5)

i A−t−B

(answer in cell A6)

j A + pt − mn

(answer in cell B6)

m ( nt + p ) k -----------------------A

(answer in cell C6)

nt + p l -------------m

(answer in cell D6)

CHAPTER 12 ALGEBRA

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Working mathematically

Communicating and reflecting

Generalised arithmetic We can use algebraic symbols to describe general laws about numbers and arithmetic. For example, if we add zero to any number, the answer is still that number. This can be written algebraically if we let N stand for any number: N+0=N What does the rule below mean in words? a+b=b+a It means that, if we add any two numbers, a and b, we will get the same answer as if we added b and a. For example, 4 + 7 is the same as 7 + 4. As a general property of numbers and arithmetic, two numbers can be added in any order. Form groups of two to four students, and complete these problems together. 1 Describe what these rules about numbers mean in words. a N×1=N c a−b≠b−a e N−0=N

b N×0=0 d a+b+c=b+a+c f ab = ba

2 Write each of these rules algebraically. a Any number divided by 1 equals itself. b Multiplying a number by 8 is the same as doubling it three times. c Any three numbers, a, b and c, can be multiplied together in any order. d Any number added to itself is the same as multiplying that number by 2. e Any number subtracted from itself equals 0. f Any number multiplied by its reciprocal equals 1. 3 Are these rules about numbers true or false? a a÷b=b÷a b N÷N=1 c 4a − a = 4 d a is a factor of a e If N is even, then N + 3 is odd

f 1--- N = N − 2

g a + (−a) = 2a i 0÷N=0

h 1 is a factor of a j N÷1=N

2

4 Explain the meanings of 2a + 1 and 2(a + 1). How are they different? 5 If k is an odd number, what is an expression for: a the previous odd number? b the next even number? 6 N ÷ 0 has no answer (it is not equal to 0). Why?

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Power plus 1 Simplify the following. a 2a + b + 3a − 4b c 3k − 2j − 5 − 4k + 6j + 9 e 4x2 + 6x − x2 − 12x g 5a + 4b + c − 3a + 5c i −d2 + 7 + 6e2 − 3d2 − 2e2 + 9

b d f h j

m − 3n − 5m − 7n a2 + 2b2 + 3b2 − a2 xy + yz + xz + 4xy + 3xz + yz −7p − 8q + 3p − 5q + 4q 5f 2 + 6f + 2 + 8f 2 − 1

2 Expand and simplify the following. a 4(x − 1) − 3(x + 1) c x(x − 3) + 5(x − 3) e −3(y − 4) + 7(y − 3) g p(p − 3q + 2r) + 2p(4r − 3p − 2q) i 12 − 5(4f − 6g − 3) k (x + 3)(x + 5)

b d f h j l

2(m + 4) − (m − 4) m(a + 4) + m(a − 7) 4(2a + b + 3c) − 2(a + 3b − c) 7k2 + 3(k2 − 3k − 1) − (6k2 − 4k + 2) 5(2x2 − 5x + 3) − 4(3x2 − 8x −25) (p − 1)(p + 10)

3 Write three algebraic expressions that can be simplified to get the following answers. d 7 e x 2 + y2 f 2k2 + 4k a 2x + 6 b 6m + 12p c x2 − x 4 Evaluate each of the following algebraic expressions if p = 4, q = −6, r = 3 and t = −1. d q2 − 6p + rt a p+q b pq c r 2 + t2 pq e ----rt q+r i ----------p+t

f 6 − 2r + p2

g 2p + 3q + 4r − 5t

p+4 h -----------2

r+7 q–t j ----------- − ---------5 5

p+3 r–2 k ------------ + ----------2 10

l q + pr − qt

5 Write, as an algebraic expression, the perimeter and the area of each of these shapes. a b c l y 2 x

b

q

m

d

e

x

p

c

f y

5

n

r

b d

m+3 a

2

6 If x is any number, simplify these expressions. a x+0 b 1×x c 0÷x f x − (−x) g 0×x h x−0

d x÷x i (−x)2

e x−x

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Chapter 12 review Worksheet 12-07

Language of maths abbreviation evaluate grouping symbols substitute

Algebra find-a-word

algebraic expression expand like terms substitution

algebraic term expanded form pronumeral term

consecutive expression simplify variable

1 There are English expressions, numerical expressions and algebraic expressions. What does the word ‘expression’ mean? 2 Explain in your own words the difference between an ‘algebraic term’ and an ‘algebraic expression’. 3 What is a non-mathematical meaning for ‘expand’? Relate this to its algebraic definition. 4 What is another name for a ‘pronumeral’? 5 How is ‘simplifying an expression’ different to ‘simplifying a fraction’?

Topic overview • • • •

Write in your own words the new things you have learnt about algebra. What parts of this topic did you like? Write any rules you have learnt. What parts of this topic did you find difficult or did you not understand? Talk to a friend or your teacher about them. • Give examples of algebra in use. • Copy this overview into your workbook and complete it. If necessary, refer to the ‘Language of maths’ section for keywords. Expanding

Expression

Adding/ subtracting

Substitution

ALGEBRA Like terms

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Chapter revision

Topic test 12

1 Write an algebraic expression for the each of following diagrams. Let m stand for the number of marbles in each cup. ( stands for one marble.) a b c

Exercise 12-01

2 Simplify each of these expressions. a 5×a×2 b 8×c×b×3×a

Exercise 12-02

3 Write each of these in expanded form. a 5m b 4mnk

c 2 × m × 4 × w × 10 × w Exercise 12-02

c 6ab

d abc

4 Write a mathematical expression for each of these. Use N to represent any number. a 3 times the number b the difference between the number and 5 c the next consecutive number d one-third of the number

Exercise 12-03

5 If A, B and C represent any three numbers, write expressions for: a the sum of A and C b the product of all three numbers c the difference between B and C, where C is greater than B

Exercise 12-03

6 Write expressions for: a the sum of M and 3

Exercise 12-03

b 5 more than B

7 Simplify these expressions. a 5p + 3p b 3y + 2y − y

c 2H decreased by k Exercise 12-04

c 7d − 4d + 3e − 3e

d −7x + 2x − 3y − y

8 Write the like terms in each of these sets of terms. a 8ab, 3x, 2a, 2g, 3a b 3mn, 7abc, 8nm, 3bc, 2a c 2xy, yx, y2, 4yx, y2x d 3a2b, 2ab, 4a2b, ba2, b2a

Exercise 12-05

9 Simplify each of these expressions by adding or subtracting like terms. a 5a + 3a + 2a b 4mn + 2nm c abc + 3abc − 2bca d 8w − 12 + 12w + 20 e 6a − 3b + 5a − 8b f 7a + 4b − 9a − 6b 2 2 g −2x + 7y − 5y − 3x h 9a + 6a − 5a − 3a i yz + xy − yz + xy

Exercise 12-05

10 Simplify the following. a 3b × 5d

Exercise 12-06

b −2h × 6n

c a × (−10ab)

11 Expand the following. a 7(e + 4) b 5(k − 8) e −3(j + 2) f −2(t − 4)

Exercise 12-07

c y(2 − y) g 9a(a − 2)

12 Expand and simplify the following. a 5w + 2(w + 3) b 6(h − 1) + 12 d 14 − (a + 3) e 3(y + 4) − 2(y + 5) g 5(m − 3) − 3(m − 8) h 24c − 4(c − 2)

d m(m + 2) h 2k(5 − k) Exercise 12-08

c 7(x + 2) + 3(x + 1) f 10(d + 2) − 5d + 5 i 3(2p + 5) + 5(3p − 2)

13 Find the value of these expressions if a = 2, b = −3, c = 5 and d = 6. a 2a + 2b − c b 5d + 3a − 2c c d 2 − c2 d cd ÷ a

Exercise 12-09

14 Evaluate each expression by substituting the values given. Values

2a + 3b

3a

4b

Exercise 12-09

2ab

5a − 2b

a a = 3 and b = 4 b a = 5 and b = −1

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