Chapter12 Algebra

12

Patterns and algebra

Algebra

Scientists put a satellite into space; a business person orders materials for a factory; extra dinner guests mean that ingredients in a recipe must be increased; and engineers build a bridge. To understand these ideas, formulas and patterns, you must know how algebra works.

In this chapter you will: ■ ■ ■ ■ ■ ■ ■

use concrete materials, such as cups and counters, to model variables and algebraic expressions translate from words to algebraic symbols use algebraic abbreviations and recognise equivalent algebraic expressions add and subtract like terms to simplify algebraic expressions recognise the role of grouping symbols, and expand expressions involving them simplify algebraic expressions involving addition, subtraction, multiplication and expanding substitute into algebraic expressions.

Wordbank ■ ■ ■ ■ ■ ■

algebraic expression An expression that describes a quantity, using variables and numerals, for example 4m + 2. like terms Terms with exactly the same variables, for example 6ab and 3ab. simplify (an algebraic expression) To write an algebraic expression in the shortest way. expand To rewrite an algebraic expression without grouping symbols. substitute To replace a variable with a numeral. evaluate To find the value of an algebraic expression after substitution.

Think! Can you think of a quick way of calculating 7 × 102? ■ What about 8 × 99? ■ What about 6 × (2a + 5)? ■

ALGEBRA

389

CHAPTER 12

Start up Worksheet 12-01 Brainstarters 12

1 If = 8, find the value of: +3 a + d g ÷2 j 2× +5

b e h k

2 If = 20, find the value of: a −5 +9 d g 60 − j 6+2×

b 3× e × h 25 − + 6 k 100 ÷ − 5

c ÷4 f 2× i + 11 − 9 l − 10 × 2

3 Find the value of: a 3 × (−4) d −7 + 5 g −4 × (−5)

b −2 × 5 e −4 − 10 h −1 × (−6)

c 8 × (−2) f 3 − (−9) i 2 + (−3)

4+ 3× 32 ÷ 11 −

+1

c −6 f ×7 i 9 + − 12 l ÷8−3

Algebraic 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 of paperclips. Find an expression to represent the number of paperclips in this diagram:

?

Solution Let y stand for the number of paperclips in the envelope. So: 1 envelope of paperclips + 3 paperclips = y + 3 paperclips

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

Solution

k marbles

k marbles

3 cups of marbles + 5 marbles = 3 lots of k + 5 =3×k+5 = 3k + 5

390

NEW CENTURY MATHS 7

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 on the previous page.

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

m coins

p coins

p coins

Solution m + 2p

Exercise 12-01 1 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

Example 1

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

Example 2

a

b

c

d

e

f

g

h

i

j

ALGEBRA

391

CHAPTER 12

3 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 an envelope.

Example 3

a

b

c

d

e

f

g

h

i

j

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

b

c

d

e

f

g

h

i

j

k

l

m

n

5 a Which of the expressions you wrote for Question 4 mean the same amount? b Which expression in each group you listed do you think is the simplest? Why?

392

NEW CENTURY MATHS 7

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 is a a simpler way of writing 2 × m, and --- is a simpler version of a ÷ 2. 2 In more complicated expressions, such as 2m + 3w − 4k, we call each part of the expression a term. algebraic expressions

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 c 5×B×A e 3×m×2×w g m×m×m

b d f h

m×w 8×a×n×b×m k×k 2×y×y

Solution Simplifying an algebraic expression means writing the expression in the shortest way. a 3 × m = 3m (or wm since multiplication can be done in any order) b m × w = mw c 5 × B × A = 5AB (or 5BA) Note that we always write the number at the start of the term, usually 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 ALGEBRA

393

CHAPTER 12

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 m 7m2 + 3n − 2m2 + 6n n 12cde − 5def o j2 p 12f − 4g + 3f + 7g + 10f − 9g Example 4

CAS 12-01 Simplifying expressions

2 Simplify each of the following expressions: a 5+5+5+5 c a+a+a+a+a+a e w+w g 8h + 3h + 2h + h i 23k + 17k k 7p + 5p − 2p m a+a+a+b+b o m + 2m − 3m

b d f h j l n p

2+2+2+2+2+2+2+2+2 e+e+e+e 5m + 2m + 6m 5m − 2m 12w − 4w − 5w 6n + 3n − 10n 3d − 2d + 5d 12q − 4q + q

3 Simplify: a 4×w c 3×c×a×d e 2×w×3×h g 2×w×c×3×m i the product of m and w k a×a m 4×f×f o 4×h×h×h q 2×m×4×m s c×c×e×e×e u y×z×y×z×y

b d f h j l n p r t v

8×a×b a×c×b×5 g×2×f×3 2×m×s×4×h×b 2×y×3×r×q×5×p×d d×d×d g×2×g l×4×l×2 6×r×r×6×r 5×j×7×j×k 4×n×3×p×n×2×p×n

4 Computer Algebra Software can be used to simplify expressions that you make up yourself. Use this link to go to an activity that will show you how to use TI InterActive! to simplify algebraic expressions. 5 What confusion would be caused by removing the multiplication sign from 5 × 3? 6 What does the expression AB mean?

394

NEW CENTURY MATHS 7

7 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

Example 5

d 3mnabcdef h q3 l 9m3n2

8 What does the word ‘abbreviation’ mean? Use it in a sentence to show its meaning. Is this word used the same way in mathematics?

From words to algebraic expressions 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 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

Worksheet 12-02

Skillsheet 12-01 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

Solution

a m+5 c m + 10 e 2m

b the double of m d the difference between N and Q f the quotient of m and 2 b 2 × m or 2m d N−Q 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 ALGEBRA

Example 6

395

CHAPTER 12

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? Example 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 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.

5 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 increased by 2

396

NEW CENTURY MATHS 7

b d f h j

3 less than B 8 increased by D X decreased by F m increased by m A decreased by B.

Like terms Earlier we did some concrete exercises involving algebra. In Example 3 and Exercise 12-01, we used p to represent the number of coins in each cup of coins in each envelope

and m to represent the number

.

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

+ p +

m

m

=

b

2p

+

2m

and + p +

m

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 in the right column. a

A + p +

m

m

+

2m

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 ALGEBRA

397

CHAPTER 12

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

−

a + p

m

A

−

m

2m

−

b +

m

B

−

4

2

−

c + p

2m

p

2p

− +

2p

D

−

6

4

p

−

e +

2m

E

−

2p

2m

f +

2m

F

−

3m

+

2p

− 2p

2

C

−

d

+

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: a p +

m

+ p

−

b +

m

−

2p

p

c

+

− + p

3m

−

+

m

2p

d 2p

+

2

+

p +

e

− +

2m

p +

m

+

−

4

f

398

3

+ 2

NEW CENTURY MATHS 7

+

− −

m

p +

2

− m

−

1

g

− m

+

2p

+

−

3

2p

−

h +

3p

−

2m

− −

m

p

−

i p +

m

+ p +

m

+

4

−

−

j +

−

4p

4 Simplify these expressions: a 3m + 2m c m+m−m e 3m + 4p + m + 2p g 6m + 4p − 2m i 12m + 6p − 3m − 2p

b d f h j

3

−

− 3m

−

2m

m

−

2p

6p − p 2m + 3p + 2p 2m + p + m + 3p 7p + 5m − 3m 6p − 4p + 3m − m

Simplifying algebraic expressions with like terms The following pairs of terms are called ‘like terms’ because each set of terms has the same pronumeral(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 or pronumerals. 3x and 5m 8wm and 2wq 5ab and 2abc 2p and 3p2

Worksheet 12-03 Collecting like terms Skillsheet 12-02 Algebra using diagrams

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

Example 8 Simplify these expressions: a 3m + 5m 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 d 42mw − 17wm b 3ab + 2ba = 5ab (ab and ba are like terms) d 42mw − 17wm = 25mw (mw and wm are like terms) ALGEBRA

399

CHAPTER 12

Example 9 Simplify these expressions by adding or subtracting like terms: b 24 + 5ab − 14 − 2ba a 3m + 5w + 2m + 3w

Solution a 3m + 5w + 2m + 3w like terms 5w + 3w

+ 5w

3m

+ 2m

+ 3w

= 5m + 8w

− 14

− 2ba

= 10 + 3ab

like terms 3m + 2m

b 24 + 5ab − 14 − 2ba like terms 24 − 14

24

+ 5ab

like terms 5ab − 2ba

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 Example 8

SkillBuilder 8-06 Like terms

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 Simplify each of these expressions by adding like terms: b 4k + 7k a 2m + 5m 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 3 Simplify each of these expressions by subtracting like terms: b 8d − 3d a 5m − 3m c 12mk − 7mk d 45abc − 12abc e 45fg − 13fg f 48mn − 29mn g 30cd − 12cd h 12xy − 7yx − xy i 8de − 12de j 6k2 − 3k2 2 2 k 4w − w l 5xy2 − 2xy2 − xy2

400

NEW CENTURY MATHS 7

4 Simplify each of these by adding or subtracting like terms: b 12a − 4a − 5a a 5m + 2m − 3m d 12f + 15f − 18f c 3x + 5x + 7x f 5mn + 6mn − 4nm e 8s2 + 7s2 − 6s2 h 11abc + 3abc − 5abc g 3pq − 2pq + pq j 3d 2 − 8d 2 + 2d 2 i 8p − 3p − 5p k − 4x + 7x − 3x l 8de + 12de − 5de + de Example 9

5 Simplify: a 3x + 4 + 5x + 6 c 2mn + 3f + 5mn + 7f e 23xy + 23ab + 17xy − 17ab g 2p + 3 − 3p + 8 i 13 + 2ab − 6ab − 1 k 3p + 12m − 6p − 4m m 4y2 + 2y + y2 + 5y o 9g2 − 12 + 2g2 + 7

b d f h j l n p

4m − 6 + 4m + 10 10k − 4 − 6k + 12 15r + 15 − 15r − r 5k − 11 − 8k + 6 5ab + 2p − 7ab − 2p −10de + 3d + 5de − 2d 7m2 + 2n2 − 4m2 + 3n2 x2 − 3x + 4x2 − 7x

6 Simplify: a 6m + 3m c 12mx + 4xm e 2f + 3g + 4f + 7g g 11p − p i 8abc − 4abc − abc k 3a + 2b + 6a + 7b m 2y + 1 + 2y + 2 + y + 3 o 12 + 6x − 4x − 6 q 2m2 + 4m2 − 6 s 20x + 3y2 − 7x − 2x

b d f h j l n p r t

k + 2 + 3k 2d + 9 + 3d 4y + 3x + 2x + 8y 7 + 5q − 3 4k + 5 + 17 + 11k x2 + 3x2 + 9x2 7r + 6s − 2r − 4s 12a − a − 2a − 6a 12pq − pq − 7qp 12ab + 3cd − 4ab + 7cd

7 Computer Algebra Software can simplify expressions by adding and subtracting like terms. The accompanying activity will teach you how to use TI InterActive! to do this.

CAS 12-02 Collecting like terms

Just for the record Brahmagupta Much of our ancient mathematics was passed down to us through the Arabs over one thousand years ago. The Arabs first met these ideas studying Greek and Hindu writings. One of the most famous Indian astronomers and mathematicians was Brahmagupta (AD 598–665). Brahmagupta wrote all his discoveries about mathematics and astronomy in a long poem, called Brahma-sphuta-siddhanta (The Opening of the Universe) in AD 628. It was a very long poem, with 25 chapters. Two chapters were devoted to mathematics (algebra, geometrical proofs, areas of triangles, and volumes of solids). The other 23 chapters dealt with astronomy.

Find out what type of numbers Brahmagupta introduced into mathematics.

ALGEBRA

401

CHAPTER 12

Multiplying algebraic terms Example 10 Simplify each of the following: b 4x × (−2) a 2 × 3a

c 7k × 3k × 2

Solution a 2 × 3a = 2 × 3 × a = 6a b 4x × (−2) = 4 × x × (−2) = 4 × (−2) × x = −8x c 7k × 3k × 2 = 7 × 3 × 2 × k × k = 42k2 When multiplying algebraic terms, multiply the numbers first, then multiply the variables. Variables are usually written in alphabetical order.

Exercise 12-06 Example 10

1 Simplify each of the following: a 3 × 2y c 4t × 3 e 4×2×d g 12c × 4d i 3m × 2m k 12f × 9f m 10 × 2m × 6n o 4x × 7x × 3

b d f h j l n p

7 × 8q 3k × 9 4a × 3b 6p × 9q 5k × 11k 6g × 3 × 4h 4a × 3 × 7a 5m × 2 × 3m

2 Simplify: a 2 × 4a × 9b c x × xy × y e 5rst × 2rs g p×9 i hjk × hj × jk k 2xy × 6xy × 9yx

b d f h j l

2mn × 4 2j × 3k × 7 3ab × 6cd m × n × 2m 4q2 × 6p2 2m × 3n × mn

3 Simplify each of the following: a −6 × 3x c −1 × p e 6 × (−m) g −2 × (−2n) i −1 × (−r) k −s × (−s)

b d f h j l

−2 × 3k −j × 4 4 × (−3y) −k × (−5) −8x × (−7) −3d × (−d)

402

NEW CENTURY MATHS 7

Expanding an expression 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 m

m

Skillsheet 12-02

m

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

b

Expandominoes

Algebra using diagrams

m

=

Worksheet 12-04

=

m

m

m

m

m

m

=

+ 3

6m m

m

m

m

m

m

What is happening? What do the parentheses (brackets) mean? 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 breadth 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.

X

expanding

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

403

CHAPTER 12

Animation 12-01 Expanding brackets

Example 11 Expand each of the following expressions: b 3(x − 4) a 2(x + 3)

c 5(2x − m)

Solution a 2(x + 3) b 3(x − 4) 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: a −2(x − 3)

b −(5 − 4x)

Solution a −2(x − 3) b −(5 − 4x)

−2(x − 3) = −2 × x − 2 × (−3) = −2x + 6 −1(5 − 4x) = −1 × 5 − 1 × (−4x) = −5 + 4x

−(5 − 4x) is the same as −1(5 − 4x)

Example 13 Expand: a m(c + 1)

b f(2m − 3r)

Solution a m(c + 1) 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 Example 11

SkillBuilder 8-10 Using brackets

1 Expand each of the following expressions: a 3(a + 2) b 2(h + 2) e 4(2m + 3) d 4(x + 6) h 5(a + 2) g 12(2p + 5) k 10(4p + 2q) j 12(2m + n)

404

NEW CENTURY MATHS 7

c f i l

2(m + 3) 7(a + b) 6(3x + 4) 3(2a + 4b)

12_NCM7_SB_TXT.fm Page 405 Friday, September 26, 2003 3:42 PM

2 Expand: a 4(x − 2) e 2(m − p) i 6(3x − 4)

b 3(m − 7) f 7(3f − 2g) j 3(1 − k)

c 8(k − 3) g 4(3m − 5) k 6(2 − 3p)

d 5(y − 5) h 5(3m − 3) l 10(2 − 2m)

3 Expand: a −3(x + 2) e − 4(6 − y) i −(6 − 2x)

b −7(p + 1) f −9(a + 4) j −3(4m + 5)

c −2(m − 3) g −(k + 3) k −5(3y − 6)

d −5(k − 4) h −(m − 2) l −(4 − 7x)

4 Expand: a m(n + s) e y(2x − 3) i 2m(3k + 1) m 5w(7x − 2m)

b f j n

x(y − 2) f(4k − 7) 3k(5x − 2) 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: a x(x + 4) e −r(2r − 1) i −3v(v + 7) m mn(2m + 3n)

b f j n

a(2a + 5) p(p + q) pq(p + q) −(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)

Example 12

Example 13

6 Computer Algebra Software can simplify expressions by expanding brackets. This link will take you to an activity that teaches you how to use TI InterActive! to do this.

CAS 12-03 Expanding expressions

Expanding and simplifying 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

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

Example 15 Expand and simplify: a 2(m + 1) + 3(m − 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) b 6(p + 2) − 3(p − 4) = 6 × p + 6 × 2 − 3 × p − 3 × (− 4) = 6p + 12 − 3p + 12 = 3p + 24

ALGEBRA

405

CHAPTER 12

Exercise 12-08 Example 14

1 Expand each of the following and simplify by collecting like terms: b 7(a + 2) + 12 a 3p + 2(p + q) d 12 + 2(n − 2) c 3(k − 5) + 7 f 12(v + 4) + 4v e 6k + 11(k + 2) h 5a − 2(a + 1) g 16(t − 2) − 8t j 13x − 3(x + 1) i 7x − 3(x + 2) l a − (2 + a) k 2(x + 1) − 5 n 3(x + 5) − 2x + 7 m b − (b − 4) p 5(3 − 2x) + 5x − 3 o 3(4 − 2x) − 10

Example 15

2 Expand and simplify: 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

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)

3 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

b 6x + (4 − 5x) = 6x + 4 − 5x =x+4

c 2x − (8 − x) = 2x − 8 + x = 3x − 8

d 4m − (3 + 2m) = 4m − 3 − 2m = 2m − 3

e 5x − (4 − 2x) + 3x = 5x − 4 + 2x + 3x = 10x − 4

f

g 2p − 3 − (3 − 4p) = 2p − 3 − 3 − 4p = −2p − 6

h 9x − 1 + (3x − 5) = 9x − 1 − 3x + 5 = 6x + 4

i

406

10 − (3x − 2) + (5x − 3) = 10 − 3x − 2 + 5x − 3 = 5 + 2x

NEW CENTURY MATHS 7

j

8 + (4x + 5) − 2x = 8 + 4x + 5 − 2x = 2x + 13

8x − (2x − 5) − (4x + 4) = 8x − 2x + 5 − 4x − 4 = 2x + 1

Skillbank 12

SkillTest 12-01 Multiplying by 9, 11, 99 or 101

Multiplying by 9, 11, 99 or 101 We can use expanding when multiplying by a number near 10 or near 100. 1 Examine these examples: a 25 × 11 = 25 × (10 + 1) = 25 × 10 + 25 × 1 = 250 + 25 = 275 c 32 × 12 = 32 × (10 + 2) = 32 × 10 + 32 × 2 = 320 + 64 = 384 e 27 × 101 = 27 × (100 + 1) = 27 × 100 + 27 × 1 = 2700 + 27 = 2727

b 14 × 9 = 14 × (10 − 1) = 14 × 10 + 14 × −1 = 140 − 14 = 126 d 7 × 99 = 7 × (100 − 1) = 7 × 100 + 7 × −1 = 700 − 7 = 693 f 18 × 8 = 18 × (10 − 2) = 18 × 10 + 18 × (−2) = 180 − 36 = 144

2 Now simplify these: a 16 × 11 c 29 × 9 e 62 × 11 g 18 × 101 i 19 × 8 k 21 × 102

b d f h j l

33 × 11 45 × 9 7 × 101 36 × 99 45 × 12 6 × 98

Algebraic 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.

Worksheet 12-05 Substitution

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

Worksheet 12-06 Formula 1 game

‘Evaluate’ means to find the value of.

Example 16 Evaluate k − 9 when k = 15.

Solution

k − 9 = 15 − 9 =6

ALGEBRA

407

CHAPTER 12

Animation 12-02 Substitution

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 Substitute 5 for m in the expression and substitute 7 for n. m(n − 3) = 5 × (7 − 3) = 5 × (4) (always do brackets first) = 20

Exercise 12-09 Example 16

1 Evaluate k + 3 when: b k = 18 a k=2

c k =119

d k = −21

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

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: b k = 10 a k = −3

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

408

NEW CENTURY MATHS 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 Evaluate these expressions if a = 0, b = 2, c = 5, d = 10, e = 16. b 3d + c a 2a + b d c+d+e c 4d − e f 3e + 2c − 3d e 4b − a − c h 5a + 2b − 3c + 4d g 4b − 3d + 8c j b(2c + 3d) i 3e − 2d + 6c − 400a l bc + de k 2d(19a + 3c − d) 9 Computer Algebra Software can simplify expressions by substituting numerical values for the variables. This link will take you to an activity that uses TI InterActive! to do this.

Substituting into an expression

10 Copy and complete this table using the different substitution values given: Values

2x + y

xy

8x

3x − 2y

37 − 4y

a

x = 3 and y = 4

10

12

24

1

21

b

x = 4 and y = 0

c

x = −2 and y = −1

d

x = 5 and y = 6

e

x=

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

1 --2

3 --4

and y =

and y =

CAS 12-04

1 --2

2 --5

j

x=

k

x = 3 and y = −4

l

x = 0.7 and y = 0.2

11 Use this table of values to evaluate the following expressions:

a c e g i k m o

a

b

c

d

x

y

z

K

P

M

7

3

−2

8

7

0

−20

1

100

0

x+y 4d + 3c z + 3y 5 × (x + c) ab + Kz zd − P Ka + Pab z2

b d f h j l n p

K+P+b P − 5a 2K + 3K + x 2 × (a − x) 7Pxy db − xy y2 a2 − c2

ALGEBRA

409

CHAPTER 12

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. b N×0=0 a N×1=N d a+b+c=b+a+c c a−b≠b−a f ab = ba e N−0=N 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? b N÷N=1 a a÷b=b÷a d a is a factor of a c 4a − a = 4 f 1--2- N = N − 2 e If N is even, then N + 3 is odd g a + (−a) = 2a i 0÷N=0

h 1 is a factor of a j N÷1=N

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?

410

NEW CENTURY MATHS 7

Power plus 1 Simplify: 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: 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: a 2x + 6 b 6m + 12p c x2 − x d 7 e x2 + y2 f 2k2 + 4k 4 Evaluate each of the following algebraic expressions, if p = 4, q = −6, r = 3 and t = −1: a p+q b pq c r2 + t2 pq d -----e q2 − 6p + rt f 6 − 2r + p2 rt p+4 q+r g 2p + 3q + 4r − 5t h -----------i ----------2 p+t r+7 q–t p+3 r–2 k ------------ + ----------l q + pr − qt j ----------- − ---------5 5 10 2 5 Write, as an algebraic expression, the perimeter and the area of each of these shapes: l y 2 a b c x

b

q x

m

d

r

c

e

y

5

n

p

f b

m+3

d 2

6 If x is any number, simplify these expressions: a x+0 b 1×x d x÷x e x−x g 0×x h x−0

a

c 0÷x f x − (−x) i (−x)2

ALGEBRA

411

CHAPTER 12

Language of maths abbreviation evaluate grouping symbols substitute

Worksheet 12-07 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 Distinguish between simplifying an expression and 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

412

NEW CENTURY MATHS 7

Chapter 12

Review

Topic test Chapter 12

1 Simplify each of these expressions: a 5×a×2 b 8×c×b×3×a 2 Write in expanded form: a 5m b 4mnk

Ex 12-02

c 2 × m × 4 × w × 10 × w Ex 12-02

c 6ab

d abc

3 Write a mathematical expression for each of the following. 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

Ex 12-03

4 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

Ex 12-03

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

Ex 12-03

b 5 more than B

c 2H decreased by k

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

Ex 12-05

7 Simplify each of these expressions by adding or subtracting like terms: b 4mn + 2nm c abc + 3abc − 2bca a 5a + 3a + 2a d 17a − 3a e 9ab − 3ba f 10fg − 4fg + 3fg g 3k + 4k − 2k h 12mn + 3mn − 5mn i 5x + 12 + 9x − 5 j 8w − 12 + 12w + 20 k 6a − 3b + 5a − 8b l 7a + 4b − 9a − 6b 2 2 m −2x + 7y − 5y − 3x n 9a + 6a − 5a − 3a o yz + xy − yz + xy

Ex 12-05

8 Simplify: a 3b × 5d

Ex 12-06

b −2h × 6n

c a × (−10ab)

9 Expand: a 7(e + 4) d m(m + 2) g 9a(a − 2)

b 5(k − 8) e −3(j + 2) h 2k(5 − k)

c y(2 − y) f −2(t − 4) i 3m(m − 5)

b 6(h − 1) + 12 e 3(y + 4) − 2(y + 5) h 24c − 4(c − 2)

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

10 Expand and simplify: a 5w + 2(w + 3) d 14 − (a + 3) g 5(m − 3) − 3(m − 8)

Ex 12-07

Ex 12-08

11 Find the value of these expressions if a = 2, b = −3, c = 5 and d = 6: a 5a − 2 b 2a + 2b − c c 5d + 3a − 2c d 7a + 4b + d e d 2 − c2 f cd ÷ a

Ex 12-09

12 Evaluate each expression for the different substitution values given: Values

a

2a + 3b

3a

4b

2ab

Ex 12-09

5a − 2b

a = 3 and b = 4

b a = 5 and b = −1 ALGEBRA

413

CHAPTER 12