Chap 05

5

44eTHINKING

Algebra

Ships use the speed of sound in water to help find the water’s depth. A sonar pulse from a ship is sent to the bottom of the ocean floor. The time taken for the pulse to hit the ocean floor and return to the ship is used to calculate the distance. If the sonar pulse returns in 1.5 seconds, what is the ocean depth?

Assume that the speed of sound in water is 1470 metres per second.

How could you set up a procedure to quickly calculate the ocean depth for any time measurement?

This chapter looks at using pronumerals to represent quantities in different situations. You will learn how to form and use algebraic expressions and how to express them in simpler forms.

198 M a t h s Q u e s t 8 f o r V i c t o r i a

READY?areyou

Are you ready?Try the questions below. If you have difficulty with any of them, extra help can be

obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon

next to the question on the Maths Quest 8 CD-ROM or ask your teacher for a copy.

Alternative expressions used to describe the four operations

1 Write expressions for the following:

a the difference between M and C

b the amount of money earned by selling B lamingtons for $2 each

c the product of X and Y

d 12 more than H

e the cost of 10 oranges if each orange costs D cents.

Order of operations II

2 Find the value of each of the following using the order of operations rule.

a 12 + 18 ÷ 6 b 14 − 16 × 4 ÷ 8 c 7 × 3 − 5 × 12

Order of operations with brackets

3 Find the value of each of the following using the order of operations rule.

a 5 + (10 × 2) − 13 b (2 × 12) + (18 ÷ 6) − (2 + 7)

Operations with directed numbers

4 Perform each of the calculations.

a −8 + −10 b 8 − −17 c 26 × −2 d −32 ÷ −4

Combining like terms

5 Simplify the following expressions by adding or subtracting like terms.

a 3g + 4g b y + 2y + 3y c 6gy − 3yg

d 20x − 19x + 11 e 7g + 8g + 8 + 9 f 7h + 4t − 3h

Simplifying fractions

6 Simplify the following.

a b c

Highest common factor

7 Find the highest common factor for each of the following pairs of numbers.

a 8, 28 b 21, 35 c 18, 27

5.1

5.2

5.3

5.4

5.5

5.6

7

21------

12

30------

15

20------

5.7

C h a p t e r 5 A l g e b r a 199

Using pronumeralsThe basic purpose of algebra is to solve mathematical

problems involving an unknown. Equations where an

unknown quantity is replaced with a letter (for

example, x) can be used to solve problems like:

At what speed should I ride my bicycle to arrive at

school on time?

How do I convert a recipe for different numbers of

guests?

What volume of concrete is needed to build a path?

A pronumeral is a letter that is used in place of a

number. In Year 7 we saw that pronumerals could be

used to make expressions and equations. Often a pro-

numeral is used to represent one particular number. For

example, in the equation

x + 1 = 7

the pronumeral x has the value 6.

Pronumerals can also be used to show a relationship

between two or more numbers, for example.

a + b = 10

Can you find some different pairs of values for a and b

that fit this rule?

Algebra allows us to show complex rules in a more

simple way and to solve problems involving

unknown numbers.

In algebraic expressions pronumerals are known as variables because the value of

the pronumeral may be varied to represent any number. With equations, pronumerals

are referred to as unknowns because the pronumeral represents a specific value that is

not yet known.

Worked example 1 shows some of the ways pronumerals can be used.

Suppose we use b to represent the number of ants in a nest.

a Write an expression for the number of ants in the nest if 25 ants died.

b Write an expression for the number of ants in the nest if the original ant population

doubled.

c Write an expression for the number of ants in the nest if the original population increased

by 50.

d What would it mean if we said that a nearby nest contained b + 100 ants?

e What would it mean if we said that another nest contained b − 1000 ants?

f Another nest in very poor soil contains ants. How much smaller than the original is

this nest?

Continued over page

b

2---

1WORKEDExample


Using pronumerals

1 Suppose we use x to represent the number of ants in a nest.

a Write an expression for the number of ants in the nest if 420 ants were born.

b Write an expression for the number of ants in the nest if the original ant population

tripled.

c Write an expression for the number of ants in the nest if the original ant population

decreased by 130.

d What would it mean if we said that a nearby nest contained x + 60 ants?

e What would it mean if we said that a nearby nest contained x − 90 ants?

f Another nest in very poor soil contains ants. How much smaller than the original

is this nest?

THINK WRITE

a The original number of ants (b) must be

reduced by 25.

a b − 25

b The original number of ants (b) must be

multiplied by 2. It is not necessary to

show the × sign.

b 2b

c 50 must be added to the original number

of ants (b).

c b + 50

d This expression tells us that the nearby

nest has 100 more ants.

d The nearby nest has 100 more ants.

e This expression tells us that the nest has

1000 fewer ants.

e This nest has 1000 fewer ants.

f The expression means b ÷ 2, so this

nest is half the size of the original nest.

f This nest is half the size of the original nest.b

2---

1. A pronumeral is a letter that is used in place of a number.

2. Pronumerals may represent a single number, or they may be used to show a

relationship between two or more numbers.

remember

5AWORKED

Example

1

x

4---


2 Suppose x people are in attendance at the start of a football match.

a If a further y people arrive during the first quarter, write an expression for the

number of people at the ground.

b Write an expression for the number of people at the ground if a further 260 people

arrive prior to the second quarter commencing.

c At half-time 170 people leave. Write an expression for the number of people at the

ground after they have left.

d In the final quarter a further 350 people leave. Write an expression for the number

of people at the ground after they have left.

3 The canteen manager at Browning Industries orders m Danish pastries each day. Write

a paragraph which could explain the table below:

4 Imagine that your cutlery drawer contains a knives, b forks and c spoons.

a Write an expression for the total number of knives and forks you have.

b Write an expression for the total number of items in the drawer.

c You put 4 more forks in the drawer. Write an expression for the number of forks

now.

d Write an expression for the number of knives in the drawer after 6 knives are

removed.

5 If y represents a certain number, write expressions for the following numbers.

a A number 7 more than y

b A number 8 less than y

c A number that is equal to five times y

d The number formed when y is subtracted from 14

e The number formed when y is divided by 3

f The number formed when y is multiplied by 8 and 3 is added to the result

6 Using a and b to represent numbers, write expressions for:

a the sum of a and b

b the difference between a and b

c three times a subtracted from two times b

d the product of a and b

e twice the product of a and b

f the sum of 3a and 7b

g a multiplied by itself

h a multiplied by itself and the result divided by 5.

Time Number of Danish pastries

9.00 am m

9.15 am m – 1

10.45 am m – 12

12.30 pm m – 12

1.00 pm m – 30

5.30 pm m – 30

SkillSHEET

5.1

Alternativeexpressions

used todescribethe four

operations


7 If tickets to a basketball match cost $27 for adults and $14 for children, write an

expression for the cost of:

a y adult tickets b d child tickets c r adult and h child tickets.

8 Naomi is now t years old.

a Write an expression for her age in 2 years’ time.

b Write an expression for Steve’s age if he is g years older than Naomi.

c How old was Naomi 5 years ago?

d Naomi’s father is twice her age. How old is he?

9 James is travelling into town one particular evening and observes that there are t

passengers in his carriage. He continues to take note of the number of people in his

carriage each time the train departs from a station, which occurs every 3 minutes. The

table below shows the number of passengers.

a Write a paragraph explaining what happened.

b When did passengers first begin to alight the train?

Time Number of passengers

7.10 pm t

7.13 pm 2t

7.16 pm 2t + 12

7.19 pm 4t + 12

7.22 pm 4t + 7

7.25 pm t

7.28 pm t + 1

7.31 pm t − 8

7.34 pm t − 12


c At what time did the carriage have the most number of passengers?

d At what time did the carriage have the least number of passengers?

10 A microbiologist places m bacteria onto an agar plate. She counts the number of

bacteria at approximately 3 hour intervals. The results are shown in the table below:

a Explain what happens to the number of

bacteria in the first 5 intervals.

b What might be causing the number of

bacteria to increase in this way?

c What is different about the last bacteria

count?

d What may have happened to cause this?

11 n represents an even number:

a Is the number n + 1 odd or even?

b Is 3n odd or even?

c Write expressions for:

i the next three even numbers that are greater than n

ii the even number that is 2 less than n.

TimeNumber of

bacteria

9.00 am m

12 noon 2m

3.18 pm 4m

6.20 pm 8m

9.05 pm 16m

12 midnight 32m – 1240

MA

TH

S Q

UEST

CHALL

EN

GE

CHALL

EN

GE

MA

TH

S Q

UEST

1 Licia has bought her lunch from the school canteen for $3.00. It con-sisted of a roll, a carton of milk and a piece of fruit. She paid 60 centsmore for the milk than the fruit and 30 cents more for the roll than themilk. How much did the roll cost her?

2 Find at least two 2-digit numbers that are equal to 7 times the sum of their digits.

3 Find 5 consecutive numbers that add to 120.

4 I’m thinking of a number. If I multiply it by 5 and subtract 4, I get the same number as when I multiply it by 4 and add 2. What is the number?

5

If this pattern continues, how manycubes will it take to make 10 layers?


SubstitutionWhen a pronumeral is replaced by a number, we say that the number is substituted for

the pronumeral. If the value of the pronumeral (or pronumerals) is known, it is possible

to evaluate (work out the value of) an expression by using substitution — that is,

replace the pronumeral by a number.

For example, if we know that x = 2 and y = 3, the expression x + y can be evaluated

as shown:

x + y = 2 + 3

= 5

When writing expressions with pronumerals, it is important to remember the

following points:

1. The multiplication sign is omitted.

For example: 8n means ‘8 × n’ and 12ab means ‘12 × a × b’.

2. The division sign is rarely used.

For example, y ÷ 6 is shown as .

When substituting pronumerals, replace the multiplication signs, as shown in the

worked example that follows.

y

6---

Find the value of the following expressions if a = 3 and b = 15.

a 6a b

THINK WRITE

a Substitute the pronumeral (a) with

its correct value and replace the

multiplication sign.

a 6a = 6 × 3

Evaluate and write the answer. = 18

b Substitute each pronumeral with its

correct value and replace the

multiplication signs.

b =

Perform the first multiplication. =

Perform the next multiplication. =

Perform the division. = 21 − 10

Perform the subtraction and write

the answer.

= 11

7a2b

3------–

1

2

1 7a2b

3------– 7 3

2 15×

3---------------–×

2 212 15×

3---------------–

3 2130

3------–

4

5

2WORKEDExample


The same methods are used when substituting into a formula or rule.

Substitution

1 Find the value of the following expressions, if a = 2 and b = 5.

a 3a b 7a c 6b d

e a + 7 f b − 4 g a + b h b − a

i j 3a + 9 k 2a + 3b l

m n ab o 2ab p 7b − 30

q 6b − 4a r s + t –

The formula for finding the area (A) of a rectangle of

length l and width w is A = l × w. Use this formula to

find the area of the rectangle at right.

THINK WRITE

Write the formula. A = l × w

Substitute each pronumeral with its

correct value.

= 270 × 32

Perform the multiplication and state the

correct units.

= 8640 m2

1

2

3

3WORKEDExample

270 m

32 m

1. Replacing a pronumeral with a number is called substitution.

2. When writing expressions with pronumerals, it is important to remember the

following points:

(a) The multiplication sign is omitted.

For example: 8n means ‘8 × n’ and 12ab means ‘12 × a × b’.

(b) The division sign is rarely used.

For example, y ÷ 6 is shown as . y

6---

remember

5BWORKED

Example

2

SkillSHEET

5.2

Order ofoperations II

a

2---

5b

5---+

8

a---

25

b------

ab

5------

15

a------

7

b---

9

a---

3

b---


2 Substitute x = 6 and y = 3 into the following expressions and evaluate.

a 6x + 2y b c 3xy d

e f 3x − y g 2.5x h

i 3.2x + 1.7y j 11y − 2x k l

m 4.8x – 3.5y n 8.7y – 3x o 12.3x – 9.6x p –

3 Evaluate the following expressions, if d = 5 and m = 2.

a d + m b m + d c m − d d d − m

e 2m f md g 5dm h

i −3d j −2m k 6m + 5d l

m 25m − 2d n o 4dm − 21 p

4 The formula for

finding the

perimeter (P) of

a rectangle of

length l and

width w is

P = 2l + 2w. Use

this formula to

find the

perimeter of the

rectangular

swimming pool

at right.

5 The formula F = C + 32 is used to convert temperatures measured in degrees Celsius

to an approximate Fahrenheit value. F represents the temperature in degrees Fahrenheit

(°F) and C the temperature in degrees Celsius (°C).

a Find F when C = 100°C.

b Convert 28° Celsius to Fahrenheit.

c Water freezes at 0° Celsius. What is the freezing temperature of water in Fahrenheit?

6 The formula for the perimeter (P) of a square of side length l is P = 4l. Use this formula

to find the perimeter of a square of side length 2.5 cm.

Mat

hcad

Substitution

EXCE

L Spreadsheet

Substitution

EXCE

L Spreadsheet

Substitutiongame

x

3---

y

3---+

24

x------

9

y---–

12

x------ 4 y+ +

7x

2------

13y

3--------- 2x–

4xy

15---------

3x

9------

y

12------

md

10-------

3md

2-----------

7d

15------

15

d------ m–

WORKED

Example

3

9

5---

50 m

25 m


7 The formula c = 0.1a + 42 is used to calculate the

cost in dollars (c) of renting a car for one day from

Poole’s Car Hire Ltd, where a is the number of

kilometres travelled on that day. Find the cost of

renting a car for one day if the distance travelled is

220 kilometres.

8 The formula D = 0.6T can be used to convert dis-

tances in kilometres (T) to the approximate equiv-

alent in miles (D). Use this rule to convert the

following distances to miles:

a 100 kilometres

b 248 kilometres

c 12.5 kilometres.

9 The area (A) of a rectangle of length l and width w can be found using the formula

A = lw. Find the area of the rectangles below:

a length 12 cm, width 4 cm

b length 200 m, width 42 m

c length 4.3 m, width 104 cm.

Working with bracketsBrackets are grouping symbols. For example, the expression 3(a + 5) can be thought of

as ‘three groups of (a + 5)’, or (a + 5) + (a + 5) + (a + 5).

When substituting into an expression with brackets, remember to place a multipli-

cation (×) sign next to the brackets. For example,

3(t + 2) means 3 × (t + 2).

6(h − 4) means 6 × (h − 4).

g(2 + 3k) means g × (2 + 3k).

(3 + 2k)4 means (3 + 2k) × 4.

(x + y)(6 − 2p) means (x + y) × (6 − 2p).

We evaluate expressions inside brackets first and then multiply by the value

outside the brackets.

GAME time

Algebra— 001

a Substitute r = 4 and s = 5 into the expression 5(s + r) and evaluate.

b Substitute t = 4, x = 3 and y = 5 into the expression 2x(3t − y) and evaluate.

Continued over page

THINK WRITE

a Place the multiplication sign back

into the expression.

a 5(s + r) = 5 × (s + r)

Substitute the pronumerals with their

correct values.

= 5 × (5 + 4)

Evaluate the expression in the pair of

brackets first.

= 5 × 9

Perform the multiplication and write

the answer.

= 45

1

2

3

4

4WORKEDExample


Working with brackets

1 Substitute r = 5 and s = 7 into the following expressions and evaluate.

a 3(r + s) b 2(s − r) c 7(r + s) d 9(s − r)

e s(r + 3) f s(2r − 5) g 3r(r + 1) h rs(3 + s)

i 11r(s − 6) j 2r(s − r) k s(4 + 3r) l 7s(r − 2)

m s(3rs + 7) n 5r(24 − 2s) o 5sr(sr + 3s) p 8r(12 − s)

2 Evaluate each of the expressions below, if x = 3, y = 5 and z = 9.

a xy(z − 3) b c

d (x + y) (z − y) e (z − 3)4x f zy(17 − xy)

g h (8 − y) (z + x) i

j k l 2x(xyz − 105)

m 12(y − 1) (z + 3) n o –2(4x + 1)

p –3(2y – 11)

THINK WRITE

b Place the multiplication signs back

into the expression.

b 2x(3t − y) = 2 × x × (3 × t − y)


correct values.

= 2 × 3 × (3 × 4 − 5)

Perform the multiplication inside the

pair of brackets.

= 2 × 3 × (12 − 5)

Perform the subtraction inside the

pair of brackets.

= 2 × 3 × 7

Perform the multiplication and write

the answer.

= 42

1

2

3

4

5

1. Brackets are grouping symbols.

2. When substituting into an expression with brackets, remember to place a

multiplication (×) sign next to the brackets.

3. Work out the brackets first.

remember

5C

SkillSH

EET 5.3

Order of operations with brackets

WORKED

Example

4

Mat

hcad

Substitution(brackets)

12

x------ z y–( ) z

3---

2y

10------ x 2–+

y

5--- 7 x– 3+( ) 7

12

x------–

4y

6

x--- xz y 3–+( ) y 2+( ) z

x--

3x 7–( ) 27

x------ 7+

36

z------ 3–

z

x-- 8+


3 The formula for the perimeter (P) of a rectangle of

length l and width w is P = 2l + 2w. This rule can

also be written as P = 2(l + w). Use the rule to find

the perimeter of rectangular comic covers with the

following measurements.

a l = 20 cm, w = 11 cm

b l = 27.5 cm, w = 21.4 cm

4

When a = 8 and b = 12 are substituted into the

expression (15 − b + 9), the expression is

equal to

A 32 B 16 C 21 D 24 E 27

5 A rule for finding the sum of the interior

angles in a many-sided figure such as a

pentagon is S = 180(n − 2), where S represents

the sum of the angles inside the figure and

n represents the number of sides. The diagram at

right shows the interior angles in a pentagon.

Use the rule to find the sum of the interior angles for

the following figures:

a a hexagon (6 sides) b a pentagon

c a triangle d a quadrilateral (4 sides)

e a 20-sided figure.

Substituting positive and negative numbers

If the pronumeral you are substituting has a negative value, simply remember the

following rules for directed numbers:

1. For addition and subtraction, signs that occur together can be combined.

Same signs positive for example, 7 + +3 = 7 + 3

and 7 − −3 = 7 + 3

Different signs negative for example, 7 − +3 = 7 − 3

and 7 + −3 = 7 − 3

2. For multiplication and division.

Same signs positive for example, +7 × +3 = +21

and −7 × −3 = +21

Different signs negative for example, +7 × −3 = −21

and −7 × +3 = −21

multiple choice

a

6---

1

3---

21.4 cm

27.5

cm

11 cm

20 c

m


a Substitute m = 5 and n = −3 into the expression m − n and evaluate.

b Substitute m = −2 and n = −1 into the expression 2n − m and evaluate.

c Substitute a = 4 and b = −3 into the expression 5ab − and evaluate.

THINK WRITE

a Substitute the pronumerals with their

correct value.

a m − n = 5 − −3

Combine the two negative signs and add. = 5 + 3

Write the answer. = 8

b Replace the multiplication sign. b 2n − m = 2 × n − m


correct values.

= 2 × −1 − −2

Perform the multiplication. = −2 − −2

Combine the two negative signs and add. = −2 + 2

Write the answer. = 0

c Replace the multiplication signs.c 5ab − = 5 × a × b −


correct values.

= 5 × 4 × −3 −

Perform the multiplications. = −60 −

Perform the division. = −60 − −4

Combine the two negative signs and add. = −60 + 4

Write the answer. = −56

12

b------

1

2

3

1

2

3

4

5

1 12

b------

12

b------

212

3–------

312

3–------

4

5

6

5WORKEDExample

When substituting, if the pronumeral you are replacing has a negative value,

simply remember the rules for directed numbers:

1. For addition and subtraction, signs that occur together can be combined.

Same signs positive for example, 7 + +3 = 7 + 3

and 7 − −3 = 7 + 3

Different signs negative for example, 7 − +3 = 7 − 3

and 7 + −3 = 7 − 3

2. For multiplication and division.

Same signs positive for example, +7 × +3 = +21

and −7 × −3 = +21

Different signs negative for example, +7 × −3 = −21

and −7 × +3 = −21

remember


Substituting positive and negative numbers

1 Substitute m = 6 and n = −3 into the following expressions and evaluate.

a m + n b m − n c n − m d n + m e 3n

f −2m g 2n − m h n + 5 i 2m + n − 4 j 11n + 20

k −5n − m l m n o

p q r 6mn − 1 s t

2 Substitute x = 8 and y = −3 into the following expressions and evaluate.

a 3(x − 2) b x(7 + y) c 5y(x − 7) d 2(3 − y)

e (y + 5)x f xy(7 − x) g (3 + x) (5 + y) h 5(7 − xy)

i j k l

3 Substitute a = −4 and b = −5 into the following expressions and evaluate.

a a + b b a − b c b − 2a d 2ab e 12 − ab

f −2(b − a) g a − b − 4 h 3a(b + 4) i j

k l m 45 + 4ab n 8ab − 3b o

p 2.5b q 11a + 6b r (a − 5)(8 − b) s (9 − a)(b − 3) t 1.5b + 2a

A rule of thumb is a rule or pattern that people use to estimate things. They obtain

this rule by observing a pattern.

1 Write an algebraic expression for each of the following rules of thumb. Explain

what each pronumeral represents in your expressions.

a Your adult height will be twice your height when you were 2.

b To estimate the number of kilometres you are from a thunderstorm,

count the number of seconds between the lightning and the thunder and

divide by 3.

c To approximately convert temperature in degrees Celsius to degrees

Fahrenheit, double it and add 30.

2 Write a question that could be solved for each of the algebraic expressions

found and clearly show how you would solve it.

3 How would you go about verifying the accuracy of these rules of thumb?

4 If the expression for accurately converting temperature in degrees Celsius (C) to

degrees Fahrenheit (F) is F = C + 32, investigate at which temperatures the

rule of thumb expression gives the best results.

5DSkillSHEET

5.4

Operationswith directed

numbers

WORKED

Example

5a

m

2----

mn

9-------

4m

n 5–------------

4m

n-------

12

2n------

9

n---

m

2----+ 3n

2------– 1.5+ 14

mn

9-------–

WORKED

Example

5b

Mathcad

Substitution(positive/negative)

x

2--- 5 y–( ) x

4--- 1–

2y

6------ 4+

9

y--- 6 x–( ) 3 x 1–( )

y

3--- 2+

WORKED

Example

5c

WorkS

HEET 5.1

4

b---

8

a---–

16

4a------–

6b

5------

a

2---

3b

5------+

COMMUNICATION Rules of thumb

9

5---

I’m now in Australia!

a + b =

Stop Stop

Stop

Start

51

74

47

19

86

64

81

66 57

68

37

421

8025

1110

54

77

46

72

6388

10075

35

2796

7

30

913

36

17

32

95

41

34

26

28

1670

12

552

24

62

49

833

22

23

45

99 2

15

3 18

21

55 61

43 38

31

91

40

60

4 48

8439

90

29

50

6

65

4456

14

69

53

8520

cx =

=y—x

bcy =

8x =

y2 =

Start

7x =

5(x + y) =

b + c + x =

y – a + b =

7c + 2x =

3bc =

x – b =

xy =

2y + c =

x(y – 1) =

y – bc =

x(y + b) =

12x – c =

Start

2bx =

x(2y – c) =

11c =

6(b + c) =

Start

y – c =

12(x + c) =

9c =

20x =

10c =

30 + 2b =

b(y + c) =

cy – b =

2x – 3b =

Start

13x =

bc =

9y – x =

3cy =

cy – a =

10(x + b) – 4 =

7x + b =

7bc =

3x + 5 =

xy – a =

6y + b =

2(y + b) =

a + b =

Start

a2 =

=4cy——

x

27——

c

=11(x + c)————

b

Start

11(a + b + c) =

=xy

—–b

7(a + b) =

4by =

11x =

c + 8 =

x(b + c) =

xy – c =

bcb =

b + c =

4(c + y) =

4(x + b) =

8b =

x(y + a + c) =

ax =

y(c + x) =

Stop

Stop

4b + 2c =

Start

20c + y – a =

6x – 4b =

4y – x – b =

bcx + b =

8x + b – a =

4x + 2c =

Stop

Colouring guide:

Join these points with thick lines.

black

2bx – a =

c + x =

7y – cx – b =

a + 9y =

Orange

6c + xy =

7(c + x) =

11y – 4c + a =

2y – c =

Start

6(x + a) =

y + x – b =

3c =

Stop

Start

40b + 2b =

6(x + c) =

y – x – a =

Stop

80 + x + c =

Start

7(a + 4b) =

12x + y + b =

c + 2b + 3a =

Stop

8(c + x) =

Start

9y – 2x + a =

7y + 8b =

12c + 3y =

Stop

12x – 3c =

Start

10x + 8c =

3cx + b =

Stop

Stop

Stop

Stop

Start

10y – x =

= 8y——

b – a =

Start

8y – c =

Stop

9x + a =

Start

20b + c =

8x + 7c =

Stop

7x + c =

cxy + 4c————

c=

5y + b + c————

x=

Join the dots nextto the values of the expressionsin the orders given below using:

a = 1, b = 2, c = 3, x = 5and y = 10.



1 If a kilogram of oranges cost $2.89 and a kilogram of

apples cost $4.99, what is the cost of p kg of oranges

and q kg of apples?

2 If d represents a certain number, write an expression for

the number formed when d is divided by 5.

3 True or false? If y = 4 and z = 1, then .

4 The area of a circle is p × r2, where p = 3.14 and

r = radius of the circle. Find the area of the circle

when r = 0.5 cm.

5 If p = 1, what is the value of q when pq(5p − 2) = 9?

6 Evaluate if r = 4 and s = 6.

7

When m = 7 and n = 4 are substituted into the expression , the value is:

A 21 B 22 C 22.25 D 25 E 28

8 Substitute p = 7 and q = −2 into .

9 From the list −2, 1, 3, 4 choose the value of a and b when .

10 Substitute x = −3 and y = −5 into the expression and evaluate.

Number laws and pronumeralsWhen dealing with any type of number, particular rules must be obeyed. This exercise

will investigate whether these rules also apply to pronumerals.

Commutative Law

The Commutative Law refers to the order in which two numbers may be added, sub-

tracted, multiplied or divided.

1

12

y------ 3z+ 4=

r = 0.5 cm

12

s------ rs 4 s–+( )

multiple choice

3mn

4---+

14

p------ 1–

pq 3+( )

a

2---

b

4---+ 0=

12

x------– 3y+


Find the value of the following expressions if x = 4 and y = 7. Comment on the results

obtained.

a i x + y ii y + x

b i x − y ii y − x

c i x × y ii y × x

d i x ÷ y ii y ÷ x

THINK WRITE

a iii Substitute each pronumeral with

its correct value.

a iii x + y = 4 + 7

Evaluate and write the answer. x + y = 11

iii Substitute each pronumeral with

its correct value.

iii y + x = 7 + 4

Evaluate and write the answer. x + y = 11

Compare the result with the

answer obtained in part a i.

The same result is obtained; therefore,

order is not important when adding two

terms.

b iii Substitute each pronumeral with

its correct value.

b iii x − y = 4 − 7

Evaluate and write the answer. x − y = −3


its correct value.

iii y − x = 7 − 4

Evalute and write the answer. x − y = 3


answer obtained in part b i.

Two different results are obtained;

therefore, order is important when

subtracting two terms.

c iii Substitute each pronumeral with

its correct value.

c iii x × y = 4 × 7

Evaluate and write the answer. x × y = 28


its correct value.

iii y × x = 7 × 4

Evalute and write the answer. x × y = 28


answer obtained in part c i.


order is not important when multiplying

two terms.

d iii Substitute each pronumeral with

its correct value.

d iii x ÷ y = 4 ÷ 7

Evaluate and write the answer. x ÷ y = (≈ 0.57)


its correct value.

iii y ÷ x = 7 ÷ 4

Evaluate and write the answer.x ÷ y = (1.75)


answer obtained in part d i.



dividing two terms.

1

2

1

2

3

1

2

1

2

3

1

2

1

2

3

1

24

7---

1

2 7

4---

3

6WORKEDExample


From worked example 6 we can see that, in general,

1. x + y = y + x

2. x − y ≠ y − x

3. x × y = y × x

4. x ÷ y ≠ y ÷ x

The Commutative Law holds true for addition (and multiplication) because the order in

which two numbers or pronumerals are added (or multiplied) does not affect the result.

It does not hold true for subtraction or division because different results are obtained.

Associative Law

The Associative Law refers to the order in which three numbers may be added, sub-

tracted, multiplied or divided, taking two at a time.

Find the value of the following expressions if x = 12, y = 6 and z = 2. Comment on the

results obtained.

a i x + (y + z) ii (x + y) + z

b i x − (y − z) ii (x − y) − z

c i x × (y × z) ii (x × y) × z

d i x ÷ (y ÷ z) ii (x ÷ y) ÷ z

Continued over page

THINK WRITE

a iii Substitute each pronumeral with

its correct value.

a iii x + (y + z) = 12 + (6 + 2)

Evaluate the expression in the

pair of brackets.

x + (y + z) = 12 + 8

Perform the addition and write

the answer.

x + (y + z) = 20


its correct value.

iii (x + y) + z = (12 + 6) + 2


pair of brackets.

x + (y + z) = 18 + 2

Perform the addition and write

the answer.

x + (y + z) = 20


answer obtained in part a i.


order is not important when adding

3 terms.

b iii Substitute each pronumeral with

its correct value.

b iii x − (y − z) = 12 − (6 − 2)


pair of brackets.

x − (y − z) = 12 − 4


the answer.

x − (y − z) = 8

1

2

3

1

2

3

4

1

2

3

7WORKEDExample


Note the similarities between the Commutative Law and the Associative Law.

From worked example 7 we can see that, in general,

1. x + (y + z) = (x + y) + z

2. x − (y − z ≠ (x − y) − z

3. x × (y × z) = (x × y) × z

4. x ÷ (y ÷ z) ≠ (x ÷ y) ÷ z

THINK WRITE


its correct value.

iii (x − y) − z = (12 − 6) − 2


pair of brackets.

x − (y − z) = 6 − 2


the answer.

x − (y − z) = 4


answer obtained in part b i.



subtracting 3 terms.

c iii Substitute each pronumeral with

its correct value.

c iii x × (y × z) = 12 × (6 × 2)


pair of brackets.

x × (y × z) = 12 × 12

Perform the multiplication and

write the answer.

x × (y × z) = 144


its correct value.

iii (x × y) × z = (12 × 6) × 2


pair of brackets.

x × (y × z) = 72 × 2

Perform the multiplication and

write the answer.

x × (y × z) = 144


answer obtained in part c i.


order is not important when multiplying

3 terms.

d iii Substitute each pronumeral with

its correct value.

d iii x ÷ (y ÷ z) = 12 ÷ (6 ÷ 2)


pair of brackets.

x ÷ (y ÷ z) = 12 ÷ 3

Perform the division and write

the answer.

x ÷ (y ÷ z) = 4


its correct value.

iii (x ÷ y) ÷ z = (12 ÷ 6) ÷ 2


pair of brackets.

x ÷ (y ÷ z) = 2 ÷ 2

Perform the division and write

the answer.

x ÷ (y ÷ z) = 1


answer obtained in part d i.



dividing 3 terms.

1

2

3

4

1

2

3

1

2

3

4

1

2

3

1

2

3

4


The Associative Law holds true for addition (and multiplication) because the order in

which three numbers or pronumerals, taking two at a time, are added (or multiplied)

does not affect the result. It does not hold true for subtraction or division because dif-

ferent results are obtained.

Other laws that hold true for addition and multiplication but not subtraction and div-

ision are the Identity Law and the Inverse Law.

Identity LawThe Identity Law for addition states that when zero is added to any number, the original

number remains unchanged. For example, 5 + 0 = 0 + 5 = 5. Similarly, the Identity Law

for multiplication states that when any number is multiplied by one, the original

number remains unchanged. For example, 3 × 1 = 1 × 3 = 3.

Therefore, in general, x + 0 = 0 + x = x

x × 1 = 1 × x = x

Inverse LawThe Inverse Law for addition states that when a number is added to its opposite, the

result is zero. Similarly, the Inverse Law for multiplication states that when a number is

multiplied by its reciprocal, the result is one.

Therefore, in general, x + −x = −x + x = 0

x × = × x = 11

x---

1

x---

1. When dealing with numbers and pronumerals, particular rules must be obeyed.

2. The Commutative Law holds true for addition (and multiplication) because the

order in which two numbers or pronumerals are added (or multiplied) does not

affect the result. Therefore, in general,

(a) x + y = y + x

(b) x − y ≠ y − x

(c) x × y = y × x

(d) x ÷ y ≠ y ÷ x

3. The Associative Law holds true for addition (and multiplication) because the

order in which three numbers or pronumerals, taking two at a time, are added

(or multiplied) does not affect the result. Therefore, in general,

(a) x + (y + z) = (x + y) + z

(b) x − (y − z ≠ (x − y) − z

(c) x × (y × z) = (x × y) × z

(d) x ÷ (y ÷ z) ≠ (x ÷ y) ÷ z

4. The Identity Law states that, in general, x + 0 = 0 + x = x

x × 1 = 1 × x = x

5. The Inverse Law states that, in general, x + −x = −x + x = 0

x × = × x = 11

x---

1

x---

remember


Number laws and pronumerals

1 Find the value of the following expressions if x = 3 and y = 8. Comment on the results

obtained.

a i x + y ii y + x

b i 3x + 2y ii 2y + 3x

c i 5x + 2y ii 2y + 5x

d i 8x + y ii y + 8x

e i x − y ii y − x

f i 2x − 3y ii 3y − 2x

g i 4x − 5y ii 5y − 4x

h i 3x − y ii y − 3x

2 Find the value of the following expressions if x = −2 and y = 5. Comment on the results

obtained.

a i x × y ii y × x

b i 6x × 3y ii 3y × 6x

c i 4x × y ii y × 4x

d i 7x × 5y ii 5y × 7x

e i x ÷ y ii y ÷ x

f i 10x ÷ 4y ii 4y ÷ 10x

g i 6x ÷ 3y ii 3y ÷ 6x

h i 7x ÷ 9y ii 9y ÷ 7x

3 Indicate whether each of the following is true or false for all values of the pronumerals.

a a + 5b = 5b + a b 6x − 2y = 2y − 6x c 7c + 3d = −3d + 7c

d 5 × 2x × x = 10x2 e 4x × −y = −y × 4x f 4 × 3x × x = 12x × x

g = h −7i − 2j = 2j + 7i i −3y ÷ 4x = 4x ÷ −3y

j −2c + 3d = 3d − 2c k = l 15 × − = × −15

4 Find the value of the following expressions if x = 3, y = 8 and z = 2. Comment on the

results obtained.

a i x + (y + z) ii (x + y) + z

b i 2x + (y + 5z) ii (2x + y) + 5z

c i 6x + (2y + 3z) ii (6x + 2y) + 3z

d i x − (y − z) ii (x − y) − z

e i x − (7y − 9z) ii (x − 7y) − 9z

f i 3x − (8y − 6z) ii (3x − 8y) − 6z

5 Find the value of the following expressions if x = 8, y = 4 and z = −2. Comment on the

results obtained.

a i x × (y × z) ii (x × y) × z

b i x × (−3y × 4z) ii (x × −3y) × 4z

c i 2x × (3y × 4z) ii (2x × 3y) × 4z

d i x ÷ (y ÷ z) ii (x ÷ y) ÷ z

e i x ÷ (2y ÷ 3z) ii (x ÷ 2y) ÷ 3z

f i −x ÷ (5y ÷ 2z) ii (−x ÷ 5y) ÷ 2z

5EWORKED

Example

6a, b

WORKED

Example

6c, d

5 p

3r------

3r

5 p------

0

3s-----

3s

0-----

2x

3------

2x

3------

WORKED

Example

7a, b

WORKED

Example

7c, d



a a − 0 = 0 b a × 1 000 000 = 0 c 15t × − = 1

d 3d × = 1 e ÷ = 1 f = 0

7

The value of the expression x × (−3y × 4z) when x = 4, y = 3 and z = −3 is:

A 108 B −432 C 432 D 112 E −108

8

The value of the expression (x − 8y) − 10z when x = 6, y = 5 and z = −4 is:

A −74 B 74 C −6 D 6 E −36

Simplifying expressionsExpressions can often be written in a more simple form. For example, the expression

3x + 4x can be written more simply as 7x.

Notice that the expression was simplified (put into a more simple form) even though

we did not know the value of the pronumeral (x).

When simplifying expressions, we can collect (add or subtract) only like terms.

Like terms are terms that contain the same pronumeral parts.

For example:

3x and 4x are like terms. 3x and 3y are not like terms.

3ab and 7ab are like terms. 7ab and 8a are not like terms.

2bc and 4cb are like terms. 8a and 3a2 are not like terms.

3g2 and 45g2 are like terms.

1

15t--------

1

3d------

8x

9y------

8x

9y------

11t

0--------

multiple choice

multiple choice

Simplify the following expressions.

a 3a + 5a b 7ab − 3a − 4ab c 2c − 6 + 4c + 15

Continued over page

THINK WRITE

a Write the expression and check that the two

terms are like terms, that is, they contain the

same pronumerals.

a 3a + 5a

Add the like terms and write the answer. = 8a

b Write the expression and check for like

terms.

b 7ab − 3a − 4ab

Rearrange the terms so that the like terms are

together. Remember to keep the correct sign

in front of each term.

= 7ab − 4ab − 3a

Subtract the like terms and write the answer. = 3ab − 3a

1

2

1

2

3

8WORKEDExample


Simplifying expressions

1

Simplifying 3a + 9a gives:

A 12 B 12a C 6a D 12a2 E 6a2

2

Simplifying 6x − 2x gives:

A 4 B 4x2 C 8x D –4x E 4x

3

Simplifying 6a + 6b gives:

A 12ab B 6ab C 36ab

D 12a E The expression cannot be simplified.

4 Simplify the following expressions.

a 4c + 2c b 2c − 5c c 3a + 5a − 4a

d 6q − 5q e −h − 2h f 7x − 5x

g 3a − 7a − 2a h −3f + 7f i 4p − 7p

j −3h + 4h k 11b + 2b + 5b l 7t − 8t + 4t

m 9m + 5m − m n x − 2x o 7z + 13z

p 5p + 3p + 2p q 9g + 12g − 4g r 18b − 4b − 11b

s 13t − 4t + 5t t −11j + 4j u −12l + 2l − 5l

v 13m − 2m − 4m + m w m + 3m − 4m x t + 2t − t + 8t

5 Simplify the following expressions.

a 3x + 7x − 2y b 3x + 4x − 12 c 11 + 5f − 7f

d 3u − 4u + 6 e 2m + 3p + 5m f −3h + 4r − 2h

g 11a − 5b + 6a h 9t − 7 + 5 i 12 − 3g + 5

j 6m + 4m − 3n + n k 5k − 5 + 2k − 7 l 3n − 4 + n − 5

m 2b − 6 − 4b + 18 n 11 − 12h + 9 o 12y − 3y – 7g + 5g − 6

THINK WRITE

c Write the expression and check for

like terms.

c 2c − 6 + 4c + 15

Rearrange the terms so that the like

terms are together. Remember to

keep the correct sign in front of each

term.

= 2c + 4c − 6 + 15

Simplify by collecting like terms and

write the answer.

= 6c + 9

1

2

3

1. When simplifying expressions, we can collect (add or subtract) only like terms.

2. Like terms are terms that contain the same pronumeral parts.

remember

5F

SkillSH

EET 5.5

Combining like terms

multiple choice

Mat

hcad

Simplifying ex-expressions

multiple choice

multiple choice

WORKED

Example

8a

WORKED

Example

8b, c


p 8h − 6 + 3h − 2 q 11s − 6t + 4t − 7s r 2m + 13l − 7m + l

s 3h + 4k − 16h − k + 7 t 13 + 5t − 9t − 8 u 2g + 5 + 5g − 7

v 17f − 3k + 2f − 7k


a x2 + 2x2 b 3y2 + 2y2 c a3 + 3a3

d d2 + 6d2 e 7g2 − 8g2 f 3y3 + 7y3

g 2b2 + 5b2 h 4a2 − 3a2 i g2 − 2g2

j a2 + 4 + 3a2 + 5 k 11x2 − 6 + 12x2 + 6 l 12s2 − 3 + 7 − s2

m 3a2 + 2a + 5a2 + 3a n 11b − 3b2 + 4b2 + 12b o 6t2 − 6g − 5t2 + 2g − 7

p 11g3 + 17 − 3g3 + 5 − g2 q 12ab + 3 + 6ab r 14xy + 3xy − xy − 5xy

s 4fg + 2s − fg + s t 11ab + ab − 5

u 18ab2 – 4ac + 2ab2 – 10ac

Multiplying pronumeralsWhen multiplying pronumerals, remember that order is not important. For example:

3 × 6 = 6 × 3

6 × w = w × 6

a × b = b × a

Also keep in mind that the multiplication sign (×) is usually left out:

3 × g × h = 3gh

2 × x2 × y = 2x2y

Although order is not important, the pronumerals in each term are usually written in

alphabetical order. For example:

2 × b2 × a × c = 2ab2c.

Simplify the following:

a 5 × 4g b −3d × 6ab × 7.

THINK WRITE

a Write the expression and replace the

hidden multiplication signs.

a 5 × 4g

= 5 × 4 × g

Multiply the numbers. = 20 × g

Remove the multiplication sign. = 20g

b Write the expression and replace the

hidden multiplication signs.

b −3d × 6ab × 7

= −3 × d × 6 × a × b × 7

Place the numbers at the front. = −3 × 6 × 7 × d × a × b

Multiply the numbers. = −126 × d × a × b

Remove the multiplication signs and place

the pronumerals in alphabetical order.

= −126abd

1

2

3

1

2

3

4

9WORKEDExample

When multiplying pronumerals:

1. the order is not important. For example, d × e = e × d.

2. place the numbers at the front of the expression and leave out the × sign.

remember


Multiplying pronumerals


a 4 × 3g b 7 × 3h c 4d × 6 d 3z × 5

e 6 × 5r f 5t × 7 g 4 × 3u h 7 × 6p

i 7gy × 3 j 2 × 11ht k 4x × 6g l 10a × 7h

m 9m × 4d n 3c × 5h o 9g × 2x p 2.5t × 5b

q 13m × 12n r 6a × 12d s 2ab × 3c t 4f × 3gh

u 2 × 8w × 3x v 11ab × 3d × 7 w 16xy × 1.5 x 3.5x × 3y

y 11q × 4s × 3 z 4a × 3b × 2c


a 3 × −5f b −6 × −2d c 11a × −3g

d −9t × −3g e −5t × −4dh f 6 × −3st

g −3 × −2w × 7d h −4a × −3b × 2c × e i 11ab × −3f

j 3as × −3b × −2x k −5h × −5t × −3q l 4 × −3w × −2 × 6p

m −7a × 3b × g n 17ab × −3gh o −3.5g × 2h × 7

p 5h × 8j × −k q 75x × 1.5y r 12rt × −3z × 4p

s 2ab × 3c × 5 t −4w × 34x × 3 u –3ab × –5cd × –6ae


a 2a × a b –5p × −5p c –5 × 3x × 2x

d ab × 7a e 3b2 × 2cd f –5xy × 4 × 8x

g 7pq × 3p × 2q h 5m × n × 6nt × –t i –3 × xyz × –3z × –2y

j –7a × –3b × –2c2 k 2mn × –3 × 2n × 0 l w2x × –9z2 × 2xy2

At the start of the chapter, we introduced the situation where a sonar pulse took

1.5 s to travel from the ship to the ocean floor and back again. (The speed of sound

in water is assumed to be 1470 m/s.) Let us look at this problem again.

1 Draw a diagram to show this situation.

2 How far does the sonar pulse travel in:

a 1 second? b 2 seconds? c 1.5 seconds?

3 Calculate the ocean depth when the pulse took 1.5 seconds to return.

4 Write a rule to find the ocean depth for any time measurement. Explain what

each pronumeral represents.

5 Use the rule found in part 4 to calculate the ocean depth for the following pulse-

return time measurements.

a 1.8 seconds b 4.22 seconds c 0.64 seconds

6 The speed of sound in water is about 5 times the speed of sound in air. A person

standing on the deck of the ship sends a sonar pulse through the air to a nearby

cliff face. If the pulse takes 3 seconds to travel to the cliff face and return,

calculate the distance to the cliff face. Write a rule to represent this situation.

5GWORKED

Example

9

THINKING Sonar measurements


Dividing pronumeralsWhen dividing pronumerals, rewrite the expression as a fraction and simplify by cancelling.

Remember that when the same pronumeral appears on both the top and bottom lines

of the fraction, it may be cancelled. Follow the worked examples given below.

a Simplify . b Simplify 15n ÷ 3n.

THINK WRITE

a Write the expression. a =

Simplify the fraction by cancelling

16 with 4 (divide both by 4).

=

No need to write the denominator

since we are dividing by 1.

= 4f

b Write the expression and then

rewrite it as a fraction.b 15n ÷ 3n =

=

Simplify the fraction by cancelling

15 with 3 and n with n.

=

No need to write the denominator

since we are dividing by 1.

= 5

16 f

4----------

116 f

4---------

16 f4

41

-----------

24 f

1------

3

1 15n

3n---------

15n5

3n1

-----------

25

1---

3

10WORKEDExample

Simplify −12xy ÷ 27y.

THINK WRITE

Write the expression and then rewrite it

as a fraction.

−12xy ÷ 27y = −

= −

Simplify the fraction by cancelling 12

with 27 (divide both by 3) and y with y.

=

112xy

27y------------

12xy4

27y9

--------------

24x

9------–

11WORKEDExample

1. When dividing pronumerals, rewrite the expression as a fraction and simplify it

by cancelling.

2. When the same pronumeral appears on both the top and bottom lines of the

fraction, it may be cancelled.

remember


Dividing pronumerals


a b c d 9g ÷ 3

e 10r ÷ 5 f 4x ÷ 2x g 8r ÷ 4r h

i 14q ÷ 21q j k l 50g ÷ 75g

m n 35x ÷ 70x o 24m ÷ 36m p y ÷ 34y

q 27h ÷ 3h r s t 81l ÷ 27l


a b 12cd ÷ 4 c d 24cg ÷ 24

e f g h

i j 55rt ÷ 77t k l 36bc ÷ 27c

m 13xy ÷ x n o 14abc ÷ 7bc p 3gh ÷ 6h

q r s 18adg ÷ 45ag t


a b c 60jk ÷ −5k d −3h ÷ −6dh

e f −12xy ÷ 48y g h

i −4xyz ÷ 6yz j k −5mn ÷ 20n l −14st ÷ −28

m 34ab ÷ −17ab n o p −60mn ÷ 55mnp

q r −72xyz ÷ 28yz s t −

5H

SkillSH

EET 5.6

Simplifying fractions

WORKED

Example

10 8 f

2------

6h

3------

15x

3---------

16m

8m----------

3x

6x------

12h

14h---------

8 f

24 f---------

20d

48d---------

64q

44q---------

15 fg

3------------

8xy

12---------

11xy

11x------------

9 pq

18q----------

21ab

28b------------

9dg

12g---------

5 jk

kj--------

10mxy

35mx----------------

16cd

40cd------------

132mnp

60np--------------------

11ad

66ad------------

bh

7h------

WORKED

Example

11 4a–

8---------

11ab–

33b---------------

32g–

40gl------------

12ab

14ab–---------------

6 fgh

30ghj--------------

rt–

6rt-------

ab–

3a–---------

7dg–

35gh------------

WorkS

HEET 5.2

28def

18d---------------

54 pq

36 pqr----------------

121oc

132oct-----------------


1 If Betty is now x years old, how old was Betty 6 years ago?

2 Find the area of a rectangle with length of 225 cm and width of 1.3 m.

3 Evaluate if p = 4, q = 2 and r = 7.

4

If m = −6 and n = −3 are substituted into the expression , it would have a value

of:

A −2 B −3 C −4 D −5 E –6

5 Simplify 11x − 8y − 9x + 4y − 3.

6 Simplify 10z2 − 5y − 3z2 + 4y + 4.

7 True or false? −6p × −4q × r × 2t = 48pqrt

8 Simplify .

9 From the list −2, −4, 12pq, −48pq, find the missing term to replace ∇ in

.

10 Simplify .

Expanding bracketsWe have seen that the expression 3(a + 5) means

3 × (a + 5) or (a + 5) + (a + 5) + (a + 5).

Simplifying this expression further gives us the expression 3a + 15:

(a + 5) + (a + 5) + (a + 5) = a + a + a + 5 + 5 + 5

= 3a + 15

Look at the pattern below:

With brackets Expanded form

1. 3 × (2 + 1) 3 × 2 + 3 × 1

= 3 × 3 = 6 + 3

= 9 = 9

2. 4 × (3 + 2) 4 × 3 + 4 × 2

= 4 × 5 = 12 + 8

= 20 = 20

2

r 10+( )p

q---

multiple choice

m

2----

6n

9------+

30ab

18abc---------------

∇

12 pr–---------------

4q

r------=

9 p–

36 pq–----------------


Removing brackets from an expression is called expanding the expression. The rule

that we have used to expand the expressions above is called the Distributive Law.

The Distributive Law states: a (b + c) = a ¥ b + a ¥ c

= ab + ac

Expanding and collecting like termsSome expressions can be simplified further by collecting like terms after any brackets

have been expanded.

Use the Distributive Law to expand the following expressions.

a 3(a + 2) b x(x − 5)

THINK WRITE

a Write the expression. a 3(a + 2) = 3(a + 2)

Use the Distributive Law to expand

the brackets.

= 3 × a + 3 × 2

Simplify by multiplying. = 3a + 6

b Write the expression.

Use the Distributive Law to expand

the brackets.

Simplify by multiplying.

b x(x − 5) = x(x − 5)

= x × x + x × −5

= x2 − 5x

1

2

3

1

2

3

12WORKEDExample

Expand the expressions below and then simplify by collecting any like terms.

a 3(x − 5) + 4 b 4(3x + 4) + 7x + 12

c 2x(3y + 3) + 3x(y + 1) d 4x(2x − 1) − 3(2x − 1)

THINK WRITE

a Write the expression. a 3(x − 5) + 4

Expand the brackets. = 3 × x + 3 × −5 + 4

= 3x − 15 + 4

Collect the like terms (−15 and 4). = 3x − 11

b Write the expression. b 4(3x + 4) + 7x + 12

Expand the brackets. = 4 × 3x + 4 × 4 + 7x + 12

= 12x + 16 + 7x + 12

Rearrange so that the like terms are

together. (Optional)

= 12x + 7x + 16 + 12

Collect the like terms. = 19x + 28

1

2

3

1

2

3

4

13WORKEDExample


Expanding brackets

1 Use the Distributive Law to expand the following expressions.

a 3(d + 4) b 2(a + 5) c 4(x + 2)

d 5(r + 7) e 6(g + 6) f 2(t + 3)

g 7(d + 8) h 9(2x + 6) i 12(4 + c)

j 7(6 + 3x) k 45(2g + 3) l 1.5(t + 6)

m 11(t − 2) n 3(2t − 6) o t(t + 3)

p x(x + 4) q g(g + 7) r 2g(g + 5)

s 3f(g + 3) t 6m(n − 2m)

2 Expand the following.

a 3(3x − 2) b 3x(x − 6y) c 5y(3x − 9y)

d 50(2y − 5) e −3(c + 3) f −5(3x + 4)

g −5x(x + 6) h −2y(6 + y) i −6(t − 3)

j −4f(5 − 2f) k 9x(3y − 2) l −3h(2b − 6h)

m 4a(5b + 3c) n −3a(2g − 7a) o 5a(3b + 6c)

p −2w(9w − 5z) q 12m(4m + 10) r −3k(−2k + 5)

THINK WRITE

c Write the expression. c 2x(3y + 3) + 3x(y + 1)

Expand the brackets. = 2x × 3y + 2x × 3 + 3x × y + 3x × 1

= 6xy + 6x + 3xy + 3x

Rearrange so that the like terms are

together. (Optional)

= 6xy + 3xy + 6x + 3x

Simplify by collecting the like terms. = 9xy + 9x

d Write the expression. d 4x(2x − 1) − 3(2x − 1)

Expand the brackets. Take care with

negative terms.

= 4x × 2x + 4x × −1 − 3 × 2x − 3 × −1

= 8x2 − 4x − 6x + 3

Simplify by collecting the like terms. = 8x2 − 10x + 3

1

2

3

4

1

2

3

1. Brackets are grouping symbols

2. Removing brackets from an expression is called expanding the expression.

3. When expanding brackets, put the × sign before the bracket.

4. The rule that is used to expand brackets is called the Distributive Law.

5. After expanding brackets, collect any like terms.

remember

5IWORKED

Example

12

Mathcad

Expandingbrackets

EXCEL Spreadsheet

Expandingbrackets

GC program

– TI

Expandingbrackets

GC

program–

Casio

Expandingbrackets


3 Expand the expressions below and then simplify by collecting any like terms.

a 7(5x + 4) + 21 b 3(c − 2) + 2

c 2c(5 − c) + 12c d 6(v + 4) + 6

e 3d(d − 4) + 2d2 f 3y + 4(2y + 3)

g 24r + r(2 + r) h 5 − 3g + 6(2g − 7)

i 4(2f − 3g) + 3f − 7 j 3(3x − 4) + 12

k −2(k + 5) − 3k l 3x(3 + 4r) + 9x − 6xr

m 12 + 5(r − 5) + 3r n 12gh + 3g(2h − 9) + 3g

o 3(2t + 8) + 5t − 23 p 24 + 3r(2 − 3r) − 2r2 + 5r

4 Expand the following and then simplify by collecting like terms.

a 3(x + 2) + 2(x + 1) b 5(x + 3) + 4(x + 2)

c 2(y + 1) + 4(y + 6) d 4(d + 7) − 3(d + 2)

e 6(2h + 1) + 2(h − 3) f 3(3m + 2) + 2(6m − 5)

g 9(4f + 3) − 4(2f + 7) h 2a(a + 2) − 5(a2 + 7)

i 3(2 − t2) + 2t(t + 1) j m(n + 4) − mn + 3m

5 Simplify the following expressions by removing the brackets and then collecting like

terms.

a 3h(2k + 7) + 4k(h + 5) b 6n(3y + 7) − 3n(8y + 9)

c 4g(5m + 6) − 6(2gm + 3) d 11b(3a + 5) + 3b(4 − 5a)

e 5a(2a − 7) − 5(a2 + 7) f 7c(2f − 3) + 3c(8 − f)

g 7x(4 − y) + 2xy − 29 h 11v(2w + 5) − 3(8 − 5vw)

i 3x(3 − 2y) + 6x(2y − 9) j 8m(7n − 2) + 3n(4 + 7m)

FactorisingFactorising is the opposite process to expanding. Factorising a number or expression

involves breaking it down into smaller factors.

3 and 2 are factors of 6, because 6 = 3 × 2

2, 4, 5 and 10 are factors of 20, because:

20 = 4 × 5 and

20 = 2 × 10.

Common factorsTwo numbers may have common factors; for example, 5 is a factor of both 15 and 20.

The numbers 9 and 12 have the common factor 3.

The numbers 14 and 21 have the common factor 7.

The numbers 4 and 8 have two common factors, 2 and 4.

Highest common factorThe highest common factor (HCF) of 4 and 8 is 4 (not 2). It is the largest factor

common to a given set of numbers or terms.

The highest common factor of 12 and 18 is 6.

The highest common factor of 8 and 20 is 4.

Algebraic terms can also be broken down into factors. For example, the factors of 3x

are 3 and x. The expression 6m can be broken down into factors as shown below:

6m = 6 × m

= 3 × 2 × m

GC pr

ogram– TI

Expanding

WORKED

Example

13

GC pr

ogram– Casio

Expanding

GAM

E time

Algebra— 002


Here are some other examples:

8x = 8 × x

= 4 × 2 × x

= 2 × 2 × 2 × x

3ab = 3 × a × b

6a2b = 6 × a × a × b

= 3 × 2 × a × a × b

To find the highest common factor of algebraic terms, follow these steps.

1. Find the highest common factor of the number parts.

2. Find the highest common factor of the pronumeral parts.

3. Multiply these together.

To factorise an expression, we place the highest common factor of the terms

outside the brackets and the remaining factors for each term inside the brackets.

Find the highest common factor of 6x and 10.

THINK WRITE

Find the highest common factor of the

number parts.

Break 6 down into factors.


The highest common factor is 2.

6 = 3 × 2

10 = 5 × 2

HCF = 2


pronumeral parts.

There isn’t one, because only the first

term has a pronumeral part. The HCF of 6x and 10 is 2.

1

2

14WORKEDExample

Find the highest common factor of 14fg and 21gh.

THINK WRITE


number parts.



The highest common factor is 7.

14 = 7 × 2

21 = 7 × 3

HCF = 7


pronumeral parts.

Break fg down into factors.

Break gh down into factors.

Both contain a factor of g.

fg = f × g

gh = g × h

HCF = g

Multiply these together. The HCF of 14 fg and 21gh is 7g.

1

2

3

15WORKEDExample


Factorise the expression 2x + 6.

THINK WRITE

Break down each term into its factors. 2x + 6

= 2 × x + 2 × 3

Write the highest common factor

outside the brackets. Write the other

factors inside the brackets.

= 2 × (x + 3)

Remove the multiplication sign. = 2(x + 3)

1

2

3

16WORKEDExample

Factorise 12gh − 8g.

THINK WRITE

Break down each term into its factors. 12gh − 8g

= 4 × 3 × g × h − 4 × 2 × g

Write the highest common factor

outside the brackets.Write the other

factors inside the brackets.

= 4 × g × (3 × h − 2)

Remove the multiplication signs. = 4g(3h − 2)

1

2

3

17WORKEDExample

1. Factorising is the opposite process to expanding.

2. Factorising a number or expression involves breaking it down into smaller

factors.

3. To find the highest common factor (HCF) of algebraic terms, follow these

steps.

(a) Find the highest common factor of the number parts.

(b) Find the highest common factor of the pronumeral parts.

(c) Multiply these together.

4. To factorise an expression we place the highest common factor of the

terms outside the brackets and the remaining factors for each term inside the

brackets.

remember


Factorising

1

a The highest common factor of 12 and 16 is:

b The highest common factor of 10 and 18 is:

c The highest common factor of 4 and 16 is:

d The highest common factor of 2x and 8xy is:

e The highest common factor of 4f and 12fg is:

2 Find the highest common factor of the following.

a 4 and 6 b 6 and 9 c 12 and 18 d 13 and 26

e 14 and 21 f 2x and 4 g 3x and 9 h 12a and 16

3 Find the highest common factor of the following.

a 2gh and 6g b 3mn and 6mp c 11a and 22b

d 4ma and 6m e 12ab and 14ac f 24 fg and 36gh

g 20dg and 18ghq h 11gl and 33lp i 16mnp and 20mn

j 28bc and 12c k 4c and 12cd l x and 3xz

4 Factorise the following expressions.

a 3x + 6 b 2y + 4 c 5g + 10

d 8x + 12 e 6f + 9 f 12c + 20

g 2d + 8 h 2x − 4 i 12g − 18

j 11h + 121 k 4s − 16 l 8x − 20

m 12g − 24 n 14 − 4b o 16a + 64

p 48 − 12q q 16 + 8f r 12 − 12d

5 Factorise the following.

a 3gh + 12 b 2xy + 6y

c 12pq + 4p d 14g − 7gh

e 16jk − 2k f 12eg + 2g

g 12k + 16 h 7mn + 6m

i 14ab + 7b j 5a − 15abc

k 8r + 14rt l 24mab + 12ab

m 4b − 6ab n 12fg − 16gh

o ab − 2bc p 14x − 21xy

q 11jk + 3k r 3p + 27pq

s 12ac − 4c + 3dc t 4g + 8gh − 16

u 28s + 14st v 15uv + 27vw

A 12 B 4 C 8 D 2 E 3

A 4 B 10 C 2 D 9 E 180

A 4 B 16 C 2 D 20 E 8

A 2 B x C 2x D 16x2y E 8

A 2 B fg C 48f2g D 4f E 2f

5JSkillSHEET

5.7

Highestcommon

factor

multiple choice

Mathcad

Factorising

EXCEL Spreadsheet

Findingthe HCF

WORKED

Example

14

WORKED

Example

15

WORKED

Example

16

WORKED

Example

17

WorkS

HEET 5.3


Copy the sentences below. Fill in the gaps by choosing the correct word or

expression from the word list that follows.

1 A is a letter that is used in place of a number.

2 Replacing a pronumeral with a number is called .

3 When dividing pronumerals, the sign (÷) is rarely used.

Normally we rewrite the expression as a and simplify it by

cancelling.

4 When multiplying pronumerals, leave out the × sign. The term 3y means

.

5 Brackets are symbols. For example, 3(x + 4) means 3 × (x + 4)

or 3 × x + 3 × 4.

6 If the pronumeral you are substituting is negative, the rules for directed

numbers must be followed. For addition and subtraction, signs that occur

together can be . Same signs produce a sign,

while different signs produce a ___________ sign.

7 The Commutative Law holds true for and . It

does not hold true for and .

8 The Law refers to refers to the order in which

numbers may be added, subtracted, multiplied or divided, taking two at a

time.

9 When simplifying an expression, terms may be collected only if they are

.

10 Expanding an expression involves brackets. For example

3(x + 2) = 3x + 6.

11 The Law gives the rule for expanding expressions.

12 Factorising an expression means breaking it down into smaller

, or putting brackets back into the expression.

summary

W O R D L I S T

three

substitution

combined

Distributive

positive

3 × y

Associative

factors

negative

pronumeral

removing

subtraction

grouping

addition

division

division

like terms

fraction

multiplication


1 Using x and y to represent numbers, write expressions for:

a the sum of x and y

b the difference between y and x

c five times y subtracted from three times x

d the product of 5 and x

e twice the product of x and y

f the sum of 6x and 7y

g y multiplied by itself

h twice a number is decreased by 7

i the sum of p and q is tripled.

2 If tickets to the school play cost $15 for

adults and $9 for children, write an

expression for the cost of:

a x adult tickets

b y child tickets

c k adult tickets and m child tickets.

3 Jake is now m years old.

a Write an expression for his age in 5 years’

time.

b Write an expression for Jo’s age if she is p

years younger than Jake.

c Jake’s mother is 5 times his age. How old is she?

4 Find the value of the following expressions if a = 2 and b = 6.

a 2a b 6a c 5b d

e a + 8 f b − 2 g a + b h b − a

i j 3a + 7 k 2a + 3b l

m 3b – 2a n o p –3ab

q r

5 The formula C = 2.2k + 4 can be used to calculate the cost in dollars, C, of travelling by taxi

for a distance of k kilometres. Find the cost of travelling 4.5 km by taxi.

6 The area (A) of a rectangle of length l and width w can be found using the formula A = lw.

Find the width of a rectangle if A = 65 cm2 and l = 13 cm.

5A

CHAPTERreview

5A

5A

5Ba

2---

5b

2---+

20

a------

b

a---

a

b---

5a

b------

2b

9a------

5B

5B


7 Substitute r = 3 and s = 5 into the following expressions and evaluate.

a 2(r + s) b 2(s − r) c 5(r + s) d 8(s − r)

e s(r + 4) f s(2r − 3) g 2r(r + 1) h rs(7 + s)

i r2(5 – r) j s2(s + 15) k 4r(s + r) l 12r(r – s)

8 Find the value of the following expressions if a = 2 and b = −5.

a a + b b b + a c ab d

e 2ab f 5 − a g 12 − ab h a2 − 2

i 3(a + 2) j b(a − 4) k 12 − a(b − 3) l 5a + 6b


a 3a + 5b = 5b + 2a b 7x − 10y = 10y − 7x c 8c + d = d + 8c

d 16 × 2x × x = 32x2 e 9x × −y = −y × 9x f −4 × 3x × x = 12x × x

g = h 7i + 2j = 2j + 7i i −3y ÷ 7x = 7x ÷ −3y

j −8c + 5d = 5d − 8c k = l 21 × − = × −21

10 Simplify the following by collecting like terms.

a 4d + 3d b 3c − 5c

c 3d + 5a − 4a d 6g − 4g

e 4x + 11 − 2x f 2g + 5 − g − 6

g 2xy + 7xy h 12t2 + 3t + 3t2 − t


a 3 × 7g b 6 × 3y

c 7d × 6 d −3z × 8


a b

c 6rt ÷ −2t d −3gh ÷ −6g

e f −36xy ÷ −12y

g h

13 Use the Distributive Law to expand the following expressions.

a 2(x + 3) b 5(2x − 1) c −2(f + 7)

d 3m(b − m) e −3y(7 − y) f 9b(c − 2)

14 Expand the following and then simplify by collecting like terms.

a 3(4v + 5) − 15 b 6t + 5(2t − 7) c 23 + 5(3p − 4) + 2p

d 2(x + 5) + 5(x + 1) e 2g(g − 6) + 3g(g − 7) f 3(3t − 4) − 6(2t − 9)

15 Factorise the following expressions.

a 3g + 12 b xy + 5y c 5n − 20

d 12mn + 4pn e 12g − 6gh f 12xy − 36yz

5C

5Dab

5------

5E

11 p

5r---------

5r

11 p---------

0

5k------

5k

0------

7x

3------

7x

3------

5F

5G

5H2a

8------

11b

44b---------

32t

40stv-------------

12ab

14ab–---------------

5egh

30ghj--------------

5I

5I

testtest

CHAPTER

yourselfyourself

5

5J

Chap 05

Documents

original number of ants

original nest

b lamingtons

nearby nest

nest i

fewer ants

particular number

single number