Algebra

34
4 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. Algebra

Transcript of Algebra

4Ships 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.

Algebra

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

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 beused 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 cement is needed to build a path?

A pronumeral is a letter that is used in place of a number. In Year 7 we saw thatpronumerals could be used to make expressions and equations. Often a pronumeral isused to represent one particular number. For example, in the equation

x + 1 = 7the pronumeral x has the value 6.

Pronumerals can also be used to show a relationship between two or more numbers,for example

a + b = 10Can you find some different pairs of values for a and b which fit this rule?

Algebra allows us to show complex rules in a more simple way, and to solve problems involving unknown numbers.

MQ 8 Ch 04 Page 118 Thursday, December 7, 2000 1:48 PM

C h a p t e r 4 A l g e b r a 119The worked example below 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?

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 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.

b2---

b2---

1WORKEDExample

remember1. 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

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

Using pronumerals

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

ground after they have left.

2 The canteen manager at Browning Industries orders m vanilla slices each day. Write aparagraph which could explain the table below:

3 Imagine that your cutlery drawer contains a knives, b forksand 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

drawerc 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.

4 If y represents a certain number, write expressions for thefollowing numbers.a A number 7 more than yb A number 8 less than yc A number which is equal to five times yd The number formed when y is subtracted from 14e The number formed when y is divided by 3.

5 Using a and b to represent numbers, write expressions for:a the sum of a and bb the difference between a and bc three times a subtracted from two times bd the product of a and be twice the product of a and bf the sum of 3a and 7bg a multiplied by itself.

Time Number of vanilla slices

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

4AWWORKEDORKEDEExamplexample

1

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C h a p t e r 4 A l g e b r a 1216 If tickets to a Brisbane Bullets/Melbourne Tigers basketball match cost $27 for adults

and $14 for children, write an expression for the cost of:a y adult ticketsb d child ticketsc r adult and h child tickets.

7 If 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?

8 Charles places p coins into a poker machine. He plays the machine and counts hiscoins every 3 minutes. The table below shows how many coins he has.

a Write a paragraph explaining what happened.b When did Charles start to lose money?c If he used $1 coins, how much did Charles win or lose, overall?

Time Number of coins

7.10 pm p

7.13 pm 2p

7.16 pm 2p + 12

7.19 pm 4p + 12

7.21 pm 4p + 7

7.24 pm p

7.27 pm p + 1

7.30 pm p − 8

7.33 pm p − 12

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

MA

TH

SQUEST

C H A L L

EN

GE

MA

TH

SQUEST

C H A L L

EN

GE

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 areequal to 7 times the sum of their digits.

3 Find 5 consecutive numbers thatadd to 120.

4 I’m thinking of a number. If Imultiply it by 5 and subtract4, I get the same number aswhen I multiply it by 4 andadd 2. What is the number?

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

5

9 A microbiologist places m bacteria onto anagar plate. She counts the number ofbacteria at approximately 3 hour intervals.The results are shown in the table below:

a Explain what happens to the number ofbacteria in the first 5 intervals.

b What might be causing the number ofbacteria to increase in this way?

c What is different about the last bacteriacount?

d What may have happened to cause this?

10 If 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 which are greater than nii the even number which is 2 less than n.

TimeNumber of

bacteria

9.00 am m

12.00 pm 2m

3.18 pm 4m

6.20 pm 8m

9.05 pm 16m

12.00 am 32m – 1240

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C h a p t e r 4 A l g e b r a 123

SubstitutionWhen a pronumeral is replaced by a number, we say that the number is substituted forthe pronumeral. If the value of the pronumeral (or pronumerals) is known, it is possibleto evaluate (work out the value of) an expression.

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

x + y = 2 + 3= 5

When writing expressions with pronumerals:1. We leave out the multiplication sign.

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 theworked example below.

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

y6---

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

Multiply. = 18

b Substitute each pronumeral with its correct value and replace the multiplication signs.

b

=

Do the first multiplication. =

Do the next multiplication. =

Do the division. = 21 − 10Do the subtraction. = 11

7a 2b3

------–

1

2

1 7a2b3

------–

7 3 2 15×3

---------------–×

2 21 2 15×3

---------------–

3 21 303

------–

4

5

2WORKEDExample

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

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

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

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 down the formula. A = l × w

Substitute each pronumeral with its correct value.

Α = 270 × 32

Multiply to find A and state the correct units.

A = 8640 m2

1

2

3

3WORKEDExample270 m

32 m

remember1. Replacing a pronumeral with a number is called substituting.2. When writing expressions with pronumerals:

(a) We leave out the multiplication signs.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 .

y6---

remember

4B

SkillSH

EET 4.1 WWORKEDORKEDEExamplexample

2 a2---

Mathca

d

Substitution

5b5---+ 8

a---

25b

------

ab5

------

EXCEL

Spreadsheet

Substitutionx3--- y

3---+ 24

x------ 9

y---–

12x

------ 4 y+ + 7x2

------

EXCEL

Spreadsheet

Substitution game 13y

3--------- 2x–

4xy15

---------

MQ 8 Ch 04 Page 124 Thursday, December 7, 2000 1:48 PM

C h a p t e r 4 A l g e b r a 1253 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 theperimeter (P) of a rectangleof length l and width w isP = 2l + 2w. Use this for-mula to find the perimeterof the rectangular swim-ming pool at right.

5 The formula F = 2c + 30 is used to convert temperatures measured in degrees Celsiusto an approximate Fahrenheit value. F represents the temperature in degrees Fahrenheitand c the temperature in degrees Celsius.a Find F when c = 100.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 length 2.5 cm.

7 The formula C = 0.1a + 42 is used to calculate the cost in dollars (C) of renting a carfor one day from Poole’s Car Hire Ltd, where a is the number of kilometres travelledon that day. Find the cost of renting a car for one day if the distance travelled is220 kilometres.

8 Distances in the USA and Canada are often expressed in both miles and kilometres.The formula D = 0.6T can be used to convert distances in kilometres (T) to the approxi-mate equivalent in miles (D). Use this rule to convert the following distances to miles:a 100 kilometresb 248 kilometresc 12.5 kilometres.

9 The area (A) of a rectangle of length l and width w can be found using the formulaA = lw. Find the area of the rectangles below:a length 12 cm, width 4 cmb length 200 m, width 42 mc length 4.3 m, width 104 cm.

md10-------

3md2

-----------

7d15------ 15

d------ m–

WWORKEDORKEDEExamplexample

3

50 m25 m

GAMEtime

Algebra— 001

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

Working with bracketsBrackets are ‘grouping’ symbols. For example, the expression 3(a + 5) can be thoughtof 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 a bracket first, then multiply by the value outside the bracket.

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.

THINK WRITE

a Put the multiplication sign back into the expression.

a 5(s + r)= 5 × (s + r)

Substitute the pronumerals with their correct values.

= 5 × (5 + 4)

Work out the bracket first. = 5 × 9Complete the multiplication. = 45

b Put the multiplication signs back into the expression.

b 2x(3t − y)= 2 × x × (3 × t − y)

Substitute the pronumerals with their correct values.

= 2 × 3 × (3 × 4 − 5)

Do the multiplication inside the brackets.

= 2 × 3 × (12 − 5)

Do the subtraction inside the brackets.

= 2 × 3 × 7

Do the final multiplication. = 42

1

2

3

4

1

2

3

4

5

4WORKEDExample

remember1. 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

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C h a p t e r 4 A l g e b r a 127

21.4 cm

27.5 cm

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

3 The formula for the perimeter (P) of a rectangleof length l and width w is P = 2l + 2w. Thisrule can also be written as P = 2(l + w). Usethe rule to find the perimeter of rectangularcomic covers with the following measure-ments.a l = 20 cm, w = 11 cmb l = 27.5 cm, w = 21.4 cm

4 A rule for finding the sum of the interiorangles in a many-sided figure such as apentagon is S = 180(n − 2) where S represents thesum of the angles inside the figure and n representsthe number of sides. The diagram at right shows theinterior angles in a pentagon.

Use the rule to find the sum of the interior angles forthe following figures:a a hexagon (6 sides) b a pentagonc a triangle d a quadrilateral (4 sides)e a 20-sided figure.

4CWWORKEDORKEDEExamplexample

4 SkillSH

EET 4.2

Mathcad

Substitution(brackets)

12x

------ z y–( ) z3--- 2y

10------ x 2–+

y5--- 7 x– 3+( ) 7 12

x------–

4y

6x--- xz y 3–+( ) y 2+( ) z

x--

3x 7–( ) 27x

------ 7+

MQ 8 Ch 04 Page 127 Thursday, December 7, 2000 1:48 PM

I’m now in Australia!

a + b =

StopStop StopStop

StopStop

StartStart

51

74

47

19

86

64

81

66 57

6837

421

8025

1110

54

77

46

72

6388

10075

352796

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

9140

60

4 48

8439

90

29

50

6

65

4456

14

69

53

8520

cx =

=y—x

bcy =

8x =

y2 =StartStart

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 =StartStart

2bx =

x(2y – c) =

11c =

6(b + c) =StartStart

y – c =

12(x + c) =

9c =

20x =

10c =

30 + 2b =

b(y + c) =

cy – b =

2x – 3b =

StartStart

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 =

StartStart

a2 =

=4cy——x

27——c

=11(x + c)————b

StartStart

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) =StopStop

StopStop

4b + 2c =StartStart

20c + y – a =

6x – 4b =

4y – x – b =

bcx + b =

8x + b – a =

4x + 2c =

StopStop

Coloring guide:Coloring guide:

Join these points with thick lines.Join these points with thick lines.

black

2bx – a =

c + x =

7y – cx – b =

a + 9y =

OrangeOrange

6c + xy =

7(c + x) =

11y – 4c + a =

2y – c =StartStart

6(x + a) =

y + x – b =

3c =StopStop

StartStart

40b + 2b =

6(x + c) =

y – x – a =StopStop

80 + x + c =StartStart

7(a + 4b) =

12x + y + b =

c + 2b + 3a =StopStop

8(c + x) =StartStart

9y – 2x + a =

7y + 8b =

12c + 3y =StopStop

12x – 3c =StartStart

10x + 8c =

3cx + b =StopStop

StopStop

StopStop

StartStart

10y – x =

= 8y——b – a =

StartStart

8y – c =

StopStop

9x + a =

StartStart

20b + c =

8x + 7c =StopStop

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.

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

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C h a p t e r 4 A l g e b r a 129

Substituting positive and negative numbers

If the pronumeral you are substituting has a negative value, simply remember thefollowing 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 + 3Different signs negative for example, 7 − +3 = 7 − 3

and 7 + −3 = 7 − 32. 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

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 Replace the pronumerals with their correct value.

a m − n= 5 − −3

Combine the two negative signs and add.

= 5 + 3= 8

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

Substitute the pronumerals with their correct values.

= 2 × −1 − −2

Do the multiplication. = −2 − −2Combine the two negative signs and add.

= −2 + 2= 0

c Replace the multiplication signs. c 5ab −

= 5 × a × b −

Substitute the pronumerals with their correct values.

= 5 × 4 × −3 −

Do the multiplications. = −60 −

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

= −60 + 4= −56

12b

------

1

2

1

2

34

112b

------12b

------

212

3–------

312

3–------

45

5WORKEDExample

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

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 + me 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 2abe 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

rememberWhen 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 − 32. 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

4D

SkillSH

EET 4.3 WWORKEDORKEDEExamplexample

5a

m2----

Mathca

d

Substitution (positive/negative)

mn9

------- 4mn 5–------------ 4m

n------- 12

2n------

9n--- m

2----+ 3n

2------– 1.5+ 14 mn

9-------–

WorkS

HEET 4.1 WWORKEDORKEDEExamplexample

5b

x2--- 5 y–( )

x4--- 1–

2y6

------ 4+ 9

y--- 6 x–( ) 3 x 1–( )

y3--- 2+

WWORKEDORKEDEExamplexample

5c

4b--- 8

a---– 16

4a------–

6b5

------

a2--- 3b

5------+

MQ 8 Ch 04 Page 130 Thursday, December 7, 2000 1:48 PM

C h a p t e r 4 A l g e b r a 131

1 If a kilogram of oranges cost $0.89 and a kilogram of carrots cost $0.99, what isthe cost of p kg of oranges and q kg of carrots.

2 If d represents a certain number, write an expression for the number formed whend 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 thearea 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.

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

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.

‘Rules of thumb’A ‘rule of thumb’ is a rule or pattern which 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 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 accurate expression for 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.

A 21 B 22 C 22.25 D 25 E 28

95---

1

12y

------ 3z+ 4=

12s

------ rs 4 s–+( )

mmultiple choiceultiple choice3m

n4---+

14p

------ 1– pq 3+( )

a2--- b

4---+ 0=

12x

------– 3y+

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

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.

Simplify the following expressions.a 3a + 5ab 7ab − 3a − 4abc 2c − 6 + 4c + 15

THINK WRITE

a Write down the expression and check that the pronumeral parts of the 2 terms are the same. They are.

a 3a + 5a

Add the 2 terms. = 8a

b Write down the expression. 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

Simplify by subtracting the like terms.

= 3ab − 3a

c Write down the expression. 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 the like terms. = 6c + 9

1

2

1

2

3

1

2

3

6WORKEDExample

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C h a p t e r 4 A l g e b r a 133

Simplifying expressions

1Simplifying 3a + 9a gives:

2Simplifying 6x − 2x gives:

3Simplifying 6a + 6b gives:

4 Simplify the following expressions.a 4c + 2c b 2c − 5c c 3a + 5a − 4ad 6q − 5q e −h − 2h f 7x − 5xg 3a − 7a − 2a h −3f + 7f i 4p − 7pj −3h + 4h k 11b + 2b + 5b l 7t − 8t + 4tm 9m + 5m − m n x − 2x o 7z + 13zp 5p + 3p + 2p q 9g + 12g − 4g r 18b − 4b − 11bs 13t − 4t + 5t t −11j + 4j u −12l + 2l − 5lv 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 − 12c 11 + 5f − 7f d 3u − 4u + 6e 2m + 3p + 5m f −3h + 4r − 2hg 11a − 5b + 6a h 9t − 7 + 5i 12 − 3g + 5 j 6m + 4m − 3n + nk 5k − 5 + 2k − 7 l 3n − 4 + n − 5m 2b − 6 − 4b + 18 n 11 − 12h + 9o 12y − 3y – 7g + 5g − 6 p 8h − 6 + 3h − 2q 11s − 6t + 4t − 7s r 2m + 13l − 7m + ls 3h + 4k − 16h − k + 7 t 13 + 5t − 9t − 8u 2g + 5 + 5g − 7 v 17f − 3k + 2f − 7k

A 12 B 12a C 6a D 12a2 E The expression cannot be simplified.

A 4 B 4x2 C 4x D 2x E The expression cannot be simplified.

A 12ab B 6ab C 36ab D 12a E The expression cannot be simplified.

remember1. When simplifying expressions, we can collect (add or subtract) only like terms.2. Like terms are terms that contain the same pronumeral parts.

remember

4EMathcad

Simplifyingexpressions

mmultiple choiceultiple choice

mmultiple choiceultiple choice

mmultiple choiceultiple choice

WWORKEDORKEDEExamplexample

6a

WWORKEDORKEDEExamplexample

6b, c

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

6 Simplify the following.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 − 7p 11g3 + 17 − 3g3 + 5 − g2 q 12ab + 3 + 6ab r 14xy + 3xy − xy − 5xys 4fg + 2s − fg + s t 11ab + ab − 5

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

3 × 6 = 6 × 36 × w = w × 6a × b = b × a

Also keep in mind that the ‘×’ sign is usually left out:3 × g × h = 3gh2 × x2 × y = 2x2y

Although order is not important, the pronumerals in each term are usually written inalphabetical order.For example: 2 × b2 × a × c = 2ab2c

Simplify:a 5 × 4gb −3d × 6ab × 7.

THINK WRITE

a Write down the expression and replace the hidden multiplication signs.

a 5 × 4g= 5 × 4 × g

Multiply the numbers. = 20 × gRemove the multiplication sign. = 20g

b Write down the expression and replace the hidden multiplication signs.

b −3d × 6ab × 7= −3 × d × 6 × a × b × 7

Put the numbers at the front. = −3 × 6 × 7 × d × a × bMultiply the numbers. = −126 × d × a × bRemove the multiplication signs. = −126abd

1

23

1

234

7WORKEDExample

rememberWhen multiplying pronumerals:1. The order is not important.

For example, d × e = e × d.2. Put the numbers at the front of the expression and leave out the × sign.

remember

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C h a p t e r 4 A l g e b r a 135

Multiplying pronumerals

1 Simplify the following.a 4 × 3g b 7 × 3h c 4d × 6d 3z × 5 e 6 × 5r f 5t × 7g 4 × 3u h 7 × 6p i 7gy × 3j 2 × 11ht k 4x × 6g l 10a × 7h m 9m × 4d n 3c × 5h o 9g × 2xp 2.5t × 5b q 13m × 12n r 6a × 12ds 2ab × 3c t 4f × 3gh u 2 × 8w × 3xv 11ab × 3d × 7 w 16xy × 1.5 x 3.5x × 3y y 11q × 4s × 3 z 4a × 3b × 2c

2 Simplify the following.a 3 × −5f b −6 × −2dc 11a × −3g d −9t × −3ge −5t × −4dh f 6 × −3stg −3 × −2w × 7d h −4a × −3b × 2c × ei 11ab × −3f j 3as × −3b × −2xk −5h × −5t × −3q l 4 × −3w × −2 × 6pm −7a × 3b × g n 17ab × −3gho −3.5g × 2h × 7 p 5h × 8j × −kq 75x × 1.5y r 12rt × −3z × 4ps 2ab × 3c × 5 t −4w × 34x × 3

Sonar measurementsAt 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 secondsb 4.22 secondsc 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.

4FWWORKEDORKEDEExamplexample

7

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

History of mathematicsT H E R H I N D PA P Y R U S ( c . 1 8 5 0 B C )

The ancient Egyptians differed from the ancient Greeks in that Egyptians thought about mathematics in a practical rather than an abstract way. They didn’t like fractions which had numerators other than one (except the fraction two-thirds for reasons still unknown). They found that fractions with numerators of one, unit fractions, were easy to multiply, since the numerator would

always be one: for example × = .

The Egyptians developed ingenious methods to avoid using any fraction other than those with a numerator of one. Solutions to many Egyptian problems concerned with beer and bread were recorded on papyri. The most famous of these is the Rhind papyrus, which contains 84 problems and their solutions including the calculation of the ancient Egyptian value for pi (π) of 3.1605. A part of the papyrus is shown in the photograph above.

The Rhind papyrus was named after the Scottish Egyptologist, A. Henry Rhind, who bought the 6 m scroll in 1858. A scribe named Ahmes is believed to have copied it in around 1650 BC from a document originally written about 200 years before that. This papyrus shows a method for multiplying numbers using only addition and subtraction. Also known as the aha papyrus: aha meaning unknown quantity to be determined, an early pronumeral, it is now in the British Museum in London.

Questions1. Which numerator did the Egyptians use

in their calculations with fractions?2. Which fraction was an exception to this

rule?3. What practical problems did most of the

solutions deal with?

ResearchHow was Egyptian multiplication done with only addition and subtraction?

12--- 1

3--- 1

6---

During this time . . .

The Sumerians built the first cities, invented writing and made wheels from date palm trunks.

Papyrus reeds were used to make boats, baskets and paper.

The Bronze Age began.

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C h a p t e r 4 A l g e b r a 137

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

Remember that when the same pronumeral appears on both the top and bottom linesof the fraction, it may be cancelled. Follow the worked examples given below.

a Simplify .

b Simplify 15n ÷ 3n.

THINK WRITE

a Write down 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 down 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 f4

----------

116 f

4---------

24 f1

------

3

1

15n3n

---------

251---

3

8WORKEDExample

Simplify −12xy ÷ 27y.

THINK WRITE

Write down 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.

=

1

12xy27y

------------–

24x9

------–

9WORKEDExample

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

Dividing pronumerals

1 Simplify the following.

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

2 Simplify the following.

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

3 Simplify the following.

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

remember1. 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

4G

SkillSH

EET 4.4 WWORKEDORKEDEExamplexample

8 8 f2

------ 6h3

------ 15x3

---------

16m8m

----------

3x6x------ 12h

14h---------

8 f24 f---------

20d48d---------

15 fg3

------------ 8xy12

---------

11xy11x

------------ 9 pq18q---------- 21ab

28b------------ 9dg

12g---------

5 jkkj

-------- 10mxy35mx

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

16cd40cd------------

132mnp60np

-------------------- 11ad66ad------------ bh

7h------

WWORKEDORKEDEExamplexample

9 4a–8

--------- 11ab–33b

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

WorkS

HEET 4.232g–

40gl------------ 12ab

14ab–--------------- 6 fgh

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

rt–6rt-------

ab–3a–

--------- 7dg–35gh------------

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C h a p t e r 4 A l g e b r a 139

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.

4If 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 Find the missing term from the list −2, −4, 12pq, −48pq 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 form1. 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

Removing brackets from an expression is called expanding the expression. The rulethat we have used to expand the expressions above is called the Distributive Law.

2

r 10+( ) pq---

mmultiple choiceultiple choicem2---- 6n

9------+

30ab18abc---------------

∇12 pr–

--------------- 4qr

------=

9 p–36 pq–

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

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

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)

Use the Distributive Law to expand the following expressions.a 3(a + 2) b x(x − 5)

THINK WRITE

a Write down the expression and replace the hidden multiplication sign.

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 Repeat the steps in part a. b x(x − 5)= x × (x − 5)= x × x + x × −5= x2 − 5x

1

2

3

10WORKEDExample

remember1. Brackets are grouping symbols2. 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.

remember

4HWWORKEDORKEDEExamplexample

10

Mathca

d

Expandingbrackets

EXCEL

Spreadsheet

Expanding brackets

GCpro

gram

Expanding brackets

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C h a p t e r 4 A l g e b r a 141

History of mathematicsJOHN COATES (1945– )

During his time . . .Space travel — men walk on the moon.The Cold War ends.Ecological awareness grows.Miniaturisation of computers.

John Coates, a world-renowned Australian mathematician, was born in 1945. He attended Taree High School and studied for his Bachelor of Science at the Australian National University (ANU). After further studies in Paris, he completed a PhD at the University of Cambridge in England, where he later lectured.

He taught mathematics at Harvard and Stanford, both very prestigious universities in the United States. Later he held positions as a professor at the ANU and two institutes in France. In 1986 he returned to Cambridge as Sadleirian Professor and was appointed Head of Department.

He still works at Cambridge in arithmetical algebraic geometry and his research interests include elliptic curves, the Iwasawa theory, Fermat’s Last Theorem and explicit reciprocity laws! As well as this, his work includes the algebraic approximation of functions.

Coates is not just a brilliant mathematician and outstanding researcher, he is also praised for being a great teacher who has inspired many students to pursue careers in mathematical research. He is also known for his valuable contributions as an editor of one of the best known journals in research mathematics, Inventiones Mathematicae.

During his international career he has also received numerous awards, including election as a fellow of the Royal Society of London in 1985 and the Senior Whitehead Prize from the London Mathematical Society in 1997.

Questions1. What country did John Coates grow

up in?2. Reciprocity is about expressions

involve reciprocals. What are reciprocals?

3. What three career areas does John Coates work in?

4. What mathematical prize did John Coates win?

ResearchWhat was Fermat’s Last Theorem?

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

Expanding and collecting like termsSome expressions can be simplified further by collecting like terms after any bracketshave been expanded.

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 − 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. = 12x + 16 + 7x + 12

Rearrange so that the like terms are together. (Optional)

= 12x + 7x + 16 + 12

Collect the like terms. = 19x + 28

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

1

2

3

4

1

2

3

4

1

2

3

11WORKEDExample

rememberAfter expanding brackets, collect any like terms.

remember

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C h a p t e r 4 A l g e b r a 143

Expanding and collecting like terms

1 Expand the expressions below and then simplify by collecting any like terms.a 7(5x + 4) + 21 b 3(c − 2) + 2c 2c(5 − c) + 12c d 6(v + 4) + 6e 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) + 12k −2(k + 5) − 3k l 3x(3 + 4r) + 9x − 6xrm 12 + 5(r − 5) + 3r n 12gh + 3g(2h − 9) + 3go 3(2t + 8) + 5t − 23 p 24 + 3r(2 − 3r) − 2r2 + 5r

2 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

3 Simplify the following expressions by removing the brackets and then collecting liketerms.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 expressioninvolves breaking it down into smaller factors.

3 and 2 are factors of 6, because 6 = 3 × 22, 4, 5 and 10 are factors of 20, because:

20 = 4 × 5 and20 = 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 factorcommon 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.

4IGC program

Expanding

WWORKEDORKEDEExamplexample

11

GAMEtime

Algebra— 002

WorkS

HEET 4.3

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

Algebraic terms can also be broken down into factors. For example, the factors of 3xare 3 and x. The expression, 6m, can be broken down into factors as shown below:

6m = 6 × m= 3 × 2 × m

Here are some other examples:8x = 8 × x

= 4 × 2 × x = 2 × 2 × 2 × x

3ab = 3 × a × b6a2b = 6 × a × a × b

= 3 × 2 × a × a × b

To find the highest common factor, HCF, 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.

Find the highest common factor (HCF) of 6x and 10.

THINK WRITE

Find the highest common factor of the number parts.Break 6 down into factors.Break 10 down into factors.The highest common factor is 2.

6 = 3 × 210 = 5 × 2

HCF = 2Find the highest common factor of the 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

12WORKEDExample

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

THINK WRITE

Find the highest common factor of the number parts.Break 14 down into factors.Break 21 down into factors.The highest common factor is 7.

14 = 7 × 221 = 7 × 3HCF = 7

Find the highest common factor of the pronumeral parts.Break fg down into factors.Break gh down into factors.Both contain a factor of g.

fg = f × ggh = g × h

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

1

2

3

13WORKEDExample

MQ 8 Ch 04 Page 144 Thursday, December 7, 2000 1:48 PM

C h a p t e r 4 A l g e b r a 145To 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.

Factorise the expression 2x + 6.

THINK WRITE

Break down each term into factors. 2x + 6= 2 × x + 2 × 3

Write the common factor outside the brackets and the other factors inside the brackets.

= 2 × (x + 3)

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

1

2

3

14WORKEDExample

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

15WORKEDExample

remember1. 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

MQ 8 Ch 04 Page 145 Thursday, December 7, 2000 1:48 PM

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

Factorising

1a The highest common factor (HCF) of 12 and 16 is:

b The highest common factor (HCF) of 10 and 18 is:

c The highest common factor (HCF) of 4 and 16 is:

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

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

2 Find the highest common factor (HCF) of the following.a 4 and 6 b 6 and 9 c 12 and 18 d 13 and 26e 14 and 21 f 2x and 4 g 3x and 9 h 12a and 16

3 Find the highest common factor (HCF) of the following.

a 2gh and 6g b 3mn and 6mp c 11a and 22bd 4ma and 6m e 12ab and 14ac f 24 fg and 36ghg 20dg and 18ghq h 11gl and 33lp i 16mnp and 20mnj 28bc and 12c k 4c and 12cd l x and 3xz

4 Factorise the following expressions.a 3x + 6 b 2y + 4 c 5g + 10d 8x + 12 e 6f + 9 f 12c + 20g 2d + 8 h 2x − 4 i 12g − 18j 11h + 121 k 4s − 16 l 8x − 20m 12g − 24 n 14 − 4b o 16a + 64p 48 − 12q q 16 + 8f r 12 − 12d

5 Factorise the following.a 3gh + 12 b 2xy + 6yc 12pq + 4p d 14g − 7ghe 16jk − 2k f 12eg + 2gg 12k + 16 h 7mn + 6mi 14ab + 7b j 5a − 15abck 8r + 14rt l 24mab + 12abm 4b − 6ab n 12fg − 16gho ab − 2bc p 14x − 21xyq 11jk + 3k r 3p + 27pqs 12ac − 4c + 3dc t 4g + 8gh − 16u 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

4J

Mathca

d

Factorising

mmultiple choiceultiple choice

EXCEL

Spreadsheet

Finding the HCF

WWORKEDORKEDEExamplexample

12

WWORKEDORKEDEExamplexample

13

WWORKEDORKEDEExamplexample

14

WWORKEDORKEDEExamplexample

15

MQ 8 Ch 04 Page 146 Thursday, December 7, 2000 1:48 PM

The fThe factorised form of the eactorised form of the exprxpressions andessions andthe letter beside each givthe letter beside each gives the puzzle codees the puzzle code..

D = 6x – 3 =

E = 8 – 2x =

G = 15x + 10 =

H = –2x + 1 =

I = 12x + 20 =

L = 2xy – 3y =

N = 18 – 42x =

O = –2x – 2 =

P = 2xy – 8y =

S = –12x – 21 =

T = xy – 2x =

V = 56x – 35 =

E = 6xy + 3x =

L =

N =

E =

O =

P =

E =

S =

T =

O =

–20x – 25 =

8xy – 72y =

–x + 2 =

6x – 14 =

–xy + y =

3x – 12 =

–8x – 24 =

3xy – 2y =

6xy – 10y =

–49 – 28x =

H =

Doctor IDoctor I’’vve se swallowallowwed the film out of ed the film out of mmy y camera!camera!

8y(x – 9) 2(4 – x)

–7(7 + 4x)–(2x – 1) 2y(x – 4) 2(3x – 7)

6(3 – 7x) 4(3x + 5) –(x – 2)–y(x – 1) x(y – 2) –5(4x + 5)

y(2x – 3) –2(x + 1) 3(x – 4) y(3x – 2)3(2x – 1) –8(x + 3) 7(8x – 5) 3x(2y + 1)

2y(3x – 5) –3(4x + 7)

5(3x + 2)

C h a p t e r 4 A l g e b r a 147

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

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

7 Expanding an expression involves brackets. For example3(x + 2) = 3x + 6.

8 The Law gives the rule for expanding expressions.

9 Factorising an expression means breaking it down into smaller, or putting brackets back into the expression.

summary

W O R D L I S TsubstitutionDistributive3 × y

factorspronumeralremoving

groupingdivision

like termsfraction

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C h a p t e r 4 A l g e b r a 149

1 Using x and y to represent numbers, write expressions for:a the sum of x and yb the difference between y and xc five times y subtracted from three times xd the product of 5 and xe twice the product of x and yf the sum of 6x and 7yg y multiplied by itself.

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 ticketsb y child ticketsc k adult tickets and

m child tickets.

3 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 − 2g a + b h b − a

i j 3a + 7

k 2a + 3b l

4 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.

4A

CHAPTERreview

4A

4Ba2---

5b2---+

20a

------

4B

MQ 8 Ch 04 Page 149 Thursday, December 7, 2000 1:48 PM

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

5 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)

6 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 − ag 12 − ab h a2 − 2i 3(a + 2) j b(a − 4)k 12 − a(b − 3) l 5a + 6b

7 Simplify the following by collecting like terms.a 4d + 3d b 3c − 5cc 3d + 5a − 4a d 6g − 4ge 4x + 11 − 2x f 2g + 5 − g − 6g 2xy + 7xy h 12t2 + 3t + 3t2 − t

8 Simplify the following.a 3 × 7g b 6 × 3yc 7d × 6 d −3z × 8

9 Simplify the following.

a b

c 6rt ÷ −2t d −3gh ÷ −6g

e f −36xy ÷ −12y

g h

10 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)

11 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)

12 Factorise the following expressions.a 3g + 12 b xy + 5yc 5n − 20 d 12mn + 4pne 12g − 6gh f 12xy − 36yz

4C

4Dab5

------

4E

4F

4G 2a8

------ 11b44b---------

32t40stv-------------

12ab14ab–

--------------- 5egh30ghj--------------

4H

4I

4J

testtest

CHAPTERyyourselfourself

testyyourselfourself

4

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