Algebra

4
Ships use the speed of sound in water to help nd the waters depth. A sonar pulse from a ship is sent to the bottom of the ocean oor. The time taken for the pulse to hit the ocean oor 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.

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Maths Quest 8 for Victoria

Using pronumerals
The 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 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 that pronumerals could be used to make expressions and equations. Often a pronumeral 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 nd some different pairs of values for a and b which t this rule?

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

Chapter 4 Algebra

119

The worked example below shows some of the ways pronumerals can be used.

WORKED Example 1
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 b -- ants. How much smaller than the 2 original is this nest? THINK a The original number of ants (b) must be reduced by 25. b The original number of ants (b) must be multiplied by 2. It is not necessary to show the sign. c 50 must be added to the original number of ants (b). d This expression tells us that the nearby nest has 100 more ants. e This nest has 1000 fewer ants. b f The expression -- means b 2, so this 2 nest is half the size of the original nest. WRITE a b 25 b 2b

c b + 50 d The nearby nest has 100 more ants. e This nest has 1000 fewer ants. f This nest is half the size of the original nest.

remember remember
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.

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Maths Quest 8 for Victoria

4A
WORKED

Using pronumerals

Example

1 Suppose x people are in attendance at the start of a football match. a If a further y people arrive during the rst 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 a paragraph which could explain the table below: Time 9.00 am 9.15 am 10.45 am 12.30 pm 1.00 pm 5.30 pm Number of vanilla slices m m1 m 12 m 12 m 30 m 30

3 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. 4 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 which is equal to ve times y d The number formed when y is subtracted from 14 e 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 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.

Chapter 4 Algebra

121

6 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 tickets b d child tickets c 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 Steves age, if he is g years older than Naomi. c How old was Naomi 5 years ago? d Naomis father is twice her age. How old is he? 8 Charles places p coins into a poker machine. He plays the machine and counts his coins every 3 minutes. The table below shows how many coins he has. Time 7.10 pm 7.13 pm 7.16 pm 7.19 pm 7.21 pm 7.24 pm 7.27 pm 7.30 pm 7.33 pm Number of coins p 2p 2p + 12 4p + 12 4p + 7 p p+1 p8 p 12

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?

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Maths Quest 8 for Victoria

9 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: Time 9.00 am 12.00 pm 3.18 pm 6.20 pm 9.05 pm 12.00 am Number of bacteria m 2m 4m 8m 16m 32m 1240

a Explain what happens to the number of bacteria in the rst 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? 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 n ii the even number which is 2 less than n.

M AT H

GE

QUEST
1 Licia has bought her lunch from the school canteen for $3.00. It consisted of a roll, a carton of milk and a piece of fruit. She paid 60 cents more for the milk than the fruit and 30 cents more for the roll than the milk. How much did the roll cost her? 2 Find at least two 2-digit numbers that are 5 equal to 7 times the sum of their digits. 3 Find 5 consecutive numbers that add to 120. 4 Im thinking of a number. If I multiply it by 5 and subtract 4, I get the same number as If this pattern continues, how many when I multiply it by 4 and cubes will it take to make 10 layers? add 2. What is the number?

CH

AL

EN

Chapter 4 Algebra

123

Substitution
When 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. 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: 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. y For example, y 6 is shown as -- . 6 When substituting pronumerals, replace the multiplication signs, as shown in the worked example below.

Find the value of the following expressions if a = 3 and b = 15. 2b a 6a b 7a ----3 THINK a
1

WORKED Example 2

WRITE a 6a =63 = 18 2b b 7a ----3 2 15 = 7 3 -------------3 2 15 = 21 -------------3 30 = 21 ----3 = 21 10 = 11

Substitute the pronumeral (a) with its correct value and replace the multiplication sign. Multiply.

Substitute each pronumeral with its correct value and replace the multiplication signs. Do the rst multiplication. Do the next multiplication. Do the division. Do the subtraction.

3 4 5

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

124

Maths Quest 8 for Victoria

WORKED Example 3
The formula for nding the area (A) of a rectangle of length l and width w is A = l w. Use this formula to nd the area of the rectangle at right. THINK
1 2

270 m 32 m

WRITE A=lw

Write down the formula. Substitute each pronumeral with its correct value. Multiply to nd A and state the correct units.

= 270 32
A = 8640 m2

remember remember
1. 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. y For example, y 6 is shown as -- . 6

4B
SkillS
HEET

Substitution
a d -2 h ba 8 l -a

4.1

WORKED

Example

1 Find the value of the following expressions, if a = 2 and b = 5. a 3a e a+7 b i 5 + -5 25 m ----b q 6b 4a b 7a f j b4 3a + 9 c 6b

g a+b k 2a + 3b

Mat

d hca

Substitution

EXCEL

Spreadshe

Substitution

EXCEL

Spreadshe

Substitution game

n ab o 2ab p 7b 30 ab r ----5 2 Substitute x = 6 and y = 3 into the following expressions and evaluate. x y 24 9 - a 6x + 2y b -- + -c 3xy d ----- -- 3 3 x y 12 7x e ----- + 4 + y f 3x y g 2.5x h ----x 2 4xy 13y i 3.2x + 1.7y j 11y 2x k -------- 2x l -------15 3

et et

Chapter 4 Algebra

125

3 Evaluate the following expressions, if d = 5 and m = 2. a d+m b m+d c md e 2m i 3d f j n md 2m 7d ----15 g 5dm k 6m + 5d o 4dm 21

d dm md h ------10 3md l ---------2 15 p ----- m d

m 25m 2d
WORKED

Example

4 The formula for nding the perimeter (P) of a rectangle 3 of length l and width w is P = 2l + 2w. Use this formula to nd the perimeter of the rectangular swimming pool at right.

25 m 50 m

5 The formula F = 2c + 30 is used to convert temperatures measured in degrees Celsius to an approximate Fahrenheit value. F represents the temperature in degrees Fahrenheit and 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 nd 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 car for one day from Pooles 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 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 approximate equivalent 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 GAME A = lw. Find the area of the rectangles below: Algebra a length 12 cm, width 4 cm 001 b length 200 m, width 42 m c length 4.3 m, width 104 cm.
time

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Maths Quest 8 for Victoria

Working with brackets

Brackets 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 multiplication () 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 rst, 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 a
1 2 3 4

WORKED Example 4

WRITE a 5(s + r) = 5 (s + r) = 5 (5 + 4) =59 = 45 b 2x(3t y) = 2 x (3 t y) = 2 3 (3 4 5) = 2 3 (12 5) =237 = 42

Put the multiplication sign back into the expression. Substitute the pronumerals with their correct values. Work out the bracket rst. Complete the multiplication. Put the multiplication signs back into the expression. Substitute the pronumerals with their correct values. Do the multiplication inside the brackets. Do the subtraction inside the brackets. Do the nal multiplication.

1 2 3 4 5

remember remember
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 rst.

Chapter 4 Algebra

127

4C
WORKED

Working with brackets

HEET

Example

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) 4 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) d (x + y) (z y) y g -- ( 7 x + 3 ) 5 j 6 -- ( xz + y 3 ) x 12 b ----- ( z y ) x e (z 3)4x h (8 y) (z + x) z k ( y + 2 ) -x 27 n ( 3x 7 ) ----- + 7 x c f i l z 2y -- ----- + x 2 - 3 10 zy(17 xy) 7 12 4y ---- x 2x(xyz 105)

4.2

SkillS

Math

cad

Substitution (brackets)

m 12(y 1) (z + 3)

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 nd 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

27.5 cm

4 A rule for nding the sum of the interior angles in a many-sided gure such as a pentagon is S = 180(n 2) where S represents the sum of the angles inside the gure and n represents the number of sides. The diagram at right shows the interior angles in a pentagon. Use the rule to nd the sum of the interior angles for the following gures: a a hexagon (6 sides) b a pentagon c a triangle d a quadrilateral (4 sides) e a 20-sided gure.

21.4 cm

128

Maths Quest 8 for Victoria

Join the dots next to the values of the expressions in the orders given below using: a = 1, b = 2, c = 3, x = 5 and y = 10.
Start

Im now in Australia!
51 64 52 41 32 28 16 74 5 12 86 47 81 29 90 20 85 57 33 66 70 19 8 48 4 96 27 37 34 39 95 6 84 68 35 60 49 7 42 22 45 13 9 1 75 25 53 62 65 30 100 80 99 2 40 36 91 11 23 69 10 72 24 56 44 15 31 54 88 14 63 50 3 18 77 21 61 55 38 43 46 26

2y c = 6(x + a) = y+xb= 3c =
Start
Stop Start
4cy x

17

Start

Start

a+b= cx =
y x

12x c = 2bx = x(2y c) = 11c = 3cy = cy a =

10(x + b) 4 =

6(b + c) = yc= 12(x + c) = 9c = 20x = 10c = 27 = c 30 + 2b = b(y + c) = cy b =

Stop Start

black

Coloring guide: Orange

2bx a = c+x= 7y cx b = a + 9y =

a2 = a+b= 4by = 7(a + b) = 11x = c+8= x(b + c) = y(c + x) =

xy b

6c + xy = 7(c + x) = 11y 4c + a =

bcy = 8x = 7c + 2x = 3bc = xb= xy = 2y + c = x(y 1) = y bc = x(y + b) =

Stop Start

Join these points with thick lines. Start Start

8y b

a=

80 + x + c = 7(a + 4b) = 12x + y + b = c + 2b + 3a =

Stop Start

7x + b = 7bc = 3x + 5 = xy a = 6y + b = 2(y + b) =
Stop Start
11(x + c) b

40b + 2b = 6(x + c) = yxa=

Stop Start

Stop Start

11(a + b + c) =

4b + 2c = 20c + y a = 6x 4b = 4y x b = 2x 3b =
Stop Start

xy c = bcb = b+c= 4(c + y) = 4(x + b) = 8b =

x(y + a + c) =

7x + c = 20b + c = 8x + 7c =
Stop Start

8(c + x) = 9y 2x + a = 7y + 8b = 12c + 3y =
Stop Start

y2 = 7x = 5(x + y) = b+c+x= ya+b=

Stop

13x = bc = 9y x =

9x + a = 8y c =
cxy + 4c = c 5y + b + c = x

bcx + b = 10y x = 8x + b a = 4x + 2c =
Stop

12x 3c = 10x + 8c = 3cx + b =

Stop

ax =
Stop

Stop

Chapter 4 Algebra

129

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

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. 12 c Substitute a = 4 and b = 3 into the expression 5ab ----- and evaluate. b THINK a
1 2

WORKED Example 5

WRITE a mn = 5 3 =5+3 =8 b 2n m =2nm = 2 1 2 = 2 2 = 2 + 2 =0 12 c 5ab ----b 12 = 5 a b ----b 12 = 5 4 3 ----3 12 = 60 ----3 = 60 4 = 60 + 4 = 56

Replace the pronumerals with their correct value. Combine the two negative signs and add. Replace the multiplication sign. Substitute the pronumerals with their correct values. Do the multiplication. Combine the two negative signs and add. Replace the multiplication signs.

1 2 3 4

Substitute the pronumerals with their correct values. Do the multiplications. Do the division. Combine the two negative signs and add.

3 4 5

130

Maths Quest 8 for Victoria

remember remember
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

4D
SkillS
HEET

Substituting positive and negative numbers

4.3

WORKED

Example

1 Substitute m = 6 and n = 3 into the following expressions and evaluate. a m+n b mn c nm d n+m 5a e 3n f 2m g 2n m h n+5 m i 2m + n 4 j 11n + 20 k 5n m l --2 mn m -----9 9 m - q -- + --n 2 n r 4m ----------n5 6mn 1 4m o -----n s 3n ----- + 1.5 2 12 p ----2n t mn 14 -----9

Mat

d hca

Substitution (positive/ negative)

Work

ET SHE

4.1

WORKED

Example

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) 5b d 2(3 y) e (y + 5)x f xy(7 x) x g (3 + x) (5 + y) h 5(7 xy) i -- ( 5 y ) 2 j x -- 1 2y + 4 ----4 6 k 9 -- ( 6 x ) y l y 3 ( x 1 ) -- + 2 3

WORKED

Example

3 Substitute a = 4 and b = 5 into the following expressions and evaluate. a a+b b ab c b 2a d 2ab 5c e 12 ab f 2(b a) g ab4 h 3a(b + 4) 4 6b 8 16 i -j -k ----l ----b 5 a 4a m 45 + 4ab q 11a + 6b n 8ab 3b r (a 5)(8 b) a 3b o -- + ----2 5 s (9 a)(b 3) p 2.5b t 1.5b + 2a

Chapter 4 Algebra

131

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 9 degrees Fahrenheit (F) is F = -- C + 32, investigate at which temperatures the 5 rule of thumb expression gives the best results.

1
1 If a kilogram of oranges cost $0.89 and a kilogram of carrots cost $0.99, what is the 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 when d is divided by 5. 12 3 True or false? If y = 4 and z = 1 then ----- + 3z = 4 . y 4 The area of a circle is p r 2 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? 12 6 Evaluate ----- ( rs + 4 s ) if r = 4 and s = 6. s 7 multiple choice n When m = 7 and n = 4 are substituted into the expression 3m + -- , the value is: 4 A 21 B 22 C 22.25 D 25 E 28 14 8 Substitute p = 7 and q = 2 into ----- 1 ( pq + 3 ) . p a b - 9 From the list 2, 1, 3, 4 choose the value of a and b when -- + -- = 0 . 2 4 12 10 Substitute x = 3 and y = 5 into the expression ----- + 3y and evaluate. x

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Maths Quest 8 for Victoria

Simplifying expressions
Expressions 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 simplied (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. 3ab and 7ab are like terms. 2bc and 4cb are like terms. 3g2 and 45g2 are like terms.

3x and 3y are not like terms. 7ab and 8a are not like terms. 8a and 3a2 are not like terms.

WORKED Example 6
Simplify the following expressions. a 3a + 5a b 7ab 3a 4ab c 2c 6 + 4c + 15 THINK a
1

WRITE a 3a + 5a

Write down the expression and check that the pronumeral parts of the 2 terms are the same. They are. Add the 2 terms. Write down the expression. Rearrange the terms so that the like terms are together. Remember to keep the correct sign in front of each term. Simplify by subtracting the like terms. Write down the expression. Rearrange the terms so that the like terms are together. Remember to keep the correct sign in front of each term. Simplify by collecting the like terms.

= 8a b 7ab 3a 4ab = 7ab 4ab 3a

1 2

= 3ab 3a

1 2

c 2c 6 + 4c + 15 = 2c + 4c 6 + 15

= 6c + 9

Chapter 4 Algebra

133

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

4E

Simplifying expressions
Math
cad

1 multiple choice Simplifying 3a + 9a gives: A 12 B 12a C 6a 2 multiple choice Simplifying 6x 2x gives: A 4 B 4x2 C 4x 3 multiple choice Simplifying 6a + 6b gives: A 12ab B 6ab C 36ab
WORKED

D 12a2

E The expression cannot be simplied.

Simplifying expressions

D 2x

E The expression cannot be simplied.

D 12a

E The expression cannot be simplied. c f i l o r u x 3a + 5a 4a 7x 5x 4p 7p 7t 8t + 4t 7z + 13z 18b 4b 11b 12l + 2l 5l t + 2t t + 8t

Example

4 Simplify the following expressions. a 4c + 2c b 2c 5c 6a d 6q 5q e h 2h g 3a 7a 2a h 3f + 7f j 3h + 4h k 11b + 2b + 5b m 9m + 5m m n x 2x p 5p + 3p + 2p q 9g + 12g 4g s 13t 4t + 5t t 11j + 4j v 13m 2m 4m + m w m + 3m 4m b d f h j l n p r t v

WORKED

Example

5 Simplify the following expressions. a 3x + 7x 2y 6b, c c 11 + 5f 7f e 2m + 3p + 5m g 11a 5b + 6a i 12 3g + 5 k 5k 5 + 2k 7 m 2b 6 4b + 18 o 12y 3y 7g + 5g 6 q 11s 6t + 4t 7s s 3h + 4k 16h k + 7 u 2g + 5 + 5g 7

3x + 4x 12 3u 4u + 6 3h + 4r 2h 9t 7 + 5 6m + 4m 3n + n 3n 4 + n 5 11 12h + 9 8h 6 + 3h 2 2m + 13l 7m + l 13 + 5t 9t 8 17f 3k + 2f 7k

134

Maths Quest 8 for Victoria

6 Simplify the following. a x2 + 2x2 d d 2 + 6d 2 g 2b2 + 5b2 j a2 + 4 + 3a2 + 5 m 3a2 + 2a + 5a2 + 3a p 11g3 + 17 3g3 + 5 g2 s 4fg + 2s fg + s

b e h k n q t

3y2 + 2y2 7g2 8g2 4a2 3a2 11x2 6 + 12x2 + 6 11b 3b2 + 4b2 + 12b 12ab + 3 + 6ab 11ab + ab 5

c f i l o r

a3 + 3a3 3y3 + 7y3 g2 2g2 12s2 3 + 7 s2 6t2 6g 5t2 + 2g 7 14xy + 3xy xy 5xy

Multiplying pronumerals
When multiplying pronumerals, remember that order is not important. For example: 36=63 6w=w6 ab=ba Also keep in mind that the 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

WORKED Example 7
Simplify: a 5 4g b 3d 6ab 7. THINK a
1

WRITE a 5 4g =54g = 20 g = 20g b 3d 6ab 7 = 3 d 6 a b 7 = 3 6 7 d a b = 126 d a b = 126abd

2 3

Write down the expression and replace the hidden multiplication signs. Multiply the numbers. Remove the multiplication sign. Write down the expression and replace the hidden multiplication signs. Put the numbers at the front. Multiply the numbers. Remove the multiplication signs.

2 3 4

remember remember
When 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.

Chapter 4 Algebra

135

4F
WORKED

Multiplying pronumerals
b e h k n q t w z 7 3h 6 5r 7 6p 4x 6g 3c 5h 13m 12n 4f 3gh 16xy 1.5 4a 3b 2c b d f h j l n p r t c f i l o r u x 4d 6 5t 7 7gy 3 10a 7h 9g 2x 6a 12d 2 8w 3x 3.5x 3y

Example

1 Simplify the following. a 4 3g 7 d 3z 5 g 4 3u j 2 11ht m 9m 4d p 2.5t 5b s 2ab 3c v 11ab 3d 7 y 11q 4s 3 2 Simplify the following. a 3 5f c 11a 3g e 5t 4dh g 3 2w 7d i 11ab 3f k 5h 5t 3q m 7a 3b g o 3.5g 2h 7 q 75x 1.5y s 2ab 3c 5

6 2d 9t 3g 6 3st 4a 3b 2c e 3as 3b 2x 4 3w 2 6p 17ab 3gh 5h 8j k 12rt 3z 4p 4w 34x 3

Sonar measurements
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 oor 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 nd 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.

136

Maths Quest 8 for Victoria

History of mathematics
T H E R H I N D PA P Y R U S ( c . 1 8 5 0
BC)

During this time . . . The Sumerians built the rst cities, invented writing and made wheels from date palm trunks. Papyrus reeds were used to make boats, baskets and paper. The Bronze Age began.

The ancient Egyptians differed from the ancient Greeks in that Egyptians thought about mathematics in a practical rather than an abstract way. They didnt 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
1 -2

1 -3

-= 1. 6

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 () 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. Questions 1. 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? Research How was Egyptian multiplication done with only addition and subtraction?

Chapter 4 Algebra

137

Dividing pronumerals
When 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.

WORKED Example 8
16 f a Simplify --------- . 4 b Simplify 15n 3n. THINK a
1

WRITE 16 f a -------4 4f = ----1 = 4f

Write down the expression. Simplify the fraction by cancelling 16 with 4 (divide both by 4). No need to write the denominator since we are dividing by 1. Write down the expression and then rewrite it as a fraction.

b 15n 3n 15n = -------3n 5 = -1 =5

Simplify the fraction by cancelling 15 with 3 and n with n. No need to write the denominator since we are dividing by 1.

Simplify 12xy 27y. THINK

1

WORKED Example 9
WRITE 12xy 27y 12xy = ----------27y 4x = ----9 Write down the expression and then rewrite it as a fraction.

Simplify the fraction by cancelling 12 with 27 (divide both by 3) and y with y.

138

Maths Quest 8 for Victoria

remember remember
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.

4G
SkillS
HEET

Dividing pronumerals
6h b ----3 e 10r 5 16m h --------8m 12h k -------14h n 35x 70x q 27h 3h 15x -------3 4x 2x 14q 21q 50g 75g

4.4

WORKED

Example

1 Simplify the following. 8f 8 a ----2 d 9g 3 g 8r 4r j 3x ----6x 8f m -------24 f p y 34y 2 Simplify the following. 15 fg a ----------b 3 11xy e ----------f 11x 5 jk i -------j kj m 13xy x 132mnp q ------------------60np n r

c f i l

o 24m 36m r 20d -------48d

12cd 4 9 pq --------18q 55rt 77t 16cd ----------40cd 11ad -----------66ad

8xy -------12 21ab g ----------28b 10mxy k --------------35mx c o 14abc 7bc s 18adg 45ag

d 24cg 24 h l 9dg -------12g 36bc 27c

p 3gh 6h t bh ----7h

WORKED

Example

Work

ET SHE

4.2

3 Simplify the following. 4a 11ab 9 a -------b --------------8 33b 32g e ----------f 12xy 48y 40gl rt i 4xyz 6yz j ------6rt ab m 34ab 17ab n ------- 3a

60jk 5k

d 3h 6dh h l 6 fgh ------------30ghj 14st 28

12ab g -------------- 14ab k 5mn 20n 7dg o -----------35gh

p 60mn 55mnp

Chapter 4 Algebra

139

2
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. p 3 Evaluate ( r + 10 ) -- if p = 4, q = 2 and r = 7. q 4 multiple choice m 6n If m = 6 and n = 3 are substituted into the expression --- + ----- , it would have 2 9 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 30ab 8 Simplify -------------- . 18abc 9 Find the --------------- = 12 pr missing term from the list 2, 4, 12pq, 48pq to replace in 4q ----- . r

9 p 10 Simplify --------------- . 36 pq

Expanding brackets
We 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) 32+31 =33 =6+3 =9 =9 2. 4 (3 + 2) 43+42 =45 = 12 + 8 = 20 = 20 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.

140

Maths Quest 8 for Victoria

WORKED Example 10
Use the Distributive Law to expand the following expressions. a 3(a + 2) b x(x 5) THINK a
1

WRITE a 3(a + 2) = 3 (a + 2) =3a+32 = 3a + 6 b x(x 5) = x (x 5) = x x + x 5 = x2 5x

Write down the expression and replace the hidden multiplication sign. Use the Distributive Law to expand the brackets. Simplify by multiplying.

b Repeat the steps in part a.

remember remember
1. 2. 3. 4. Brackets are grouping symbols Removing brackets from an expression is called expanding the expression. When expanding brackets, put the sign before the bracket. The rule that is used to expand brackets is called the Distributive Law.

4H
WORKED

Expanding brackets

Example

Mat

d hca

Expanding brackets

EXCEL

Spreadshe

Expanding brackets

GC p

am rogr

Expanding brackets

1 Use the Distributive Law to expand the following expressions. a 3(d + 4) b 2(a + 5) c 4(x + 2) 10 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)

et

Chapter 4 Algebra

141

History of mathematics
J O H N C OAT E S ( 1 9 4 5 )

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, Fermats 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. Questions 1. 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? Research What was Fermats Last Theorem?

142

Maths Quest 8 for Victoria

Expanding and collecting like terms

Some expressions can be simplied further by collecting like terms after any brackets have been expanded.

WORKED Example 11
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) THINK a
1 2

d 4x(2x 1) 3(2x 1) WRITE a 3(x 5) + 4 = 3 (x 5) + 4 = 3x 15 + 4 = 3x 11 b 4(3x + 4) + 7x + 12 = 12x + 16 + 7x + 12 = 12x + 7x + 16 + 12 = 19x + 28 c 2x(3y + 3) + 3x(y + 1) = 2x 3y + 2x 3 + 3x y + 3x 1 = 6xy + 6x + 3xy + 3x = 6xy + 3xy + 6x + 3x = 9xy + 9x d 4x(2x 1) 3(2x 1) = 4x 2x + 4x 1 3 2x 3 1 = 8x2 4x 6x + 3 = 8x2 10x + 3

Write the expression. Expand the brackets. Collect the like terms (15 and 4). Write the expression. Expand the brackets. Rearrange so that the like terms are together. (Optional) Collect the like terms. Write the expression. Expand the brackets. Rearrange so that the like terms are together. (Optional) Simplify by collecting the like terms. Write the expression. Expand the brackets. Take care with negative terms. Simplify by collecting the like terms.

1 2 3

1 2

1 2

remember remember
After expanding brackets, collect any like terms.

Chapter 4 Algebra

143

4I
WORKED

Expanding and collecting like terms

GC pro
gram

Example

1 Expand the expressions below and then simplify by collecting any like terms. a 7(5x + 4) + 21 b 3(c 2) + 2 11 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 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) 2 i 3(2 t ) + 2t(t + 1) j m(n + 4) mn + 3m

Expanding

GAME
time

Algebra

3 Simplify the following expressions by removing the brackets and then collecting like 002 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) ET SHE 4.3 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)
Work

Factorising
Factorising 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 factors
Two 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 factor

The 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.

144

Maths Quest 8 for Victoria

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 =32m Here are some other examples: 8x = 8 x =42x =222x 3ab = 3 a b 6a2b = 6 a a b =32aab To nd 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.

WORKED Example 12
Find the highest common factor (HCF) of 6x and 10. THINK
1

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. Find the highest common factor of the pronumeral parts. There isnt one, because only the rst term has a pronumeral part!

6=32 10 = 5 2 HCF = 2

The HCF of 6x and 10 is 2.

WORKED Example 13
Find the highest common factor (HCF) of 14fg and 21gh. THINK
1

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. 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. Multiply these together.

14 = 7 2 21 = 7 3 HCF = 7 fg = f g gh = g h HCF = g The HCF of 14fg and 21gh is 7g.

Chapter 4 Algebra

145

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.

Factorise the expression 2x + 6. THINK

1

WORKED Example 14
WRITE 2x + 6 =2x+23 = 2 (x + 3) Break down each term into factors. Write the common factor outside the brackets and the other factors inside the brackets. Remove the multiplication sign.

= 2(x + 3)

Factorise 12gh 8g. THINK

1

WORKED Example 15
WRITE 12gh 8g =43gh42g = 4 g (3 h 2) Break down each term into its factors. Write the highest common factor outside the brackets. Write the other factors inside the brackets. Remove the multiplication signs.

= 4g(3h 2)

remember remember
1. Factorising is the opposite process to expanding. 2. Factorising a number or expression involves breaking it down into smaller factors. 3. To nd 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.

146

Maths Quest 8 for Victoria

4J
Mat

Factorising

d hca

Factorising

1 multiple choice a The highest common factor (HCF) of 12 and 16 is: A 12 B 4 C 8 D 2 b The highest common factor (HCF) of 10 and 18 is: A 4 B 10 C 2 D 9

E 3 E 180 E 8 E 8 E 2f d 13 and 26 h 12a and 16

EXCEL

Spreadshe

The highest common factor (HCF) of 4 and 16 is: A 4 B 16 C 2 D 20

Finding the HCF

et
WORKED

d The highest common factor (HCF) of 2x and 8xy is: A 2 B x C 2x D 16x2y e The highest common factor (HCF) of 4f and 12fg is: A 2 B fg C 48f 2g D 4f
Example

2 Find the highest common factor (HCF) of the following. a 4 and 6 b 6 and 9 c 12 and 18 12 e 14 and 21 f 2x and 4 g 3x and 9 c f i l c f i l o r b d f h j l n p r t v

3 Find the highest common factor (HCF) of the following. WORKED a 2gh and 6g b 3mn and 6mp Example d 4ma and 6m e 12ab and 14ac 13 g 20dg and 18ghq h 11gl and 33lp j 28bc and 12c k 4c and 12cd
WORKED

11a and 22b 24fg and 36gh 16mnp and 20mn x and 3xz 5g + 10 12c + 20 12g 18 8x 20 16a + 64 12 12d

Example

4 Factorise the following expressions. a 3x + 6 b 2y + 4 14 d 8x + 12 e 6f + 9 g 2d + 8 h 2x 4 j 11h + 121 k 4s 16 m 12g 24 n 14 4b p 48 12q q 16 + 8f 5 Factorise the following. a 3gh + 12 15 c 12pq + 4p e 16jk 2k g 12k + 16 i 14ab + 7b k 8r + 14rt m 4b 6ab o ab 2bc q 11jk + 3k s 12ac 4c + 3dc u 28s + 14st

WORKED

Example

2xy + 6y 14g 7gh 12eg + 2g 7mn + 6m 5a 15abc 24mab + 12ab 12fg 16gh 14x 21xy 3p + 27pq 4g + 8gh 16 15uv + 27vw

Chapter 4 Algebra

147

Doctor Ive swallowed the film out of I swallow my camera!

8y(x 9) (2x 1) 6(3 7x) y(x 1) x(y 2) 5(4x + 5) 2(4 x) 7(7 + 4x) 4(3x + 5) 2(x + 1) 2y(3x 5) 2y(x 4) (x 2) 3(x 4) 3(4x + 7) 2(3x 7) 5(3x + 2) y(3x 2)

3(2x 1) 8(x + 3) 7(8x 5) 3x(2y + 1) y(2x 3) The factorised form of the expressions and factorised expressions the letter beside each gives the puzzle code. gives code.

D = 6x 3 = E = 8 2x = G = 15x + 10 = H = 2x + 1 = I = 12x + 20 = L = 2xy 3y =

E = 6xy + 3x = H = 20x 25 = L = 8xy 72y = N = x + 2 = E = 6x 14 = O = xy + y =

N = 18 42x = O = 2x 2 = P = 2xy 8y = S = 12x 21 = T = xy 2x = V = 56x 35 = P = 3x 12 = E = 8x 24 = S = 3xy 2y = T = 6xy 10y = O = 49 28x =

148

Maths Quest 8 for Victoria

summary
Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list that follows. 1 2 3 A is a letter that is used in place of a number. . sign () is rarely used. and simplify it by Replacing a pronumeral with a number is called When dividing pronumerals, the Normally we rewrite the expression as a cancelling. 4 5 6 7 8 9

When multiplying pronumerals, leave out the sign. The term 3y means . Brackets are or 3 x + 3 4. symbols. For example, 3(x + 4) means 3 (x + 4)

When simplifying an expression, terms may be collected only if they are . Expanding an expression involves 3(x + 2) = 3x + 6. The brackets. For example

Law gives the rule for expanding expressions.

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

WORD
substitution Distributive 3y

LIST
factors pronumeral removing grouping division like terms fraction

Chapter 4 Algebra

149

CHAPTER review
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 ve 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.

4A

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.

4A

3 Find the value of the following expressions, if a = 2 and b = 6. a 2a b 6a a c 5b d -2 e a+8 f b2 g a+b h ba b i 5 + -j 3a + 7 2 k 2a + 3b l 20 ----a

4B

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.

4B

150
4C

Maths Quest 8 for Victoria

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 ab c ab d ----5 e 2ab f 5a g 12 ab h a2 2 i 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 c 3d + 5a 4a e 4x + 11 2x g 2xy + 7xy 8 Simplify the following. a 3 7g c 7d 6 9 Simplify the following. 2a a ----8 c 6rt 2t 32t e -----------40stv 12ab g -------------- 14ab b d f h 3c 5c 6g 4g 2g + 5 g 6 12t 2 + 3t + 3t 2 t

4D

4E

4F 4G

b 6 3y d 3z 8 11b b -------44b d 3gh 6g f h 36xy 12y 5egh ------------30ghj

4H 4I 4J
test yourself
CHAPTER

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 c 5n 20 e 12g 6gh b xy + 5y d 12mn + 4pn f 12xy 36yz