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5
44eTHINKING
Algebra
Ships use the speed of sound in water to help find the water’s depth. A sonar pulse from a ship is sent to the bottom of the ocean floor. The time taken for the pulse to hit the ocean floor and return to the ship is used to calculate the distance. If the sonar pulse returns in 1.5 seconds, what is the ocean depth?
Assume that the speed of sound in water is 1470 metres per second.
How could you set up a procedure to quickly calculate the ocean depth for any time measurement?
This chapter looks at using pronumerals to represent quantities in different situations. You will learn how to form and use algebraic expressions and how to express them in simpler forms.
198 M a t h s Q u e s t 8 f o r V i c t o r i a
READY?areyou
Are you ready?Try the questions below. If you have difficulty with any of them, extra help can be
obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon
next to the question on the Maths Quest 8 CD-ROM or ask your teacher for a copy.
Alternative expressions used to describe the four operations
1 Write expressions for the following:
a the difference between M and C
b the amount of money earned by selling B lamingtons for $2 each
c the product of X and Y
d 12 more than H
e the cost of 10 oranges if each orange costs D cents.
Order of operations II
2 Find the value of each of the following using the order of operations rule.
a 12 + 18 ÷ 6 b 14 − 16 × 4 ÷ 8 c 7 × 3 − 5 × 12
Order of operations with brackets
3 Find the value of each of the following using the order of operations rule.
a 5 + (10 × 2) − 13 b (2 × 12) + (18 ÷ 6) − (2 + 7)
Operations with directed numbers
4 Perform each of the calculations.
a −8 + −10 b 8 − −17 c 26 × −2 d −32 ÷ −4
Combining like terms
5 Simplify the following expressions by adding or subtracting like terms.
a 3g + 4g b y + 2y + 3y c 6gy − 3yg
d 20x − 19x + 11 e 7g + 8g + 8 + 9 f 7h + 4t − 3h
Simplifying fractions
6 Simplify the following.
a b c
Highest common factor
7 Find the highest common factor for each of the following pairs of numbers.
a 8, 28 b 21, 35 c 18, 27
5.1
5.2
5.3
5.4
5.5
5.6
7
21------
12
30------
15
20------
5.7
C h a p t e r 5 A l g e b r a 199
Using pronumeralsThe basic purpose of algebra is to solve mathematical
problems involving an unknown. Equations where an
unknown quantity is replaced with a letter (for
example, x) can be used to solve problems like:
At what speed should I ride my bicycle to arrive at
school on time?
How do I convert a recipe for different numbers of
guests?
What volume of concrete is needed to build a path?
A pronumeral is a letter that is used in place of a
number. In Year 7 we saw that pronumerals could be
used to make expressions and equations. Often a pro-
numeral is used to represent one particular number. For
example, in the equation
x + 1 = 7
the pronumeral x has the value 6.
Pronumerals can also be used to show a relationship
between two or more numbers, for example.
a + b = 10
Can you find some different pairs of values for a and b
that fit this rule?
Algebra allows us to show complex rules in a more
simple way and to solve problems involving
unknown numbers.
In algebraic expressions pronumerals are known as variables because the value of
the pronumeral may be varied to represent any number. With equations, pronumerals
are referred to as unknowns because the pronumeral represents a specific value that is
not yet known.
Worked example 1 shows some of the ways pronumerals can be used.
Suppose we use b to represent the number of ants in a nest.
a Write an expression for the number of ants in the nest if 25 ants died.
b Write an expression for the number of ants in the nest if the original ant population
doubled.
c Write an expression for the number of ants in the nest if the original population increased
by 50.
d What would it mean if we said that a nearby nest contained b + 100 ants?
e What would it mean if we said that another nest contained b − 1000 ants?
f Another nest in very poor soil contains ants. How much smaller than the original is
this nest?
Continued over page
b
2---
1WORKEDExample
200 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 we use x to represent the number of ants in a nest.
a Write an expression for the number of ants in the nest if 420 ants were born.
b Write an expression for the number of ants in the nest if the original ant population
tripled.
c Write an expression for the number of ants in the nest if the original ant population
decreased by 130.
d What would it mean if we said that a nearby nest contained x + 60 ants?
e What would it mean if we said that a nearby nest contained x − 90 ants?
f Another nest in very poor soil contains ants. How much smaller than the original
is this nest?
THINK WRITE
a The original number of ants (b) must be
reduced by 25.
a b − 25
b The original number of ants (b) must be
multiplied by 2. It is not necessary to
show the × sign.
b 2b
c 50 must be added to the original number
of ants (b).
c b + 50
d This expression tells us that the nearby
nest has 100 more ants.
d The nearby nest has 100 more ants.
e This expression tells us that the nest has
1000 fewer ants.
e This nest has 1000 fewer ants.
f The expression means b ÷ 2, so this
nest is half the size of the original nest.
f This nest is half the size of the original nest.b
2---
1. A pronumeral is a letter that is used in place of a number.
2. Pronumerals may represent a single number, or they may be used to show a
relationship between two or more numbers.
remember
5AWORKED
Example
1
x
4---
C h a p t e r 5 A l g e b r a 201
2 Suppose x people are in attendance at the start of a football match.
a If a further y people arrive during the first quarter, write an expression for the
number of people at the ground.
b Write an expression for the number of people at the ground if a further 260 people
arrive prior to the second quarter commencing.
c At half-time 170 people leave. Write an expression for the number of people at the
ground after they have left.
d In the final quarter a further 350 people leave. Write an expression for the number
of people at the ground after they have left.
3 The canteen manager at Browning Industries orders m Danish pastries each day. Write
a paragraph which could explain the table below:
4 Imagine that your cutlery drawer contains a knives, b forks and c spoons.
a Write an expression for the total number of knives and forks you have.
b Write an expression for the total number of items in the drawer.
c You put 4 more forks in the drawer. Write an expression for the number of forks
now.
d Write an expression for the number of knives in the drawer after 6 knives are
removed.
5 If y represents a certain number, write expressions for the following numbers.
a A number 7 more than y
b A number 8 less than y
c A number that is equal to five times y
d The number formed when y is subtracted from 14
e The number formed when y is divided by 3
f The number formed when y is multiplied by 8 and 3 is added to the result
6 Using a and b to represent numbers, write expressions for:
a the sum of a and b
b the difference between a and b
c three times a subtracted from two times b
d the product of a and b
e twice the product of a and b
f the sum of 3a and 7b
g a multiplied by itself
h a multiplied by itself and the result divided by 5.
Time Number of Danish pastries
9.00 am m
9.15 am m – 1
10.45 am m – 12
12.30 pm m – 12
1.00 pm m – 30
5.30 pm m – 30
SkillSHEET
5.1
Alternativeexpressions
used todescribethe four
operations
202 M a t h s Q u e s t 8 f o r V i c t o r i a
7 If tickets to a basketball match cost $27 for adults and $14 for children, write an
expression for the cost of:
a y adult tickets b d child tickets c r adult and h child tickets.
8 Naomi is now t years old.
a Write an expression for her age in 2 years’ time.
b Write an expression for Steve’s age if he is g years older than Naomi.
c How old was Naomi 5 years ago?
d Naomi’s father is twice her age. How old is he?
9 James is travelling into town one particular evening and observes that there are t
passengers in his carriage. He continues to take note of the number of people in his
carriage each time the train departs from a station, which occurs every 3 minutes. The
table below shows the number of passengers.
a Write a paragraph explaining what happened.
b When did passengers first begin to alight the train?
Time Number of passengers
7.10 pm t
7.13 pm 2t
7.16 pm 2t + 12
7.19 pm 4t + 12
7.22 pm 4t + 7
7.25 pm t
7.28 pm t + 1
7.31 pm t − 8
7.34 pm t − 12
C h a p t e r 5 A l g e b r a 203
c At what time did the carriage have the most number of passengers?
d At what time did the carriage have the least number of passengers?
10 A microbiologist places m bacteria onto an agar plate. She counts the number of
bacteria at approximately 3 hour intervals. The results are shown in the table below:
a Explain what happens to the number of
bacteria in the first 5 intervals.
b What might be causing the number of
bacteria to increase in this way?
c What is different about the last bacteria
count?
d What may have happened to cause this?
11 n represents an even number:
a Is the number n + 1 odd or even?
b Is 3n odd or even?
c Write expressions for:
i the next three even numbers that are greater than n
ii the even number that is 2 less than n.
TimeNumber of
bacteria
9.00 am m
12 noon 2m
3.18 pm 4m
6.20 pm 8m
9.05 pm 16m
12 midnight 32m – 1240
MA
TH
S Q
UEST
CHALL
EN
GE
CHALL
EN
GE
MA
TH
S Q
UEST
1 Licia has bought her lunch from the school canteen for $3.00. It con-sisted of a roll, a carton of milk and a piece of fruit. She paid 60 centsmore for the milk than the fruit and 30 cents more for the roll than themilk. How much did the roll cost her?
2 Find at least two 2-digit numbers that are equal to 7 times the sum of their digits.
3 Find 5 consecutive numbers that add to 120.
4 I’m thinking of a number. If I multiply it by 5 and subtract 4, I get the same number as when I multiply it by 4 and add 2. What is the number?
5
If this pattern continues, how manycubes will it take to make 10 layers?
204 M a t h s Q u e s t 8 f o r V i c t o r i a
SubstitutionWhen a pronumeral is replaced by a number, we say that the number is substituted for
the pronumeral. If the value of the pronumeral (or pronumerals) is known, it is possible
to evaluate (work out the value of) an expression by using substitution — that is,
replace the pronumeral by a number.
For example, if we know that x = 2 and y = 3, the expression x + y can be evaluated
as shown:
x + y = 2 + 3
= 5
When writing expressions with pronumerals, it is important to remember the
following points:
1. The multiplication sign is omitted.
For example: 8n means ‘8 × n’ and 12ab means ‘12 × a × b’.
2. The division sign is rarely used.
For example, y ÷ 6 is shown as .
When substituting pronumerals, replace the multiplication signs, as shown in the
worked example that follows.
y
6---
Find the value of the following expressions if a = 3 and b = 15.
a 6a b
THINK WRITE
a Substitute the pronumeral (a) with
its correct value and replace the
multiplication sign.
a 6a = 6 × 3
Evaluate and write the answer. = 18
b Substitute each pronumeral with its
correct value and replace the
multiplication signs.
b =
Perform the first multiplication. =
Perform the next multiplication. =
Perform the division. = 21 − 10
Perform the subtraction and write
the answer.
= 11
7a2b
3------–
1
2
1 7a2b
3------– 7 3
2 15×
3---------------–×
2 212 15×
3---------------–
3 2130
3------–
4
5
2WORKEDExample
C h a p t e r 5 A l g e b r a 205
The same methods are used when substituting into a formula or rule.
Substitution
1 Find the value of the following expressions, if a = 2 and b = 5.
a 3a b 7a c 6b d
e a + 7 f b − 4 g a + b h b − a
i j 3a + 9 k 2a + 3b l
m n ab o 2ab p 7b − 30
q 6b − 4a r s + t –
The formula for finding the area (A) of a rectangle of
length l and width w is A = l × w. Use this formula to
find the area of the rectangle at right.
THINK WRITE
Write the formula. A = l × w
Substitute each pronumeral with its
correct value.
= 270 × 32
Perform the multiplication and state the
correct units.
= 8640 m2
1
2
3
3WORKEDExample
270 m
32 m
1. Replacing a pronumeral with a number is called substitution.
2. When writing expressions with pronumerals, it is important to remember the
following points:
(a) The multiplication sign is omitted.
For example: 8n means ‘8 × n’ and 12ab means ‘12 × a × b’.
(b) The division sign is rarely used.
For example, y ÷ 6 is shown as . y
6---
remember
5BWORKED
Example
2
SkillSHEET
5.2
Order ofoperations II
a
2---
5b
5---+
8
a---
25
b------
ab
5------
15
a------
7
b---
9
a---
3
b---
206 M a t h s Q u e s t 8 f o r V i c t o r i a
2 Substitute x = 6 and y = 3 into the following expressions and evaluate.
a 6x + 2y b c 3xy d
e f 3x − y g 2.5x h
i 3.2x + 1.7y j 11y − 2x k l
m 4.8x – 3.5y n 8.7y – 3x o 12.3x – 9.6x p –
3 Evaluate the following expressions, if d = 5 and m = 2.
a d + m b m + d c m − d d d − m
e 2m f md g 5dm h
i −3d j −2m k 6m + 5d l
m 25m − 2d n o 4dm − 21 p
4 The formula for
finding the
perimeter (P) of
a rectangle of
length l and
width w is
P = 2l + 2w. Use
this formula to
find the
perimeter of the
rectangular
swimming pool
at right.
5 The formula F = C + 32 is used to convert temperatures measured in degrees Celsius
to an approximate Fahrenheit value. F represents the temperature in degrees Fahrenheit
(°F) and C the temperature in degrees Celsius (°C).
a Find F when C = 100°C.
b Convert 28° Celsius to Fahrenheit.
c Water freezes at 0° Celsius. What is the freezing temperature of water in Fahrenheit?
6 The formula for the perimeter (P) of a square of side length l is P = 4l. Use this formula
to find the perimeter of a square of side length 2.5 cm.
Mat
hcad
Substitution
EXCE
L Spreadsheet
Substitution
EXCE
L Spreadsheet
Substitutiongame
x
3---
y
3---+
24
x------
9
y---–
12
x------ 4 y+ +
7x
2------
13y
3--------- 2x–
4xy
15---------
3x
9------
y
12------
md
10-------
3md
2-----------
7d
15------
15
d------ m–
WORKED
Example
3
9
5---
50 m
25 m
C h a p t e r 5 A l g e b r a 207
7 The formula c = 0.1a + 42 is used to calculate the
cost in dollars (c) of renting a car for one day from
Poole’s Car Hire Ltd, where a is the number of
kilometres travelled on that day. Find the cost of
renting a car for one day if the distance travelled is
220 kilometres.
8 The formula D = 0.6T can be used to convert dis-
tances in kilometres (T) to the approximate equiv-
alent in miles (D). Use this rule to convert the
following distances to miles:
a 100 kilometres
b 248 kilometres
c 12.5 kilometres.
9 The area (A) of a rectangle of length l and width w can be found using the formula
A = lw. Find the area of the rectangles below:
a length 12 cm, width 4 cm
b length 200 m, width 42 m
c length 4.3 m, width 104 cm.
Working with bracketsBrackets are grouping symbols. For example, the expression 3(a + 5) can be thought of
as ‘three groups of (a + 5)’, or (a + 5) + (a + 5) + (a + 5).
When substituting into an expression with brackets, remember to place a multipli-
cation (×) sign next to the brackets. For example,
3(t + 2) means 3 × (t + 2).
6(h − 4) means 6 × (h − 4).
g(2 + 3k) means g × (2 + 3k).
(3 + 2k)4 means (3 + 2k) × 4.
(x + y)(6 − 2p) means (x + y) × (6 − 2p).
We evaluate expressions inside brackets first and then multiply by the value
outside the brackets.
GAME time
Algebra— 001
a Substitute r = 4 and s = 5 into the expression 5(s + r) and evaluate.
b Substitute t = 4, x = 3 and y = 5 into the expression 2x(3t − y) and evaluate.
Continued over page
THINK WRITE
a Place the multiplication sign back
into the expression.
a 5(s + r) = 5 × (s + r)
Substitute the pronumerals with their
correct values.
= 5 × (5 + 4)
Evaluate the expression in the pair of
brackets first.
= 5 × 9
Perform the multiplication and write
the answer.
= 45
1
2
3
4
4WORKEDExample
208 M a t h s Q u e s t 8 f o r V i c t o r i a
Working with brackets
1 Substitute r = 5 and s = 7 into the following expressions and evaluate.
a 3(r + s) b 2(s − r) c 7(r + s) d 9(s − r)
e s(r + 3) f s(2r − 5) g 3r(r + 1) h rs(3 + s)
i 11r(s − 6) j 2r(s − r) k s(4 + 3r) l 7s(r − 2)
m s(3rs + 7) n 5r(24 − 2s) o 5sr(sr + 3s) p 8r(12 − s)
2 Evaluate each of the expressions below, if x = 3, y = 5 and z = 9.
a xy(z − 3) b c
d (x + y) (z − y) e (z − 3)4x f zy(17 − xy)
g h (8 − y) (z + x) i
j k l 2x(xyz − 105)
m 12(y − 1) (z + 3) n o –2(4x + 1)
p –3(2y – 11)
THINK WRITE
b Place the multiplication signs back
into the expression.
b 2x(3t − y) = 2 × x × (3 × t − y)
Substitute the pronumerals with their
correct values.
= 2 × 3 × (3 × 4 − 5)
Perform the multiplication inside the
pair of brackets.
= 2 × 3 × (12 − 5)
Perform the subtraction inside the
pair of brackets.
= 2 × 3 × 7
Perform the multiplication and write
the answer.
= 42
1
2
3
4
5
1. Brackets are grouping symbols.
2. When substituting into an expression with brackets, remember to place a
multiplication (×) sign next to the brackets.
3. Work out the brackets first.
remember
5C
SkillSH
EET 5.3
Order of operations with brackets
WORKED
Example
4
Mat
hcad
Substitution(brackets)
12
x------ z y–( ) z
3---
2y
10------ x 2–+
y
5--- 7 x– 3+( ) 7
12
x------–
4y
6
x--- xz y 3–+( ) y 2+( ) z
x--
3x 7–( ) 27
x------ 7+
36
z------ 3–
z
x-- 8+
C h a p t e r 5 A l g e b r a 209
3 The formula for the perimeter (P) of a rectangle of
length l and width w is P = 2l + 2w. This rule can
also be written as P = 2(l + w). Use the rule to find
the perimeter of rectangular comic covers with the
following measurements.
a l = 20 cm, w = 11 cm
b l = 27.5 cm, w = 21.4 cm
4
When a = 8 and b = 12 are substituted into the
expression (15 − b + 9), the expression is
equal to
A 32 B 16 C 21 D 24 E 27
5 A rule for finding the sum of the interior
angles in a many-sided figure such as a
pentagon is S = 180(n − 2), where S represents
the sum of the angles inside the figure and
n represents the number of sides. The diagram at
right shows the interior angles in a pentagon.
Use the rule to find the sum of the interior angles for
the following figures:
a a hexagon (6 sides) b a pentagon
c a triangle d a quadrilateral (4 sides)
e a 20-sided figure.
Substituting positive and negative numbers
If the pronumeral you are substituting has a negative value, simply remember the
following rules for directed numbers:
1. For addition and subtraction, signs that occur together can be combined.
Same signs positive for example, 7 + +3 = 7 + 3
and 7 − −3 = 7 + 3
Different signs negative for example, 7 − +3 = 7 − 3
and 7 + −3 = 7 − 3
2. For multiplication and division.
Same signs positive for example, +7 × +3 = +21
and −7 × −3 = +21
Different signs negative for example, +7 × −3 = −21
and −7 × +3 = −21
multiple choice
a
6---
1
3---
21.4 cm
27.5
cm
11 cm
20 c
m
210 M a t h s Q u e s t 8 f o r V i c t o r i a
a Substitute m = 5 and n = −3 into the expression m − n and evaluate.
b Substitute m = −2 and n = −1 into the expression 2n − m and evaluate.
c Substitute a = 4 and b = −3 into the expression 5ab − and evaluate.
THINK WRITE
a Substitute the pronumerals with their
correct value.
a m − n = 5 − −3
Combine the two negative signs and add. = 5 + 3
Write the answer. = 8
b Replace the multiplication sign. b 2n − m = 2 × n − m
Substitute the pronumerals with their
correct values.
= 2 × −1 − −2
Perform the multiplication. = −2 − −2
Combine the two negative signs and add. = −2 + 2
Write the answer. = 0
c Replace the multiplication signs.c 5ab − = 5 × a × b −
Substitute the pronumerals with their
correct values.
= 5 × 4 × −3 −
Perform the multiplications. = −60 −
Perform the division. = −60 − −4
Combine the two negative signs and add. = −60 + 4
Write the answer. = −56
12
b------
1
2
3
1
2
3
4
5
1 12
b------
12
b------
212
3–------
312
3–------
4
5
6
5WORKEDExample
When substituting, if the pronumeral you are replacing has a negative value,
simply remember the rules for directed numbers:
1. For addition and subtraction, signs that occur together can be combined.
Same signs positive for example, 7 + +3 = 7 + 3
and 7 − −3 = 7 + 3
Different signs negative for example, 7 − +3 = 7 − 3
and 7 + −3 = 7 − 3
2. For multiplication and division.
Same signs positive for example, +7 × +3 = +21
and −7 × −3 = +21
Different signs negative for example, +7 × −3 = −21
and −7 × +3 = −21
remember
C h a p t e r 5 A l g e b r a 211
Substituting positive and negative numbers
1 Substitute m = 6 and n = −3 into the following expressions and evaluate.
a m + n b m − n c n − m d n + m e 3n
f −2m g 2n − m h n + 5 i 2m + n − 4 j 11n + 20
k −5n − m l m n o
p q r 6mn − 1 s t
2 Substitute x = 8 and y = −3 into the following expressions and evaluate.
a 3(x − 2) b x(7 + y) c 5y(x − 7) d 2(3 − y)
e (y + 5)x f xy(7 − x) g (3 + x) (5 + y) h 5(7 − xy)
i j k l
3 Substitute a = −4 and b = −5 into the following expressions and evaluate.
a a + b b a − b c b − 2a d 2ab e 12 − ab
f −2(b − a) g a − b − 4 h 3a(b + 4) i j
k l m 45 + 4ab n 8ab − 3b o
p 2.5b q 11a + 6b r (a − 5)(8 − b) s (9 − a)(b − 3) t 1.5b + 2a
A rule of thumb is a rule or pattern that people use to estimate things. They obtain
this rule by observing a pattern.
1 Write an algebraic expression for each of the following rules of thumb. Explain
what each pronumeral represents in your expressions.
a Your adult height will be twice your height when you were 2.
b To estimate the number of kilometres you are from a thunderstorm,
count the number of seconds between the lightning and the thunder and
divide by 3.
c To approximately convert temperature in degrees Celsius to degrees
Fahrenheit, double it and add 30.
2 Write a question that could be solved for each of the algebraic expressions
found and clearly show how you would solve it.
3 How would you go about verifying the accuracy of these rules of thumb?
4 If the expression for accurately converting temperature in degrees Celsius (C) to
degrees Fahrenheit (F) is F = C + 32, investigate at which temperatures the
rule of thumb expression gives the best results.
5DSkillSHEET
5.4
Operationswith directed
numbers
WORKED
Example
5a
m
2----
mn
9-------
4m
n 5–------------
4m
n-------
12
2n------
9
n---
m
2----+ 3n
2------– 1.5+ 14
mn
9-------–
WORKED
Example
5b
Mathcad
Substitution(positive/negative)
x
2--- 5 y–( ) x
4--- 1–
2y
6------ 4+
9
y--- 6 x–( ) 3 x 1–( )
y
3--- 2+
WORKED
Example
5c
WorkS
HEET 5.1
4
b---
8
a---–
16
4a------–
6b
5------
a
2---
3b
5------+
COMMUNICATION Rules of thumb
9
5---
I’m now in Australia!
a + b =
Stop Stop
Stop
Start
51
74
47
19
86
64
81
66 57
68
37
421
8025
1110
54
77
46
72
6388
10075
35
2796
7
30
913
36
17
32
95
41
34
26
28
1670
12
552
24
62
49
833
22
23
45
99 2
15
3 18
21
55 61
43 38
31
91
40
60
4 48
8439
90
29
50
6
65
4456
14
69
53
8520
cx =
=y—x
bcy =
8x =
y2 =
Start
7x =
5(x + y) =
b + c + x =
y – a + b =
7c + 2x =
3bc =
x – b =
xy =
2y + c =
x(y – 1) =
y – bc =
x(y + b) =
12x – c =
Start
2bx =
x(2y – c) =
11c =
6(b + c) =
Start
y – c =
12(x + c) =
9c =
20x =
10c =
30 + 2b =
b(y + c) =
cy – b =
2x – 3b =
Start
13x =
bc =
9y – x =
3cy =
cy – a =
10(x + b) – 4 =
7x + b =
7bc =
3x + 5 =
xy – a =
6y + b =
2(y + b) =
a + b =
Start
a2 =
=4cy——
x
27——
c
=11(x + c)————
b
Start
11(a + b + c) =
=xy
—–b
7(a + b) =
4by =
11x =
c + 8 =
x(b + c) =
xy – c =
bcb =
b + c =
4(c + y) =
4(x + b) =
8b =
x(y + a + c) =
ax =
y(c + x) =
Stop
Stop
4b + 2c =
Start
20c + y – a =
6x – 4b =
4y – x – b =
bcx + b =
8x + b – a =
4x + 2c =
Stop
Colouring guide:
Join these points with thick lines.
black
2bx – a =
c + x =
7y – cx – b =
a + 9y =
Orange
6c + xy =
7(c + x) =
11y – 4c + a =
2y – c =
Start
6(x + a) =
y + x – b =
3c =
Stop
Start
40b + 2b =
6(x + c) =
y – x – a =
Stop
80 + x + c =
Start
7(a + 4b) =
12x + y + b =
c + 2b + 3a =
Stop
8(c + x) =
Start
9y – 2x + a =
7y + 8b =
12c + 3y =
Stop
12x – 3c =
Start
10x + 8c =
3cx + b =
Stop
Stop
Stop
Stop
Start
10y – x =
= 8y——
b – a =
Start
8y – c =
Stop
9x + a =
Start
20b + c =
8x + 7c =
Stop
7x + c =
cxy + 4c————
c=
5y + b + c————
x=
Join the dots nextto the values of the expressionsin the orders given below using:
a = 1, b = 2, c = 3, x = 5and y = 10.
212 M a t h s Q u e s t 8 f o r V i c t o r i a
C h a p t e r 5 A l g e b r a 213
1 If a kilogram of oranges cost $2.89 and a kilogram of
apples cost $4.99, what is the cost of p kg of oranges
and q kg of apples?
2 If d represents a certain number, write an expression for
the number formed when d is divided by 5.
3 True or false? If y = 4 and z = 1, then .
4 The area of a circle is p × r2, where p = 3.14 and
r = radius of the circle. Find the area of the circle
when r = 0.5 cm.
5 If p = 1, what is the value of q when pq(5p − 2) = 9?
6 Evaluate if r = 4 and s = 6.
7
When m = 7 and n = 4 are substituted into the expression , the value is:
A 21 B 22 C 22.25 D 25 E 28
8 Substitute p = 7 and q = −2 into .
9 From the list −2, 1, 3, 4 choose the value of a and b when .
10 Substitute x = −3 and y = −5 into the expression and evaluate.
Number laws and pronumeralsWhen dealing with any type of number, particular rules must be obeyed. This exercise
will investigate whether these rules also apply to pronumerals.
Commutative Law
The Commutative Law refers to the order in which two numbers may be added, sub-
tracted, multiplied or divided.
1
12
y------ 3z+ 4=
r = 0.5 cm
12
s------ rs 4 s–+( )
multiple choice
3mn
4---+
14
p------ 1–
pq 3+( )
a
2---
b
4---+ 0=
12
x------– 3y+
214 M a t h s Q u e s t 8 f o r V i c t o r i a
Find the value of the following expressions if x = 4 and y = 7. Comment on the results
obtained.
a i x + y ii y + x
b i x − y ii y − x
c i x × y ii y × x
d i x ÷ y ii y ÷ x
THINK WRITE
a iii Substitute each pronumeral with
its correct value.
a iii x + y = 4 + 7
Evaluate and write the answer. x + y = 11
iii Substitute each pronumeral with
its correct value.
iii y + x = 7 + 4
Evaluate and write the answer. x + y = 11
Compare the result with the
answer obtained in part a i.
The same result is obtained; therefore,
order is not important when adding two
terms.
b iii Substitute each pronumeral with
its correct value.
b iii x − y = 4 − 7
Evaluate and write the answer. x − y = −3
iii Substitute each pronumeral with
its correct value.
iii y − x = 7 − 4
Evalute and write the answer. x − y = 3
Compare the result with the
answer obtained in part b i.
Two different results are obtained;
therefore, order is important when
subtracting two terms.
c iii Substitute each pronumeral with
its correct value.
c iii x × y = 4 × 7
Evaluate and write the answer. x × y = 28
iii Substitute each pronumeral with
its correct value.
iii y × x = 7 × 4
Evalute and write the answer. x × y = 28
Compare the result with the
answer obtained in part c i.
The same result is obtained; therefore,
order is not important when multiplying
two terms.
d iii Substitute each pronumeral with
its correct value.
d iii x ÷ y = 4 ÷ 7
Evaluate and write the answer. x ÷ y = (≈ 0.57)
iii Substitute each pronumeral with
its correct value.
iii y ÷ x = 7 ÷ 4
Evaluate and write the answer.x ÷ y = (1.75)
Compare the result with the
answer obtained in part d i.
Two different results are obtained;
therefore, order is important when
dividing two terms.
1
2
1
2
3
1
2
1
2
3
1
2
1
2
3
1
24
7---
1
2 7
4---
3
6WORKEDExample
C h a p t e r 5 A l g e b r a 215
From worked example 6 we can see that, in general,
1. x + y = y + x
2. x − y ≠ y − x
3. x × y = y × x
4. x ÷ y ≠ y ÷ x
The Commutative Law holds true for addition (and multiplication) because the order in
which two numbers or pronumerals are added (or multiplied) does not affect the result.
It does not hold true for subtraction or division because different results are obtained.
Associative Law
The Associative Law refers to the order in which three numbers may be added, sub-
tracted, multiplied or divided, taking two at a time.
Find the value of the following expressions if x = 12, y = 6 and z = 2. Comment on the
results obtained.
a i x + (y + z) ii (x + y) + z
b i x − (y − z) ii (x − y) − z
c i x × (y × z) ii (x × y) × z
d i x ÷ (y ÷ z) ii (x ÷ y) ÷ z
Continued over page
THINK WRITE
a iii Substitute each pronumeral with
its correct value.
a iii x + (y + z) = 12 + (6 + 2)
Evaluate the expression in the
pair of brackets.
x + (y + z) = 12 + 8
Perform the addition and write
the answer.
x + (y + z) = 20
iii Substitute each pronumeral with
its correct value.
iii (x + y) + z = (12 + 6) + 2
Evaluate the expression in the
pair of brackets.
x + (y + z) = 18 + 2
Perform the addition and write
the answer.
x + (y + z) = 20
Compare the result with the
answer obtained in part a i.
The same result is obtained; therefore,
order is not important when adding
3 terms.
b iii Substitute each pronumeral with
its correct value.
b iii x − (y − z) = 12 − (6 − 2)
Evaluate the expression in the
pair of brackets.
x − (y − z) = 12 − 4
Perform the subtraction and write
the answer.
x − (y − z) = 8
1
2
3
1
2
3
4
1
2
3
7WORKEDExample
216 M a t h s Q u e s t 8 f o r V i c t o r i a
Note the similarities between the Commutative Law and the Associative Law.
From worked example 7 we can see that, in general,
1. x + (y + z) = (x + y) + z
2. x − (y − z ≠ (x − y) − z
3. x × (y × z) = (x × y) × z
4. x ÷ (y ÷ z) ≠ (x ÷ y) ÷ z
THINK WRITE
iii Substitute each pronumeral with
its correct value.
iii (x − y) − z = (12 − 6) − 2
Evaluate the expression in the
pair of brackets.
x − (y − z) = 6 − 2
Perform the subtraction and write
the answer.
x − (y − z) = 4
Compare the result with the
answer obtained in part b i.
Two different results are obtained;
therefore, order is important when
subtracting 3 terms.
c iii Substitute each pronumeral with
its correct value.
c iii x × (y × z) = 12 × (6 × 2)
Evaluate the expression in the
pair of brackets.
x × (y × z) = 12 × 12
Perform the multiplication and
write the answer.
x × (y × z) = 144
iii Substitute each pronumeral with
its correct value.
iii (x × y) × z = (12 × 6) × 2
Evaluate the expression in the
pair of brackets.
x × (y × z) = 72 × 2
Perform the multiplication and
write the answer.
x × (y × z) = 144
Compare the result with the
answer obtained in part c i.
The same result is obtained; therefore,
order is not important when multiplying
3 terms.
d iii Substitute each pronumeral with
its correct value.
d iii x ÷ (y ÷ z) = 12 ÷ (6 ÷ 2)
Evaluate the expression in the
pair of brackets.
x ÷ (y ÷ z) = 12 ÷ 3
Perform the division and write
the answer.
x ÷ (y ÷ z) = 4
iii Substitute each pronumeral with
its correct value.
iii (x ÷ y) ÷ z = (12 ÷ 6) ÷ 2
Evaluate the expression in the
pair of brackets.
x ÷ (y ÷ z) = 2 ÷ 2
Perform the division and write
the answer.
x ÷ (y ÷ z) = 1
Compare the result with the
answer obtained in part d i.
Two different results are obtained;
therefore, order is important when
dividing 3 terms.
1
2
3
4
1
2
3
1
2
3
4
1
2
3
1
2
3
4
C h a p t e r 5 A l g e b r a 217
The Associative Law holds true for addition (and multiplication) because the order in
which three numbers or pronumerals, taking two at a time, are added (or multiplied)
does not affect the result. It does not hold true for subtraction or division because dif-
ferent results are obtained.
Other laws that hold true for addition and multiplication but not subtraction and div-
ision are the Identity Law and the Inverse Law.
Identity LawThe Identity Law for addition states that when zero is added to any number, the original
number remains unchanged. For example, 5 + 0 = 0 + 5 = 5. Similarly, the Identity Law
for multiplication states that when any number is multiplied by one, the original
number remains unchanged. For example, 3 × 1 = 1 × 3 = 3.
Therefore, in general, x + 0 = 0 + x = x
x × 1 = 1 × x = x
Inverse LawThe Inverse Law for addition states that when a number is added to its opposite, the
result is zero. Similarly, the Inverse Law for multiplication states that when a number is
multiplied by its reciprocal, the result is one.
Therefore, in general, x + −x = −x + x = 0
x × = × x = 11
x---
1
x---
1. When dealing with numbers and pronumerals, particular rules must be obeyed.
2. The Commutative Law holds true for addition (and multiplication) because the
order in which two numbers or pronumerals are added (or multiplied) does not
affect the result. Therefore, in general,
(a) x + y = y + x
(b) x − y ≠ y − x
(c) x × y = y × x
(d) x ÷ y ≠ y ÷ x
3. The Associative Law holds true for addition (and multiplication) because the
order in which three numbers or pronumerals, taking two at a time, are added
(or multiplied) does not affect the result. Therefore, in general,
(a) x + (y + z) = (x + y) + z
(b) x − (y − z ≠ (x − y) − z
(c) x × (y × z) = (x × y) × z
(d) x ÷ (y ÷ z) ≠ (x ÷ y) ÷ z
4. The Identity Law states that, in general, x + 0 = 0 + x = x
x × 1 = 1 × x = x
5. The Inverse Law states that, in general, x + −x = −x + x = 0
x × = × x = 11
x---
1
x---
remember
218 M a t h s Q u e s t 8 f o r V i c t o r i a
Number laws and pronumerals
1 Find the value of the following expressions if x = 3 and y = 8. Comment on the results
obtained.
a i x + y ii y + x
b i 3x + 2y ii 2y + 3x
c i 5x + 2y ii 2y + 5x
d i 8x + y ii y + 8x
e i x − y ii y − x
f i 2x − 3y ii 3y − 2x
g i 4x − 5y ii 5y − 4x
h i 3x − y ii y − 3x
2 Find the value of the following expressions if x = −2 and y = 5. Comment on the results
obtained.
a i x × y ii y × x
b i 6x × 3y ii 3y × 6x
c i 4x × y ii y × 4x
d i 7x × 5y ii 5y × 7x
e i x ÷ y ii y ÷ x
f i 10x ÷ 4y ii 4y ÷ 10x
g i 6x ÷ 3y ii 3y ÷ 6x
h i 7x ÷ 9y ii 9y ÷ 7x
3 Indicate whether each of the following is true or false for all values of the pronumerals.
a a + 5b = 5b + a b 6x − 2y = 2y − 6x c 7c + 3d = −3d + 7c
d 5 × 2x × x = 10x2 e 4x × −y = −y × 4x f 4 × 3x × x = 12x × x
g = h −7i − 2j = 2j + 7i i −3y ÷ 4x = 4x ÷ −3y
j −2c + 3d = 3d − 2c k = l 15 × − = × −15
4 Find the value of the following expressions if x = 3, y = 8 and z = 2. Comment on the
results obtained.
a i x + (y + z) ii (x + y) + z
b i 2x + (y + 5z) ii (2x + y) + 5z
c i 6x + (2y + 3z) ii (6x + 2y) + 3z
d i x − (y − z) ii (x − y) − z
e i x − (7y − 9z) ii (x − 7y) − 9z
f i 3x − (8y − 6z) ii (3x − 8y) − 6z
5 Find the value of the following expressions if x = 8, y = 4 and z = −2. Comment on the
results obtained.
a i x × (y × z) ii (x × y) × z
b i x × (−3y × 4z) ii (x × −3y) × 4z
c i 2x × (3y × 4z) ii (2x × 3y) × 4z
d i x ÷ (y ÷ z) ii (x ÷ y) ÷ z
e i x ÷ (2y ÷ 3z) ii (x ÷ 2y) ÷ 3z
f i −x ÷ (5y ÷ 2z) ii (−x ÷ 5y) ÷ 2z
5EWORKED
Example
6a, b
WORKED
Example
6c, d
5 p
3r------
3r
5 p------
0
3s-----
3s
0-----
2x
3------
2x
3------
WORKED
Example
7a, b
WORKED
Example
7c, d
C h a p t e r 5 A l g e b r a 219
6 Indicate whether each of the following is true or false for all values of the pronumerals.
a a − 0 = 0 b a × 1 000 000 = 0 c 15t × − = 1
d 3d × = 1 e ÷ = 1 f = 0
7
The value of the expression x × (−3y × 4z) when x = 4, y = 3 and z = −3 is:
A 108 B −432 C 432 D 112 E −108
8
The value of the expression (x − 8y) − 10z when x = 6, y = 5 and z = −4 is:
A −74 B 74 C −6 D 6 E −36
Simplifying expressionsExpressions can often be written in a more simple form. For example, the expression
3x + 4x can be written more simply as 7x.
Notice that the expression was simplified (put into a more simple form) even though
we did not know the value of the pronumeral (x).
When simplifying expressions, we can collect (add or subtract) only like terms.
Like terms are terms that contain the same pronumeral parts.
For example:
3x and 4x are like terms. 3x and 3y are not like terms.
3ab and 7ab are like terms. 7ab and 8a are not like terms.
2bc and 4cb are like terms. 8a and 3a2 are not like terms.
3g2 and 45g2 are like terms.
1
15t--------
1
3d------
8x
9y------
8x
9y------
11t
0--------
multiple choice
multiple choice
Simplify the following expressions.
a 3a + 5a b 7ab − 3a − 4ab c 2c − 6 + 4c + 15
Continued over page
THINK WRITE
a Write the expression and check that the two
terms are like terms, that is, they contain the
same pronumerals.
a 3a + 5a
Add the like terms and write the answer. = 8a
b Write the expression and check for like
terms.
b 7ab − 3a − 4ab
Rearrange the terms so that the like terms are
together. Remember to keep the correct sign
in front of each term.
= 7ab − 4ab − 3a
Subtract the like terms and write the answer. = 3ab − 3a
1
2
1
2
3
8WORKEDExample
220 M a t h s Q u e s t 8 f o r V i c t o r i a
Simplifying expressions
1
Simplifying 3a + 9a gives:
A 12 B 12a C 6a D 12a2 E 6a2
2
Simplifying 6x − 2x gives:
A 4 B 4x2 C 8x D –4x E 4x
3
Simplifying 6a + 6b gives:
A 12ab B 6ab C 36ab
D 12a E The expression cannot be simplified.
4 Simplify the following expressions.
a 4c + 2c b 2c − 5c c 3a + 5a − 4a
d 6q − 5q e −h − 2h f 7x − 5x
g 3a − 7a − 2a h −3f + 7f i 4p − 7p
j −3h + 4h k 11b + 2b + 5b l 7t − 8t + 4t
m 9m + 5m − m n x − 2x o 7z + 13z
p 5p + 3p + 2p q 9g + 12g − 4g r 18b − 4b − 11b
s 13t − 4t + 5t t −11j + 4j u −12l + 2l − 5l
v 13m − 2m − 4m + m w m + 3m − 4m x t + 2t − t + 8t
5 Simplify the following expressions.
a 3x + 7x − 2y b 3x + 4x − 12 c 11 + 5f − 7f
d 3u − 4u + 6 e 2m + 3p + 5m f −3h + 4r − 2h
g 11a − 5b + 6a h 9t − 7 + 5 i 12 − 3g + 5
j 6m + 4m − 3n + n k 5k − 5 + 2k − 7 l 3n − 4 + n − 5
m 2b − 6 − 4b + 18 n 11 − 12h + 9 o 12y − 3y – 7g + 5g − 6
THINK WRITE
c Write the expression and check for
like terms.
c 2c − 6 + 4c + 15
Rearrange the terms so that the like
terms are together. Remember to
keep the correct sign in front of each
term.
= 2c + 4c − 6 + 15
Simplify by collecting like terms and
write the answer.
= 6c + 9
1
2
3
1. When simplifying expressions, we can collect (add or subtract) only like terms.
2. Like terms are terms that contain the same pronumeral parts.
remember
5F
SkillSH
EET 5.5
Combining like terms
multiple choice
Mat
hcad
Simplifying ex-expressions
multiple choice
multiple choice
WORKED
Example
8a
WORKED
Example
8b, c
C h a p t e r 5 A l g e b r a 221
p 8h − 6 + 3h − 2 q 11s − 6t + 4t − 7s r 2m + 13l − 7m + l
s 3h + 4k − 16h − k + 7 t 13 + 5t − 9t − 8 u 2g + 5 + 5g − 7
v 17f − 3k + 2f − 7k
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 − 7
p 11g3 + 17 − 3g3 + 5 − g2 q 12ab + 3 + 6ab r 14xy + 3xy − xy − 5xy
s 4fg + 2s − fg + s t 11ab + ab − 5
u 18ab2 – 4ac + 2ab2 – 10ac
Multiplying pronumeralsWhen multiplying pronumerals, remember that order is not important. For example:
3 × 6 = 6 × 3
6 × w = w × 6
a × b = b × a
Also keep in mind that the multiplication sign (×) is usually left out:
3 × g × h = 3gh
2 × x2 × y = 2x2y
Although order is not important, the pronumerals in each term are usually written in
alphabetical order. For example:
2 × b2 × a × c = 2ab2c.
Simplify the following:
a 5 × 4g b −3d × 6ab × 7.
THINK WRITE
a Write the expression and replace the
hidden multiplication signs.
a 5 × 4g
= 5 × 4 × g
Multiply the numbers. = 20 × g
Remove the multiplication sign. = 20g
b Write the expression and replace the
hidden multiplication signs.
b −3d × 6ab × 7
= −3 × d × 6 × a × b × 7
Place the numbers at the front. = −3 × 6 × 7 × d × a × b
Multiply the numbers. = −126 × d × a × b
Remove the multiplication signs and place
the pronumerals in alphabetical order.
= −126abd
1
2
3
1
2
3
4
9WORKEDExample
When multiplying pronumerals:
1. the order is not important. For example, d × e = e × d.
2. place the numbers at the front of the expression and leave out the × sign.
remember
222 M a t h s Q u e s t 8 f o r V i c t o r i a
Multiplying pronumerals
1 Simplify the following.
a 4 × 3g b 7 × 3h c 4d × 6 d 3z × 5
e 6 × 5r f 5t × 7 g 4 × 3u h 7 × 6p
i 7gy × 3 j 2 × 11ht k 4x × 6g l 10a × 7h
m 9m × 4d n 3c × 5h o 9g × 2x p 2.5t × 5b
q 13m × 12n r 6a × 12d s 2ab × 3c t 4f × 3gh
u 2 × 8w × 3x v 11ab × 3d × 7 w 16xy × 1.5 x 3.5x × 3y
y 11q × 4s × 3 z 4a × 3b × 2c
2 Simplify the following.
a 3 × −5f b −6 × −2d c 11a × −3g
d −9t × −3g e −5t × −4dh f 6 × −3st
g −3 × −2w × 7d h −4a × −3b × 2c × e i 11ab × −3f
j 3as × −3b × −2x k −5h × −5t × −3q l 4 × −3w × −2 × 6p
m −7a × 3b × g n 17ab × −3gh o −3.5g × 2h × 7
p 5h × 8j × −k q 75x × 1.5y r 12rt × −3z × 4p
s 2ab × 3c × 5 t −4w × 34x × 3 u –3ab × –5cd × –6ae
3 Simplify the following.
a 2a × a b –5p × −5p c –5 × 3x × 2x
d ab × 7a e 3b2 × 2cd f –5xy × 4 × 8x
g 7pq × 3p × 2q h 5m × n × 6nt × –t i –3 × xyz × –3z × –2y
j –7a × –3b × –2c2 k 2mn × –3 × 2n × 0 l w2x × –9z2 × 2xy2
At the start of the chapter, we introduced the situation where a sonar pulse took
1.5 s to travel from the ship to the ocean floor and back again. (The speed of sound
in water is assumed to be 1470 m/s.) Let us look at this problem again.
1 Draw a diagram to show this situation.
2 How far does the sonar pulse travel in:
a 1 second? b 2 seconds? c 1.5 seconds?
3 Calculate the ocean depth when the pulse took 1.5 seconds to return.
4 Write a rule to find the ocean depth for any time measurement. Explain what
each pronumeral represents.
5 Use the rule found in part 4 to calculate the ocean depth for the following pulse-
return time measurements.
a 1.8 seconds b 4.22 seconds c 0.64 seconds
6 The speed of sound in water is about 5 times the speed of sound in air. A person
standing on the deck of the ship sends a sonar pulse through the air to a nearby
cliff face. If the pulse takes 3 seconds to travel to the cliff face and return,
calculate the distance to the cliff face. Write a rule to represent this situation.
5GWORKED
Example
9
THINKING Sonar measurements
C h a p t e r 5 A l g e b r a 223
Dividing pronumeralsWhen dividing pronumerals, rewrite the expression as a fraction and simplify by cancelling.
Remember that when the same pronumeral appears on both the top and bottom lines
of the fraction, it may be cancelled. Follow the worked examples given below.
a Simplify . b Simplify 15n ÷ 3n.
THINK WRITE
a Write the expression. a =
Simplify the fraction by cancelling
16 with 4 (divide both by 4).
=
No need to write the denominator
since we are dividing by 1.
= 4f
b Write the expression and then
rewrite it as a fraction.b 15n ÷ 3n =
=
Simplify the fraction by cancelling
15 with 3 and n with n.
=
No need to write the denominator
since we are dividing by 1.
= 5
16 f
4----------
116 f
4---------
16 f4
41
-----------
24 f
1------
3
1 15n
3n---------
15n5
3n1
-----------
25
1---
3
10WORKEDExample
Simplify −12xy ÷ 27y.
THINK WRITE
Write the expression and then rewrite it
as a fraction.
−12xy ÷ 27y = −
= −
Simplify the fraction by cancelling 12
with 27 (divide both by 3) and y with y.
=
112xy
27y------------
12xy4
27y9
--------------
24x
9------–
11WORKEDExample
1. When dividing pronumerals, rewrite the expression as a fraction and simplify it
by cancelling.
2. When the same pronumeral appears on both the top and bottom lines of the
fraction, it may be cancelled.
remember
224 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 s t 81l ÷ 27l
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
q r −72xyz ÷ 28yz s t −
5H
SkillSH
EET 5.6
Simplifying fractions
WORKED
Example
10 8 f
2------
6h
3------
15x
3---------
16m
8m----------
3x
6x------
12h
14h---------
8 f
24 f---------
20d
48d---------
64q
44q---------
15 fg
3------------
8xy
12---------
11xy
11x------------
9 pq
18q----------
21ab
28b------------
9dg
12g---------
5 jk
kj--------
10mxy
35mx----------------
16cd
40cd------------
132mnp
60np--------------------
11ad
66ad------------
bh
7h------
WORKED
Example
11 4a–
8---------
11ab–
33b---------------
32g–
40gl------------
12ab
14ab–---------------
6 fgh
30ghj--------------
rt–
6rt-------
ab–
3a–---------
7dg–
35gh------------
WorkS
HEET 5.2
28def
18d---------------
54 pq
36 pqr----------------
121oc
132oct-----------------
C h a p t e r 5 A l g e b r a 225
1 If Betty is now x years old, how old was Betty 6 years ago?
2 Find the area of a rectangle with length of 225 cm and width of 1.3 m.
3 Evaluate if p = 4, q = 2 and r = 7.
4
If m = −6 and n = −3 are substituted into the expression , it would have a value
of:
A −2 B −3 C −4 D −5 E –6
5 Simplify 11x − 8y − 9x + 4y − 3.
6 Simplify 10z2 − 5y − 3z2 + 4y + 4.
7 True or false? −6p × −4q × r × 2t = 48pqrt
8 Simplify .
9 From the list −2, −4, 12pq, −48pq, find the missing term to replace ∇ in
.
10 Simplify .
Expanding bracketsWe have seen that the expression 3(a + 5) means
3 × (a + 5) or (a + 5) + (a + 5) + (a + 5).
Simplifying this expression further gives us the expression 3a + 15:
(a + 5) + (a + 5) + (a + 5) = a + a + a + 5 + 5 + 5
= 3a + 15
Look at the pattern below:
With brackets Expanded form
1. 3 × (2 + 1) 3 × 2 + 3 × 1
= 3 × 3 = 6 + 3
= 9 = 9
2. 4 × (3 + 2) 4 × 3 + 4 × 2
= 4 × 5 = 12 + 8
= 20 = 20
2
r 10+( )p
q---
multiple choice
m
2----
6n
9------+
30ab
18abc---------------
∇
12 pr–---------------
4q
r------=
9 p–
36 pq–----------------
226 M a t h s Q u e s t 8 f o r V i c t o r i a
Removing brackets from an expression is called expanding the expression. The rule
that we have used to expand the expressions above is called the Distributive Law.
The Distributive Law states: a (b + c) = a ¥ b + a ¥ c
= ab + ac
Expanding and collecting like termsSome expressions can be simplified further by collecting like terms after any brackets
have been expanded.
Use the Distributive Law to expand the following expressions.
a 3(a + 2) b x(x − 5)
THINK WRITE
a Write the expression. a 3(a + 2) = 3(a + 2)
Use the Distributive Law to expand
the brackets.
= 3 × a + 3 × 2
Simplify by multiplying. = 3a + 6
b Write the expression.
Use the Distributive Law to expand
the brackets.
Simplify by multiplying.
b x(x − 5) = x(x − 5)
= x × x + x × −5
= x2 − 5x
1
2
3
1
2
3
12WORKEDExample
Expand the expressions below and then simplify by collecting any like terms.
a 3(x − 5) + 4 b 4(3x + 4) + 7x + 12
c 2x(3y + 3) + 3x(y + 1) d 4x(2x − 1) − 3(2x − 1)
THINK WRITE
a Write the expression. a 3(x − 5) + 4
Expand the brackets. = 3 × x + 3 × −5 + 4
= 3x − 15 + 4
Collect the like terms (−15 and 4). = 3x − 11
b Write the expression. b 4(3x + 4) + 7x + 12
Expand the brackets. = 4 × 3x + 4 × 4 + 7x + 12
= 12x + 16 + 7x + 12
Rearrange so that the like terms are
together. (Optional)
= 12x + 7x + 16 + 12
Collect the like terms. = 19x + 28
1
2
3
1
2
3
4
13WORKEDExample
C h a p t e r 5 A l g e b r a 227
Expanding brackets
1 Use the Distributive Law to expand the following expressions.
a 3(d + 4) b 2(a + 5) c 4(x + 2)
d 5(r + 7) e 6(g + 6) f 2(t + 3)
g 7(d + 8) h 9(2x + 6) i 12(4 + c)
j 7(6 + 3x) k 45(2g + 3) l 1.5(t + 6)
m 11(t − 2) n 3(2t − 6) o t(t + 3)
p x(x + 4) q g(g + 7) r 2g(g + 5)
s 3f(g + 3) t 6m(n − 2m)
2 Expand the following.
a 3(3x − 2) b 3x(x − 6y) c 5y(3x − 9y)
d 50(2y − 5) e −3(c + 3) f −5(3x + 4)
g −5x(x + 6) h −2y(6 + y) i −6(t − 3)
j −4f(5 − 2f) k 9x(3y − 2) l −3h(2b − 6h)
m 4a(5b + 3c) n −3a(2g − 7a) o 5a(3b + 6c)
p −2w(9w − 5z) q 12m(4m + 10) r −3k(−2k + 5)
THINK WRITE
c Write the expression. c 2x(3y + 3) + 3x(y + 1)
Expand the brackets. = 2x × 3y + 2x × 3 + 3x × y + 3x × 1
= 6xy + 6x + 3xy + 3x
Rearrange so that the like terms are
together. (Optional)
= 6xy + 3xy + 6x + 3x
Simplify by collecting the like terms. = 9xy + 9x
d Write the expression. d 4x(2x − 1) − 3(2x − 1)
Expand the brackets. Take care with
negative terms.
= 4x × 2x + 4x × −1 − 3 × 2x − 3 × −1
= 8x2 − 4x − 6x + 3
Simplify by collecting the like terms. = 8x2 − 10x + 3
1
2
3
4
1
2
3
1. Brackets are grouping symbols
2. Removing brackets from an expression is called expanding the expression.
3. When expanding brackets, put the × sign before the bracket.
4. The rule that is used to expand brackets is called the Distributive Law.
5. After expanding brackets, collect any like terms.
remember
5IWORKED
Example
12
Mathcad
Expandingbrackets
EXCEL Spreadsheet
Expandingbrackets
GC program
– TI
Expandingbrackets
GC
program–
Casio
Expandingbrackets
228 M a t h s Q u e s t 8 f o r V i c t o r i a
3 Expand the expressions below and then simplify by collecting any like terms.
a 7(5x + 4) + 21 b 3(c − 2) + 2
c 2c(5 − c) + 12c d 6(v + 4) + 6
e 3d(d − 4) + 2d2 f 3y + 4(2y + 3)
g 24r + r(2 + r) h 5 − 3g + 6(2g − 7)
i 4(2f − 3g) + 3f − 7 j 3(3x − 4) + 12
k −2(k + 5) − 3k l 3x(3 + 4r) + 9x − 6xr
m 12 + 5(r − 5) + 3r n 12gh + 3g(2h − 9) + 3g
o 3(2t + 8) + 5t − 23 p 24 + 3r(2 − 3r) − 2r2 + 5r
4 Expand the following and then simplify by collecting like terms.
a 3(x + 2) + 2(x + 1) b 5(x + 3) + 4(x + 2)
c 2(y + 1) + 4(y + 6) d 4(d + 7) − 3(d + 2)
e 6(2h + 1) + 2(h − 3) f 3(3m + 2) + 2(6m − 5)
g 9(4f + 3) − 4(2f + 7) h 2a(a + 2) − 5(a2 + 7)
i 3(2 − t2) + 2t(t + 1) j m(n + 4) − mn + 3m
5 Simplify the following expressions by removing the brackets and then collecting like
terms.
a 3h(2k + 7) + 4k(h + 5) b 6n(3y + 7) − 3n(8y + 9)
c 4g(5m + 6) − 6(2gm + 3) d 11b(3a + 5) + 3b(4 − 5a)
e 5a(2a − 7) − 5(a2 + 7) f 7c(2f − 3) + 3c(8 − f)
g 7x(4 − y) + 2xy − 29 h 11v(2w + 5) − 3(8 − 5vw)
i 3x(3 − 2y) + 6x(2y − 9) j 8m(7n − 2) + 3n(4 + 7m)
FactorisingFactorising is the opposite process to expanding. Factorising a number or expression
involves breaking it down into smaller factors.
3 and 2 are factors of 6, because 6 = 3 × 2
2, 4, 5 and 10 are factors of 20, because:
20 = 4 × 5 and
20 = 2 × 10.
Common factorsTwo numbers may have common factors; for example, 5 is a factor of both 15 and 20.
The numbers 9 and 12 have the common factor 3.
The numbers 14 and 21 have the common factor 7.
The numbers 4 and 8 have two common factors, 2 and 4.
Highest common factorThe highest common factor (HCF) of 4 and 8 is 4 (not 2). It is the largest factor
common to a given set of numbers or terms.
The highest common factor of 12 and 18 is 6.
The highest common factor of 8 and 20 is 4.
Algebraic terms can also be broken down into factors. For example, the factors of 3x
are 3 and x. The expression 6m can be broken down into factors as shown below:
6m = 6 × m
= 3 × 2 × m
GC pr
ogram– TI
Expanding
WORKED
Example
13
GC pr
ogram– Casio
Expanding
GAM
E time
Algebra— 002
C h a p t e r 5 A l g e b r a 229
Here are some other examples:
8x = 8 × x
= 4 × 2 × x
= 2 × 2 × 2 × x
3ab = 3 × a × b
6a2b = 6 × a × a × b
= 3 × 2 × a × a × b
To find the highest common factor of algebraic terms, follow these steps.
1. Find the highest common factor of the number parts.
2. Find the highest common factor of the pronumeral parts.
3. Multiply these together.
To factorise an expression, we place the highest common factor of the terms
outside the brackets and the remaining factors for each term inside the brackets.
Find the highest common factor of 6x and 10.
THINK WRITE
Find the highest common factor of the
number parts.
Break 6 down into factors.
Break 10 down into factors.
The highest common factor is 2.
6 = 3 × 2
10 = 5 × 2
HCF = 2
Find 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
14WORKEDExample
Find the highest common factor 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 × 2
21 = 7 × 3
HCF = 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 × g
gh = g × h
HCF = g
Multiply these together. The HCF of 14 fg and 21gh is 7g.
1
2
3
15WORKEDExample
230 M a t h s Q u e s t 8 f o r V i c t o r i a
Factorise the expression 2x + 6.
THINK WRITE
Break down each term into its factors. 2x + 6
= 2 × x + 2 × 3
Write the highest common factor
outside the brackets. Write the other
factors inside the brackets.
= 2 × (x + 3)
Remove the multiplication sign. = 2(x + 3)
1
2
3
16WORKEDExample
Factorise 12gh − 8g.
THINK WRITE
Break down each term into its factors. 12gh − 8g
= 4 × 3 × g × h − 4 × 2 × g
Write the highest common factor
outside the brackets.Write the other
factors inside the brackets.
= 4 × g × (3 × h − 2)
Remove the multiplication signs. = 4g(3h − 2)
1
2
3
17WORKEDExample
1. Factorising is the opposite process to expanding.
2. Factorising a number or expression involves breaking it down into smaller
factors.
3. To find the highest common factor (HCF) of algebraic terms, follow these
steps.
(a) Find the highest common factor of the number parts.
(b) Find the highest common factor of the pronumeral parts.
(c) Multiply these together.
4. To factorise an expression we place the highest common factor of the
terms outside the brackets and the remaining factors for each term inside the
brackets.
remember
C h a p t e r 5 A l g e b r a 231
Factorising
1
a The highest common factor of 12 and 16 is:
b The highest common factor of 10 and 18 is:
c The highest common factor of 4 and 16 is:
d The highest common factor of 2x and 8xy is:
e The highest common factor of 4f and 12fg is:
2 Find the highest common factor of the following.
a 4 and 6 b 6 and 9 c 12 and 18 d 13 and 26
e 14 and 21 f 2x and 4 g 3x and 9 h 12a and 16
3 Find the highest common factor of the following.
a 2gh and 6g b 3mn and 6mp c 11a and 22b
d 4ma and 6m e 12ab and 14ac f 24 fg and 36gh
g 20dg and 18ghq h 11gl and 33lp i 16mnp and 20mn
j 28bc and 12c k 4c and 12cd l x and 3xz
4 Factorise the following expressions.
a 3x + 6 b 2y + 4 c 5g + 10
d 8x + 12 e 6f + 9 f 12c + 20
g 2d + 8 h 2x − 4 i 12g − 18
j 11h + 121 k 4s − 16 l 8x − 20
m 12g − 24 n 14 − 4b o 16a + 64
p 48 − 12q q 16 + 8f r 12 − 12d
5 Factorise the following.
a 3gh + 12 b 2xy + 6y
c 12pq + 4p d 14g − 7gh
e 16jk − 2k f 12eg + 2g
g 12k + 16 h 7mn + 6m
i 14ab + 7b j 5a − 15abc
k 8r + 14rt l 24mab + 12ab
m 4b − 6ab n 12fg − 16gh
o ab − 2bc p 14x − 21xy
q 11jk + 3k r 3p + 27pq
s 12ac − 4c + 3dc t 4g + 8gh − 16
u 28s + 14st v 15uv + 27vw
A 12 B 4 C 8 D 2 E 3
A 4 B 10 C 2 D 9 E 180
A 4 B 16 C 2 D 20 E 8
A 2 B x C 2x D 16x2y E 8
A 2 B fg C 48f2g D 4f E 2f
5JSkillSHEET
5.7
Highestcommon
factor
multiple choice
Mathcad
Factorising
EXCEL Spreadsheet
Findingthe HCF
WORKED
Example
14
WORKED
Example
15
WORKED
Example
16
WORKED
Example
17
WorkS
HEET 5.3
232 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 If the pronumeral you are substituting is negative, the rules for directed
numbers must be followed. For addition and subtraction, signs that occur
together can be . Same signs produce a sign,
while different signs produce a ___________ sign.
7 The Commutative Law holds true for and . It
does not hold true for and .
8 The Law refers to refers to the order in which
numbers may be added, subtracted, multiplied or divided, taking two at a
time.
9 When simplifying an expression, terms may be collected only if they are
.
10 Expanding an expression involves brackets. For example
3(x + 2) = 3x + 6.
11 The Law gives the rule for expanding expressions.
12 Factorising an expression means breaking it down into smaller
, or putting brackets back into the expression.
summary
W O R D L I S T
three
substitution
combined
Distributive
positive
3 × y
Associative
factors
negative
pronumeral
removing
subtraction
grouping
addition
division
division
like terms
fraction
multiplication
C h a p t e r 5 A l g e b r a 233
1 Using x and y to represent numbers, write expressions for:
a the sum of x and y
b the difference between y and x
c five times y subtracted from three times x
d the product of 5 and x
e twice the product of x and y
f the sum of 6x and 7y
g y multiplied by itself
h twice a number is decreased by 7
i the sum of p and q is tripled.
2 If tickets to the school play cost $15 for
adults and $9 for children, write an
expression for the cost of:
a x adult tickets
b y child tickets
c k adult tickets and m child tickets.
3 Jake is now m years old.
a Write an expression for his age in 5 years’
time.
b Write an expression for Jo’s age if she is p
years younger than Jake.
c Jake’s mother is 5 times his age. How old is she?
4 Find the value of the following expressions if a = 2 and b = 6.
a 2a b 6a c 5b d
e a + 8 f b − 2 g a + b h b − a
i j 3a + 7 k 2a + 3b l
m 3b – 2a n o p –3ab
q r
5 The formula C = 2.2k + 4 can be used to calculate the cost in dollars, C, of travelling by taxi
for a distance of k kilometres. Find the cost of travelling 4.5 km by taxi.
6 The area (A) of a rectangle of length l and width w can be found using the formula A = lw.
Find the width of a rectangle if A = 65 cm2 and l = 13 cm.
5A
CHAPTERreview
5A
5A
5Ba
2---
5b
2---+
20
a------
b
a---
a
b---
5a
b------
2b
9a------
5B
5B
234 M a t h s Q u e s t 8 f o r V i c t o r i a
7 Substitute r = 3 and s = 5 into the following expressions and evaluate.
a 2(r + s) b 2(s − r) c 5(r + s) d 8(s − r)
e s(r + 4) f s(2r − 3) g 2r(r + 1) h rs(7 + s)
i r2(5 – r) j s2(s + 15) k 4r(s + r) l 12r(r – s)
8 Find the value of the following expressions if a = 2 and b = −5.
a a + b b b + a c ab d
e 2ab f 5 − a g 12 − ab h a2 − 2
i 3(a + 2) j b(a − 4) k 12 − a(b − 3) l 5a + 6b
9 Indicate whether each of the following is true or false for all values of the pronumerals.
a 3a + 5b = 5b + 2a b 7x − 10y = 10y − 7x c 8c + d = d + 8c
d 16 × 2x × x = 32x2 e 9x × −y = −y × 9x f −4 × 3x × x = 12x × x
g = h 7i + 2j = 2j + 7i i −3y ÷ 7x = 7x ÷ −3y
j −8c + 5d = 5d − 8c k = l 21 × − = × −21
10 Simplify the following by collecting like terms.
a 4d + 3d b 3c − 5c
c 3d + 5a − 4a d 6g − 4g
e 4x + 11 − 2x f 2g + 5 − g − 6
g 2xy + 7xy h 12t2 + 3t + 3t2 − t
11 Simplify the following.
a 3 × 7g b 6 × 3y
c 7d × 6 d −3z × 8
12 Simplify the following.
a b
c 6rt ÷ −2t d −3gh ÷ −6g
e f −36xy ÷ −12y
g h
13 Use the Distributive Law to expand the following expressions.
a 2(x + 3) b 5(2x − 1) c −2(f + 7)
d 3m(b − m) e −3y(7 − y) f 9b(c − 2)
14 Expand the following and then simplify by collecting like terms.
a 3(4v + 5) − 15 b 6t + 5(2t − 7) c 23 + 5(3p − 4) + 2p
d 2(x + 5) + 5(x + 1) e 2g(g − 6) + 3g(g − 7) f 3(3t − 4) − 6(2t − 9)
15 Factorise the following expressions.
a 3g + 12 b xy + 5y c 5n − 20
d 12mn + 4pn e 12g − 6gh f 12xy − 36yz
5C
5Dab
5------
5E
11 p
5r---------
5r
11 p---------
0
5k------
5k
0------
7x
3------
7x
3------
5F
5G
5H2a
8------
11b
44b---------
32t
40stv-------------
12ab
14ab–---------------
5egh
30ghj--------------
5I
5I
testtest
CHAPTER
yourselfyourself
5
5J