Download Linear equations

number and algebra
TOPIC 4
ONLINE PAGE PROOFS
Linear equations
4.1 Overview
Why learn this?
Looking for patterns in numbers, relationships and measurements
helps us to understand the world around us. A mathematical model is
a mathematical representation of a situation. If we can see a pattern
in a table of values or a graph that shows ordered pairs following an
approximately straight line, the model is called a linear model.
What do you know?
1 THInK List what you know about linear equations. Use a
thinking tool such as a concept map to show your list.
2 PaIr Share what you know with a partner and then with a
small group.
3 SHare As a class, create a thinking tool such as a large concept
map to show your class’s knowledge of linear equations.
Learning sequence
4.1
4.2
4.3
4.4
4.5
4.6
4.7
Overview
Solving linear equations
Solving linear equations with brackets
Solving linear equations with pronumerals on both sides
Solving problems with linear equations
Rearranging formulas
Review ONLINE ONLY
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ONLINE PAGE PROOFS
WaTCH THIS vIdeO
The story of mathematics
The mighty Roman armies
Searchlight Id: eles-1691
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number and algebra
4.2 Solving linear equations
What is a linear equation?
ONLINE PAGE PROOFS
• An equation is a mathematical statement that contains an = sign.
• For an equation, the expression on the left-hand side of the equals sign has the same
value as the expression on the right-hand side.
• Solving a linear equation means finding a value for the pronumeral that makes the
statement true.
• ‘Doing the same thing’ to both sides of the equation ensures that the two expressions
remain equal.WOrKed eXamPle 1
WOrKed eXamPle 1
For each of the following equations, determine whether x = 10 is a solution.
a
b
2x + 3 = 3x − 7
THInK
a
b
c
104
x+2
=6
3
c
x2 − 2x = 9x − 10
WrITe
x+2
3
10 + 2
=
3
12
=
3
=4
LHS =
1
Substitute 10 for x in the left-hand
side of the equation.
2
Write the right-hand side.
RHS = 6
3
Is the equation true? That is, does the
left-hand side equal the right-hand side?
LHS ≠ RHS
4
State whether x = 10 is a solution.
1
Substitute 10 for x in the left-hand side.
2
Substitute 10 for x in the right-hand side.
RHS = 3x − 7
= 3(10) − 7
= 23
3
Is the equation true?
LHS = RHS
4
State whether x = 10 is a solution.
1
Substitute 10 for x in the left-hand side.
2
Substitute 10 for x in the right-hand side.
a
x = 10 is not a solution.
b
LHS = 2x + 3
= 2(10) + 3
= 23
x = 10 is a solution.
c
LHS = x2 − 2x
= 102 − 2(10)
= 100 − 20
= 80
RHS = 9x − 10
= 9(10) − 10
= 90 − 10
= 80
Maths Quest 9
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number and algebra
3
Is the equation true?
4
State whether x = 10 is a solution.
LHS = RHS
x = 10 is a solution.
Solving one-step equations
ONLINE PAGE PROOFS
• If one operation has been performed on
a pronumeral, it is known as a one-step
equation.
• Simple equations can be solved by applying
the inverse operation.
• The inverse operation has the effect of
undoing the original operation.
Operation
Inverse operation
+
−
−
+
×
÷
÷
×
WOrKed eXamPle 2
Solve each of the following linear equations.
a x − 79 = 153
b x + 46 = 82
c
6x = 100
THInK
a
b
c
d
d
x
= 19
7
WrITe
x − 79 = 153
1
79 is subtracted from x to give 153.
2
Apply the inverse operation by adding 79 to
both sides of the equation.
x = 153 + 79
3
Write the value of x.
x = 232
1
46 is added to x to give 82.
2
Apply the inverse operation by subtracting 46
from both sides of the equation.
x = 82 − 46
3
Write the value of x.
x = 36
1
6 is multiplied by x to give 100.
2
Perform the inverse operation by dividing both
sides of the equation by 6.
3
Write the value of x.
1
x is divided by 7 to give 19.
2
Perform the inverse operation by multiplying
both sides of the equation by 7.
3
Write the value of x.
a
b
c
x + 46 = 82
6x = 100
100
x=
6
x = 1623
d
x
= 19
7
x = 19 × 7
= 133
Note: In each case the result can be checked by substituting the value obtained for x
back into the original equation and confirming that it will make the equation a true
statement.
Topic 4 • Linear equations 105
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number and algebra
Solving two-step equations
• If two operations have been performed on the pronumeral, it is known as a two-step
equation.
• To solve two-step equations, establish the order in which the operations were performed.
• Perform inverse operations in the reverse order to both sides of the equation.
• Each inverse operation must be performed one step at a time.
• This principle will apply to any equation with two or more steps, as shown in the
examples that follow.
ONLINE PAGE PROOFS
WOrKed eXamPle 3
Solve the following linear equations.
a 2y + 4 = 12
b 6−x=8
c
THInK
a
b
c
d
x
−4=2
3
d
3x
=6
5
WrITe
1
First subtract 4 from both sides.
2
Divide both sides by 2.
3
Write the value of y.
1
6 − x is the same as −x + 6. Rewrite the
equation.
2
Subtract 6 from both sides.
3
Divide both sides by −1.
4
Write the value of x.
1
Add 4 to both sides.
2
Multiply both sides by 3.
3
Write the value of x.
1
Multiply both sides by 5.
a
2y + 4 = 12
2y + 4 − 4 = 12 − 4
2y = 8
2y 8
=
2
2
y=4
b
6−x=8
−x + 6 = 8
−x + 6 − 6 = 8 − 6
−x = 2
−x
2
=
−1 −1
x = −2
c
x
−4=2
3
x
−4+4=2+4
3
x
=6
3
x
×3=6×3
3
x = 18
d
3x
=6
5
3x
×5=6×5
5
3x = 30
106
2
Divide both sides by 3.
3
Write the value of x.
3x 30
=
3
3
x = 10
Maths Quest 9
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number and algebra
WOrKed eXamPle 4
Solve the following linear equations.
x+1
7−x
a
= 11
b
= −6.3
5
2
THInK
ONLINE PAGE PROOFS
a
WrITe
1
All of x + 1 has been divided by 2.
2
Multiply both sides by 2.
a
x+1
= 11
2
x+1
× 2 = 11 × 2
2
x + 1 = 22
b
x = 21
3
Subtract 1 from both sides.
1
All of 7 − x has been divided by 5.
2
Multiply both sides by 5.
3
7 − x is the same as −x + 7.
4
Subtract 7 from both sides.
5
Divide both sides by −1.
b
7−x
= −6.3
5
7−x
× 5 = −6.3 × 5
5
7 − x = −31.5
7 − x − 7 = −31.5 − 7
−x = −38.5
x = 38.5
Algebraic fractions with the pronumeral in the
denominator
• If a pronumeral is in the denominator, there is an extra step involved in finding the
solution.
Consider the following example:
3
4
=
x
2
In order to solve this equation, we first multiply both sides of the equations by x.
3
4
×x= ×x
x
2
3x
4=
2
3x
or
=4
2
The pronumeral is now in the numerator, and the equation is easy to solve.
3x
=4
2
3x = 8
8
x=
3
Topic 4 • Linear equations 107
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number and algebra
WOrKed eXamPle 5
Solve each of the following linear equations.
3 4
5
a =
b = −2
a 5
b
THInK
ONLINE PAGE PROOFS
a
1
Multiply both sides by a.
2
Multiply both sides by 5.
3
Divide both sides by 4.
3 4
=
a 5
4a
3=
5
a
15 = 4a
15
a=
4
or a = 334
b
1
Write the equation.
2
Multiply both sides by b.
3
Divide both sides of the equation by −2.
5
= −2
b
b
5 = −2b
5
=b
−2
b = −212
Exercise 4.2 Solving linear equations
IndIvIdual PaTHWaYS
reFleCTIOn
How are linear equations
defined?
⬛
PraCTISe
⬛
Questions:
1a–f, 2a–l, 3a–h, 4, 5, 6a–f,
7a–f, 8a–f, 9a–f, 10, 11, 12, 17
COnSOlIdaTe
⬛
Questions:
1d–i, 2g–r, 3d–i, 4, 5, 6d–i, 7d–i,
8d–i, 9d–l, 10–13, 17–19
⬛ ⬛ ⬛ Individual pathway interactivity
maSTer
Questions:
1g–l, 2i–u, 3g–l, 4, 5, 6g–l, 7g–l,
8g–l, 9g–l, 10–12, 14–20
int-####
FluenCY
1
doc-6150
For each of the following equations, determine whether x = 6 is a solution.
a x+3=7
b 2x − 5 = 7
c x2 − 2 = 38
2(x + 1)
6
d
e
=2
f 3−x=9
+x=7
x
7
g x2+ 3x = 39
h 3(x + 2) = 5(x − 4)
i x2 + 2x = 9x − 6
WE1
x2 = (x + 1)2 −14
j
doc-6151
doc-6152
2
k
(x − 1)2 = 4x + 1
l
5x + 2 = x2 + 4
Solve each of the following linear equations. Check your answers by substitution.
x − 43 = 167
b x − 17 = 35
c x + 286 = 516
58 + x = 81
e x − 78 = 64
f 209 − x = 305
5x = 185
h 60x = 1200
i 5x = 250
x
x
x
=6
k
l
= 26
= 27
17
9
23
WE2
a
d
g
doc-10826
108
WrITe
j
Maths Quest 9
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number and algebra
− 16 = −31
n 5.5 + y = 7.3
p 6y = 14
q 0.2y = 4.8
y
y
s
= 4.3
t
= 23
5
7.5
3 WE3a Solve each of the following linear equations.
a 2y − 3 = 7
b 2y + 7 = 3
d 6y + 2 = 8
e 7 + 3y = 10
g 15 = 3y − 1
h −6 = 3y − 1
j 4.5y + 2.3 = 7.7
k 0.4y − 2.7 = 6.2
ONLINE PAGE PROOFS
m y
Solve each of the following linear equations.
a 3 − 2x = 1
b −3x − 1 = 5
d 1 − 3x = 19
e −5 − 7x = 2
g 9 − 6x = −1
h −5x − 4.2 = 7.4
j −3 = −6x − 8
k −1 = 4 − 4x 5 Solve each of the following linear equations.
a 7 − x = 8
b 8 − x = 7
d 5 − x = 0
e 15.3 = 6.7 − x g 9 − x = 0.1
h 140 − x = 121
j −5 = −6 − x k −x + 1 = 2
4
5y − 1 = 0
f 8 + 2y = 12
i 6y − 7 = 140
l 600y − 240 = 143
c
WE3b Solve each of the following linear equations.
x
x
a
b − 2 = −1
+ 1 = 3
4
3
x
x
d − = 5
e 5 − = −8
2
3
5x
2x
g
= 6
h
= −3
2
3
8x
2x
j −
= 6
k
= −2
7
3
7 WE4 Solve each of the following linear equations.
z−1
z+1
a
= 5
b
= 8
4
3
6−z
3−z
d
= 0
e
= 6
7
2
z − 4.4
z+2
g
= −3
h
= 1.2
2.1
7.4
−z − 0.4
z−6
j
= −0.5
k
= −4.6
2
9
8 Solve each of the following linear equations.
5x + 1
2x − 5
a
= 2
b
= 3
7
3
4x − 13
4 − 3x
d
= −5
e
= 8
9
2
−5x − 3
−10x − 4
g
= 3
h
= 1
9
3
5x − 0.7
1 − 0.5x
j
= −3.1
k
= −2.5
4
−0.3
6
y − 7.3 = 5.5
r 0.9y = −0.05
y
u = −1.04
8
o
−4x − 7 = −19
f −8 − 2x = −9
i 2 = 11 − 3x
l 35 − 13x = −5
c
5−x=5
f 5.1 = 4.2 − x i −30 − x = −4
l −2x − 1 = 0
c
WE3c, d x 1
=
8 2
x
f 4 − = 11
6
3x
i −
= −7
4
3x
1
l −
=−
10
5
c
z−4
= −4
2
−z − 50
f
= −2
22
140 − z
i
= 0
150
z + 65
l
= 1
73
c
3x + 4
= −1
2
1 − 2x
f
= −10
6
4x + 2.6
i
= 8.8
5
−3x − 8 1
l
=
14
2
c
Topic 4 • Linear equations 109
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number and algebra
Solve each of the following linear equations.
3
5 −3
2 1
−4 7
a = b = 7
c
= d =
x 2
x
x
x
2
4
0.4 9
8
−6 −4
−4 2
e
= f
= 1
g
= h
=
x
x
x
x
2
3
5
6
50 −35
1.7 1
4 −15
i
= j
= −1
k =
l
=
x
x
x
x
22
3
43
10 MC aThe solution to the equation 82 − x = 44 is:
A x = 126B
x = −126C
x = 122D
x = 38
b What is the solution to the equation 5x − 12 = −62?
A x = −14.8B
x = 14.8C
x = 10D
x = −10
x−1
c What is the solution to the equation
= 5.3?
2
A x = 9.6B
x = 10.6C
x = 11.6D
x=2
11 Solve each of the following linear equations.
a 3a + 7 = 4
b 5 − b = −5
c 4c − 4.4 = 44
2f
d−4
d
= 0
e 5 − 3e = −10 f
= 8
3
67
h+2
g 100 = 6g + 4.2
h
= 5.5
i 452i − 124 = −98
6
6j − 1
l − 5.2
12 − k
j
= 0
k
= 4
l
= 1.5
17
3.4
5
ONLINE PAGE PROOFS
9 WE5 UNDERSTANDING
Write the following worded statements as a mathematical sentence and then solve for
the unknown.
a Seven is added to the product of x and 3, which gives the result of 4.
2
b Four is divided by x and this result is equivalent to .
3
c Three is subtracted from x and this result is divided by 12 to give 25.
13 Driving lessons are usually quite expensive but a discount of $15 per lesson is given
if a family member belongs to the automobile club. If 10 lessons cost $760 (after the
discount), find the cost of each lesson before the discount.
14 Anton lives in Australia and his pen pal, Utan, lives in USA. Anton’s home town
of Horsham experienced one of the hottest days on record with a temperature of
46.7 °C. Utan said that his home town had experienced a day hotter than that, with the
temperature reaching 113 °F. The formula for converting Celsius to Fahrenheit is
F = 95C + 32. Was he correct?
12
REASONING
Santo solved the linear equation 9 = 5 − x. His second step was to divide both sides
by −1. Trudy, his mathematics buddy, said that she multiplied both sides by −1.
Explain why they are both correct.
16 Find the mistake in the following working and explain what is wrong.
x
− 1 = 2
5
x − 1 = 10
x = 11
15
Problem solving
17
Sweet-tooth Sammy goes to the corner store and buys an equal number of 25-cent and
30-cent lollies for $16.50. How many lollies did he buy?
110 Maths Quest 9
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ONLINE PAGE PROOFS
18
In a cannery, cans are filled by two machines which together produce
16 000 cans during an 8-hour shift. If the newer machine produces
340 more cans per hour than the older machine, how many cans does each
machine produce in an 8-hour shift?
General admission to an exhibition is $55 for an adult ticket, $27 for a
child and $130 for a family of two adults and two children.
a How much is saved by buying a family ticket instead of buying two
adult and two child tickets?
b Is it worthwhile buying a family ticket if the family has only one child?
20 A teacher comes across a clue shown below in a cryptic mathematics cross-number.
What is the value of n that the teacher is looking for?
19
3n – 6
5n + 2
18
150
doc-6156
CHallenge 4.1
4.3 Solving linear equations with brackets
• Consider the equation 3(x + 5) = 18.
There are two good methods for solving this equation.
Method 1:
First divide both sides by 3.
3(x + 5) 18
=
3
3
x+5=6
x=1
Method 2:
First expand the brackets.
3(x + 5) = 18
3x + 15 = 18
3x = 3
x=1
In this case, method 1 works well because 3 divides exactly into 18.
Now try the equation 7(x + 2) = 10.
Topic 4 • Linear equations 111
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number and algebra
Method 1:
First divide both sides by 7.
7(x + 2) 10
=
7
7
10
x+2=
7
4
x=−
7
Method 2:
First expand the brackets.
7(x + 2) = 10
7x + 14 = 10
7x = −4
−4
x=
7
ONLINE PAGE PROOFS
In this case, method 2 works well because it avoids fraction addition or subtraction.
Try both methods and choose the one that works best for you.WOrKed eXamPle 6
WOrKed eXamPle 6
Solve each of the following linear equations.
a 7(x − 5) = 28
b 6(x + 3) = 7
THInK
a
b
WrITe
7(x − 5) = 28
7(x − 5) 28
=
7
7
1
7 is a factor of 28, so divide both sides
by 7.
2
Add 5 to both sides.
x−5=4
3
Write the value of x.
x=9
1
6 is not a factor of 7, so it will be easier to
expand the brackets first.
2
Subtract 18 from both sides.
3
Divide both sides by 6.
a
6(x + 3) = 7
6x + 18 = 7
b
6x + 18 = 7 − 18
6x = −11
x = −11
(or −156)
6
Exercise 4.3 Solving linear equations with
brackets
IndIvIdual PaTHWaYS
reFleCTIOn
Explain why there are two
possible methods for solving
equations in factorised form.
⬛
PraCTISe
Questions:
1a–f, 2a–h, 3a–f, 4a–f,
5, 6, 8, 10
⬛
COnSOlIdaTe
⬛
Questions:
1d–i, 2d–i, 3d–i, 4d–i, 5,
7–11
⬛ ⬛ ⬛ Individual pathway interactivity
maSTer
Questions:
1g–l, 2g–l, 3g–l, 4g–l, 5, 7–12
int-####
FluenCY
1
doc-10827
112
Solve each of the following linear equations.
a 5(x − 2) = 20
b 4(x + 5) = 8
d 5(x − 41) = 75
e 8(x + 2) = 24
WE6
6(x + 3) = 18
f 3(x + 5) = 15
c
Maths Quest 9
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number and algebra
5(x + 4) = 15
h 3(x − 2) = −12
i 7(x − 6) = 0
j −6(x − 2) = 12
k 4(x + 2) = 4.8
l 16(x − 3) = 48
WE6 2
Solve each of the following equations.
a 6(b − 1) = 1
b 2(m − 3) = 3
c 2(a + 5) = 7
d 3(m + 2) = 2
e 5(p − 2) = −7
f 6(m − 4) = −8
g −10(a + 1) = 5
h −12(p − 2) = 6
i −9(a − 3) = −3
j −2(m + 3) = −1
k 3(2a + 1) = 2
l 4(3m + 2) = 5
3 Solve each of the following equations.
a 9(x − 7) = 82
b 2(x + 5) = 14
c 7(a − 1) = 28
d 4(b − 6) = 4
e 3(y − 7) = 0
f −3(x + 1) = 7
g −6(m + 1) = −30
h −4(y + 2) = −12
i −3(a − 6) = 3
j −2(p + 9) = −14
k 3(2m − 7) = −3
l 2(4p + 5) = 18
4 Solve the following linear equations. Round the answers to 3 decimal places where
appropriate.
a 2(y + 4) = −7
b 0.3(y + 8) = 1
c 4(y + 19) = −29
d 7(y − 5) = 25
e 6(y + 3.4) = 3
f 7(y − 2) = 8.7
g 1.5(y + 3) = 10
h 2.4(y − 2) = 1.8
i 1.7(y + 2.2) = 7.1
j −7(y + 2) = 0
k −6(y + 5) = −11
l −5(y − 2.3) = 1.6
MC 5
a The best first step in solving the equation 7(x − 6) = 23 would be to:
A add 6 to both sides
B subtract 7 from both sides
C divide both sides by 23
D expand the brackets
b The solution to the equation 84(x − 21) = 782 is closest to:
A x = 9.31B
x = 9.56
C x = 30.31D
x = −11.69
ONLINE PAGE PROOFS
g
UNDERSTANDING
In 1974 a mother was 6 times as old as her daughter. If the mother turned 50 in the year
2000, in what year was the mother double her daughter’s age?
7 New edging is to be placed around a rectangular children’s playground. The width of the
playground is x m and the length is 7 metres longer than the width.
a Write down an expression for the perimeter of the playground. Write your answer in
factorised form.
b If the amount of edging required is 54 m, determine the dimensions of the playground.
6
REASONING
Juanita is solving the following equation: 2(x − 8) = 10. She performs the following
operations to both sides of the equation in order: +8, ÷2. Explain why Juanita will not
find the correct value of x using her order of inverse operations, then solve the equation.
9 As your first step to solve the equation 3(2x – 7) = 18, you are given three options:
• Expand the brackets on the left-hand side.
• Add 7 to both sides.
• Divide both sides by 3.
Which of the options is your least preferred and why?
8
Topic 4 • Linear equations 113
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number and algebra
PrOblem SOlvIng
Five times the sum of 4 and a number is equal to 35. What is the number?
11 Kyle earns $55 more than Noah each week, but Callum earns three times as much as
Kyle. If Callum earns $270 a week, how much do Kyle and Noah earn each week?
12 A school wishes to hire a bus to travel to a football game. The bus will take
28 passengers, and the school will contribute $48 towards the cost of the trip.
If the hiring of the bus is $300 + 10% of the cost of all the tickets, what should be the
cost per person?
ONLINE PAGE PROOFS
10
4.4 Solving linear equations with pronumerals
on both sides
int-2764
• When solving equations, it is important to remember that whatever we do to one side of
an equation we must do to the other.
• If the pronumeral occurs on both sides of the equation, first remove it from one side, as
shown in the example below.WOrKed eXamPle 7
WOrKed eXamPle 7
Solve each of the following linear equations.
a 5y = 3y + 3
b 7x + 5 = 2 − 4x
c 3(x + 1) = 14 − 2x
d 2(x + 3) = 3(x + 7)
THInK
a
b
c
114
WrITe
1
3y is smaller than 5y. Subtract 3y from
both sides.
2
Divide both sides by 2.
1
−4x is smaller than 7x. Add 4x to both
sides.
2
Subtract 5 from both sides.
3
Divide both sides by 11.
1
Expand the bracket.
2
−2x is smaller than 3x. Add 2x to both
sides.
3
Subtract 3 from both sides.
4
Divide both sides by 5.
a
5y = 3y + 3
5y − 3y = 3y + 3 − 3y
2y = 3
y=
b
3
(or 112)
2
7x + 5 = 2 − 4x
7x + 5 + 4x = 2 − 4x + 4x
11x + 5 = 2
11x + 5 − 5 = 2 − 5
11x = −3
−3
x=
11
c
3(x + 1) = 14 − 2x
3x + 3 = 14 − 2x
3x + 3 + 2x = 14 − 2x + 2x
5x + 3 = 14
5x + 3 − 3 = 14 − 3
5x = 11
x=
11
5
Maths Quest 9
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number and algebra
ONLINE PAGE PROOFS
d
1
Expand the brackets.
2
2x is smaller than 3x.
Subtract 2x from both sides.
3
Subtract 21 from both sides.
4
Write the answer with the pronumeral
written on the left-hand side.
2(x + 3) = 3(x + 7)
2x + 6 = 3x + 21
d
2x + 6 − 2x = 3x + 21 − 2x
6 = x + 21
6 − 21 = x + 21 − 21
−15 = x
x = −15
Exercise 4.4 Solving linear equations with
pronumerals on both sides
IndIvIdual PaTHWaYS
⬛
PraCTISe
⬛
Questions:
1a–f, 2, 3a–f, 4, 5, 6a–f, 7, 8, 11
COnSOlIdaTe
⬛
Questions:
1d–i, 2, 3d–i, 4, 5, 6d–i, 7–12
⬛ ⬛ ⬛ Individual pathway interactivity
maSTer
Questions:
1g–l, 2, 3g–l, 4, 5, 6g–l, 7–14
reFleCTIOn
Draw a diagram that could
represent 2x + 4 = 3x + 1.
int-####
FluenCY
1
Solve each of the following linear equations.
5y = 3y − 2
b 6y = −y + 7
25 + 2y = −3y
e 8y = 7y − 45
7y = −3y − 20
h 23y = 13y + 200
6 − 2y = −7y
k 24 − y = 5y
WE7a
a
d
g
j
10y = 5y − 15
f 15y − 8 = −12y
i 5y − 3 = 2y
l 6y = 5y − 2
c
doc-10828
To solve the equation 3x + 5 = −4 − 2x, the first step is to:
a add 3x to both sides
b add 5 to both sides
C add 2x to both sides
d subtract 2x from both sides
b To solve the equation 6x − 4 = 4x + 5, the first step is to:
a subtract 4x from both sides
b add 4x to both sides
C subtract 4 from both sides
d add 5 to both sides
3 WE7b Solve each of the following linear equations.
a 2x + 3 = 8 − 3x
b 4x + 11 = 1 − x
c x − 3 = 6 − 2x
d 4x − 5 = 2x + 3
e 3x − 2 = 2x + 7
f 7x + 1 = 4x + 10
g 5x + 3 = x − 5
h 6x + 2 = 3x + 14
i 2x − 5 = x − 9
j 10x − 1 = −2x + 5
k 7x + 2 = −5x + 2
l 15x + 3 = 7x − 3
2
MC a
Topic 4 • Linear equations 115
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number and algebra
ONLINE PAGE PROOFS
4Solve
each of the following linear equations.
a x − 4 = 3x + 8
b 3x + 12 = 4x + 5
c 2x + 9 = 7x − 1
d −2x + 7 = 4x + 19
e −3x + 2 = −2x − 11
f 11 − 6x = 18 − 5x
g 6 − 9x = 4 + 3x h x − 3 = 18x − 1
i 5x + 13 = 15x + 3
5 MC a The solution to 5x + 2 = 2x + 23 is:
A x = 3B
x = −3
C x = 5D
x=7
b The solution to 3x − 4 = 11 − 2x is:
A x = 15B
x=7
C x = 3D
x=5
6 WE7c, d Solve each of the following.
a 5(x − 2) = 2x + 5
b 7(x + 1) = x − 11
c 2(x − 8) = 4x d 3(x + 5) = x
e 6(x − 3) = 14 − 2x f 9x − 4 = 2(3 − x)
g 4(x + 3) = 3(x − 2)
h 5(x − 1) = 2(x + 3)
i 8(x − 4) = 5(x − 6)
j 3(x + 6) = 4(2 − x)
k 2(x − 12) = 3(x − 8)
l 4(x + 11) = 2(x + 7)
UNDERSTANDING
7Aamir’s
teacher gave him an algebra problem and told him to solve it. Can you help
him? 3x + 7 = x2 + k = 7x + 15
What is the value of k?
8A classroom contained an equal number of boys and girls. Six girls left to play hockey,
leaving twice as many boys as girls in the classroom. What was the original number of
students present?
REASONING
the following information as an equation, then show that n = 29 is the
solution.
9Express
n – 36
n – 36
10
–98
n – 36
20n + 50
150 – 31n
Explain what the problem is in solving the equation 4(3x – 5) = 6(2x + 3) .
Problem solving
11 This
year Tom is 4 times as old as his daughter, while in 5 years’ time he will be only
3 times as old as his daughter. Find the ages of Tom and his daughter now.
12 If you multiply an unknown number by 6 and then add 5, the result is 7 less than the
unknown number plus 1 multiplied by 3. Find the unknown number.
116 Maths Quest 9
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number and algebra
ONLINE PAGE PROOFS
13
You are investigating getting a business card printed for your new game store. A local
printing company charges $250 for the cardboard used and an hourly rate for labour
of $40.
Address: 123 The Street
Melbourne
VIC 3000
Phone no: 03 1234 5678
If h is the number of hours of labour required to print the cards, construct an
equation for the cost of the cards, C.
b You have budgeted $1000 for the printing job. How many hours of labour
can you afford? Give your answer to the nearest minute.
c The company estimates that it can print 1000 cards per hour of labour.
How many cards will you get printed with your current budget?
d An alternative to printing is photocopying. The company charges
15 cents per side for the first 10 000 cards and then 10 cents per side for
the remaining cards. Which is the cheaper option for 18 750 single-sided
cards and by how much?
14 A local pinball arcade offers its regular customers the following deal. For a
monthly fee of $40 players get 25 $2 pinball games. Additional games cost $2
each. After a player has played 50 games in a month, all further games are $1.
a If Tom has $105 to spend in a month, how many games can he play if he
takes up the special deal?
b How much did Tom save by taking up the special deal.
a
4.5 Solving problems with linear equations
Converting worded sentences to algebraic equations
• An important skill in mathematics is the ability to translate written problems into
algebraic equations in order to solve problems.
WOrKed eXamPle 8
Write linear equations for each of the following statements, using x to represent
the unknown. (Do not attempt to solve the equations.)
a When 6 is subtracted from a certain number, the result is 15.
b Three more than seven times a certain number is zero.
c When 2 is divided by a certain number, the answer is 4 more than the number.
THInK
a
WrITe
1
Let x be the number.
2
Write x and subtract 6. This expression
equals 15.
a
x = unknown number
x − 6 = 15
Topic 4 • Linear equations 117
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number and algebra
b
ONLINE PAGE PROOFS
c
118
1
Let x be the number.
2
7 times the number is 7x. Three more than 7x
equals 7x + 3. This expression equals 0.
1
Let x be the number.
2
Write the term for 2 divided by a certain
number.
3
b
x = unknown number
7x + 3 = 0
c
x = unknown number
2
x
Write the expression for 4 more than the
number.
x+4
Write the equation.
2
=x+4
x
WOrKed eXamPle 9
In a basketball game, Hao scored 5 more points than Seve. If they scored a total of
27 points between them, how many points did each of them score?
THInK
WrITe
1
Define a pronumeral.
Let Seve’s score be x.
2
Hao scored 5 more than Seve.
Hao’s score is x + 5.
3
Between them they scored a total of
27 points.
x + (x + 5) = 27
4
Solve the equation.
5
Since x = 11, this is Seve’s score.
Write Hao’s score.
Hao’s score = x + 5
= 11+ 5
= 16
6
Write the answer in words.
Seve scored 11 points and Hao
scored 16 points.
2x + 5 = 27
2x = 22
x = 11
Maths Quest 9
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number and algebra
WOrKed eXamPle 10
ONLINE PAGE PROOFS
Taxi charges are $3.60 plus $1.38 per kilometre for any trip in Melbourne.
If Elena’s taxi fare was $38.10, how far did she travel?
THInK
WrITe
Let x = distance travelled
(in kilometres).
1
The distance travelled by Elena has to be
found. Define the pronumeral.
2
It costs 1.38 to travel 1 kilometre, so the cost
Total cost = 3.60 + 1.38x
to travel x kilometres = 1.38x. The fixed cost is
$3.60. Write an expression for the total cost.
3
Let the total cost = 38.10.
4
Solve the equation.
5
State the solution in words.
3.60 + 1.38x = 38.10
1.38x = 34.50
34.50
x=
1.38
= 25
Elena travelled 25 kilometres.
Exercise 4.5 Solving problems with
linear equations
IndIvIdual PaTHWaYS
⬛
PraCTISe
Questions:
1–4, 7, 9, 11–14
⬛
COnSOlIdaTe
⬛
Questions:
1–5, 7–10, 12–15
⬛ ⬛ ⬛ Individual pathway interactivity
maSTer
Questions:
1–16
int-####
reFleCTIOn
Why is it important to define
the pronumeral used in forming
a linear equation to solve a
problem?
Topic 4 • Linear equations 119
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number and algebra
FluenCY
Write linear equations for each of the following statements, using x to represent
the unknown. (Do not attempt to solve the equations.)
a When 3 is added to a certain number, the answer is 5.
b Subtracting 9 from a certain number gives a result of 7.
c Seven times a certain number is 24.
d A certain number divided by 5 gives a result of 11.
e Dividing a certain number by 2 equals −9.
f Three subtracted from five times a certain number gives a result of −7.
g When a certain number is subtracted from 14 and this result is then multiplied by 2,
the result is −3.
h When 5 is added to three times a certain number, the answer is 8.
i When 12 is subtracted from two times a certain number, the result is 15.
j The sum of 3 times a certain number and 4 is divided by 2, which gives a result of 5.
2 MC Which equation matches the following statement?
a A certain number, when divided by 2, gives a result of −12.
−12
a x=
b 2x = −12
2
x
x
C
d
= −12
= −2
2
12
b Dividing 7 times a certain number by −4 equals 9.
x
−4x
a
b
=9
=9
7
−4
7+x
7x
=9
=9
C
d
−4
−4
c Subtracting twice a certain number from 8 gives 12.
a 2x − 8 = 12
b 8 − 2x = 12
C 2 − 8x = 12
d 8 − (x + 2) = 12
d When 15 is added to a quarter of a number, the answer is 10.
x
a 15 + 4x = 10
b 10 = + 15
4
x + 15
4
C
= 10
d 15 + = 10
x
4
1
ONLINE PAGE PROOFS
doc-10826
120
WE8
underSTandIng
When a certain number is added to 3 and the result is multiplied by 4, the answer is the
same as when the same number is added to 4 and the result is multiplied by 3. Find the
number.
x
4 WE9 John is three times as old as his son Jack, and
the sum of their ages is 48. How old is John?
5 In one afternoon’s shopping Seedevi spent half as
x+5
much money as Georgia, but $6 more than Amy. If
30
the three of them spent a total of $258, how much did
20
Seedevi spend?
6 These rectangular blocks of land have the same area.
Find the dimensions of each block, and the area.
3
Maths Quest 9
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number and algebra
REASONING
7A
square pool is surrounded by a paved area that is 2 metres wide. If the area of the
paving is 72 m2, what is the length of the pool?
ONLINE PAGE PROOFS
2m
8Maria
is paid $11.50 per hour, plus $7 for each jacket that she sews.
If she earned $176 for one 8-hour shift, how many jackets did she sew?
9Mai hired a car for a fee of $120 plus $30 per day. Casey’s rate was $180 plus $26 per
day. If their final cost was the same, how long was the rental period?
10 WE10 The cost of producing music CDs is quoted as $1200 plus $0.95 per disk. If
Maya’s recording studio has a budget of $2100, how many CDs can she have made?
11 Joseph
wishes to have some flyers delivered for his grocery business. Post Quick
quotes a price of $200 plus 50 cents per flyer, while Fast Box quotes $100 plus
80 cents per flyer.
a If Joseph needs to order 1000 flyers, which distributor would be cheaper to use?
b For what number of fliers will the cost be the same for either distributor?
Topic 4 • Linear equations 121
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number and algebra
PrOblem SOlvIng
A number is multiplied by 8 and 16 is then subtracted. The result is the same as 4 times
the original number minus 8. What is the number?
13 Carmel sells three different types of healthy
drinks; herbal, vegetable and citrus fizz. One
hour she sells 4 herbal, 3 vegetable and 6
citrus fizz for $60.50. The next hour she sells
2 herbal, 4 vegetable and 3 citrus fizz. The
third hour she sells 1 herbal, 2 vegetable and
4 citrus fizz. The total amount in cash sales
for the three hours is $136.50. Carmel made
$7 less in the third hour than she did in the
second hour of sales.
Determine her sales in the fourth hour, if
Carmel sells 2 herbal, 3 vegetable and 4 citrus fizz.
Fence
14 A rectangular swimming pool is surrounded by a path which is
x+2
enclosed by a pool fence. All measurements are in metres and
are not to scale in the diagram shown.
2 5
a Write an expression for the entire fenced-off area.
x+4
b Write an expression for the area of the path surrounding the pool.
2
c If the area of the path surrounding the pool is 34 m , find the dimensions of the
swimming pool.
d What fraction of the fenced-off area is taken up by the pool?
ONLINE PAGE PROOFS
12
122
doc-6159
4.6 Rearranging formulas
• Formulas are generally written in terms of two or more pronumerals or variables.
• One pronumeral is usually written on one side of the equal sign.
• When rearranging formulas, use the same methods as for solving linear equations (use
inverse operations in reverse order).
The difference between rearranging formulas and solving linear equations is that
rearranging formulas does not require a value for the pronumeral(s) to be found.
• The subject of the formula is the pronumeral or variable that is written by itself. It is
usually written on the left-hand side of the equation.
Rearranging (transposing) formulas
• A formula is simply an equation that is used for some specific purpose. By now you will
be familiar with many mathematical or scientific formulas.
For example, C = 2πr relates the circumference of a circle to its radius. When the
formula is shown in this order, C is called the subject of the formula. The formula can be
transposed (rearranged) to make r the subject.
C = 2πr
Divide both sides by 2π.
2πr
C
=
2π
2π
C
=r
2π
C
Now r is the subject.
or r =
2π
Maths Quest 9
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number and algebra
WOrKed eXamPle 11
Rearrange each formula to make x the subject.
a y = kx + m
b 6(y + 1) = 7(x − 2)
THInK
ONLINE PAGE PROOFS
a
b
WrITe
1
Subtract m from both sides.
2
Divide both sides by k.
3
Rewrite the equation so that x is
on the left-hand side.
1
Expand the brackets.
2
Add 14 to both sides.
3
Divide both sides by 7.
4
Rewrite the equation so that x is
on the left-hand side.
a
y = kx + m
y − m = kx
y − m kx
=
k
k
y−m
=x
k
y−m
x=
k
b
6(y + 1) = 7(x − 2)
6y + 6 = 7x − 14
6y + 20 = 7x
6y + 20
=x
7
x=
6y + 20
7
WOrKed eXamPle 12
WOrKed eXamPle 12
For each of the following make the variable shown in brackets the subject of the
formula.
a g = 6d − 3 (d)
v−u
b a=
(v)
t
THInK
a
b
WrITe
1
Add 3 to both sides.
2
Divide both sides by 6.
3
Rewrite the equation so that d
is on the left-hand side.
1
Multiply both sides by t.
a
g = 6d − 3
g + 3 = 6d
g+3
=d
6
g+3
6
v−u
a=
t
d=
b
at = v − u
2
Add u to both sides.
3
Rewrite the equation so that v
is on the left-hand side.
at + u = v
v = at + u
Topic 4 • Linear equations 123
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number and algebra
Exercise 4.6 Rearranging formulas
IndIvIdual PaTHWaYS
reFleCTIOn
How does rearranging
formulas differ to solving linear
equations?
⬛
PraCTISe
Questions:
1a–f, 2a–f, 3, 6
⬛
COnSOlIdaTe
ONLINE PAGE PROOFS
⬛ ⬛ ⬛ Individual pathway interactivity
124
⬛
Questions:
1e–h, 2e–h, 3–6, 8
maSTer
Questions:
1g–l, 2g–n, 3–10
int-####
FluenCY
Rearrange each formula to make x the subject.
a y = ax
b y = ax + b
c y = 2ax − b
d y + 4 = 2x − 3
e 6(y + 2) = 5(4 − x)
f x(y − 2) = 1
g x(y − 2) = y + 1
h 5x − 4y = 1
i 6(x + 2) = 5(x − y)
j 7(x − a) = 6x + 5a
k 5(a − 2x) = 9(x + 1)
l 8(9x − 2) + 3 = 7(2a −3x)
2 WE12 For each of the following, make the variable shown in brackets the subject of
the formula.
9c
a g = 4P − 3
(P)
b f=
(c)
5
9c
+ 32 (c)
c f=
d V = IR
(I)
5
e v = u + at
(t)
f d = b2 − 4ac
(c)
y−k
y−a
g m=
(y)
h m=
(y)
h
x−b
y−a
y−a
i m=
(a)
j m=
(x)
x−b
x−b
2π
k C=
(r)
l f = ax + by
(x)
r
GMm
1
m s = ut + at2 (a)
n F=
(G)
2
r2
1
doc-10829
WE11
underSTandIng
3
The cost to rent a car is given by the formula C = 50d + 0.2k, where d = the number of
days rented and k = the number of kilometres driven. Lin has $300 to spend on car
rental for her 4-day holiday. How far can she travel on this holiday?
eles-0113
Maths Quest 9
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number and algebra
cyclist pumps up a bike tyre that has a slow leak. The volume of air (in cm3) after
t minutes is given by the formula:
4A
ONLINE PAGE PROOFS
V = 24 000 − 300t
What is the volume of air in the tyre when it is first filled?
b Write an equation and solve it to work out how long it takes the tyre to go
completely flat.
a
REASONING
total surface area of a cylinder is given by the formula T = 2πr2 + 2πrh, where
r = radius and h = height. A car manufacturer wants the engine’s cylinders to have a
radius of 4 cm and a total surface area of 400 cm2. Show that the height of the cylinder
is approximately 11.92 cm, correct to 2 decimal places. (Hint: Express h in terms of
T and r.)
6If B = 3x − 6xy, write x as the subject. Explain the process by showing all working.
5The
Problem solving
7Use
algebra to show that
fv
1 1 1
.
= − can also be written as u =
v u f
v+f
the formula d = "b2 − 4ac.
Rearrange the formula to make a the subject.
9Find values for a and b, such that:
ax + b
3
4
−
=
x + 1 x + 2 (x + 1) (x + 2)
8Consider
10
A new game has been created by students for the school fair. To win the game you
need to hit the target with 5 darts in the shaded region.
x
R
a
Write an expression for the area of the shaded region.
Topic 4 • Linear equations 125
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number and algebra
b
If R = 7.5 cm and x = 4 cm, find the area of the game board, correct to 2 decimal
places.
c
Show that R =
A + x2
by transposing the formula found in part a.
Å π
ONLINE PAGE PROOFS
CHallenge 4.2
126
Maths Quest 9
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number and algebra
ONLINE ONLY
4.7 Review
www.jacplus.com.au
ONLINE PAGE PROOFS
The Maths Quest Review is available in a customisable format
for students to demonstrate their knowledge of this topic.
The Review contains:
• Fluency questions — allowing students to demonstrate the
skills they have developed to efficiently answer questions
using the most appropriate methods
• Problem Solving questions — allowing students to
demonstrate their ability to make smart choices, to model
and investigate problems, and to communicate solutions
effectively.
A summary on the key points covered and a concept
map summary of this chapter are also available as digital
documents.
Review
questions
Download the Review
questions document
from the links found in
your eBookPLUS.
Language
int-0686
int-0700
algebraic equation
algebraic fraction
alternative
decomposed
define
expand
expression
fixed
forensic science
formula
inverse operation
justify
linear equation
one-step equation
solution
solve
subject
two-step equation
int-3204
Link to assessON for
questions to test your
readiness FOr learning,
your progress aS you learn and your
levels OF achievement.
assessON provides sets of questions
for every topic in your course, as well
as giving instant feedback and worked
solutions to help improve your mathematical
skills.
www.assesson.com.au
Topic 4 • Linear equations 127
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number and algebra
<InveSTIgaTIOn>
InveSTIgaTIOn
FOr rICH TaSK Or <number and algebra> FOr PuZZle
rICH TaSK
ONLINE PAGE PROOFS
Forensic science
128
Maths Quest 9
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ONLINE PAGE PROOFS
number
number and
and algebra
algebra
Imagine the following situation.
A decomposed body was found in the bushland. A team of
forensic scientists suspects that the body could be the remains
of either Alice Brown or James King; they have been missing for
several years. From the description provided by their Missing
Persons file, Alice is 162 cm tall and James’ file indicates that
he is 172 cm tall. The forensic scientists hope to identify the
body based on the length of the body’s humerus.
1 Complete the following tables for both males and females,
using the equations on the previous page. Calculate the body
height to the nearest centimetre.
Table for males
Table for females
2 On a piece of graph paper, draw the first quadrant of a
3
4
5
6
Cartesian plane. Since the length of the humerus is the
independent variable, place it on the x -axis. Place the
dependent variable, body height, on the y -axis.
Plot the points from the two tables representing both male and
female bodies from question 1 onto the set of axes drawn in
question 2. Join the points with straight lines, using different colours to represent males and females.
Describe the shape of the two graphs.
Measure the length of your humerus. Use your graph to predict your height. How accurate is the
measurement?
The two lines of your graph will intersect if extended. At what point does this occur? Comment on this
value.
The forensic scientists measured the length of the humerus of the bone remains and found it
to be 33 cm.
7 Using methods covered in this activity, identify the body, justifying your decision with mathematical
evidence.
Topic 4 • Linear equations 129
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<InveSTIgaTIOn>
number
and algebra
FOr rICH TaSK Or <number and algebra> FOr PuZZle
COde PuZZle
The driest place
ONLINE PAGE PROOFS
Solve the equations given and colour in the block containing
each answer. The letters in the remaining blocks will spell out
the puzzle’s answer.
18 – 2x = 10
6 – 3w = –27
4(15 – 3a ) = 0
3(7+ 5x ) = –9
5x + 8 – 7x = 26
17 + 8x
= –1
7
–1 = 5 – 2x
3
8 + 3e
=2
19
–3 = 3 + 2(5 – x )
5 = –7 + 4f
105 – 12e = 21
2(7 – 2b ) = 34
3=
7 – 8f = 95
5–m
4
25 – 6c = 13
1 – 7y = 85
130
M
7
A
3
I
5
D E
–4 21
T
1
17 + 4x = 41
H E M
–8 16 –3
A
4
S
17
A
0
T
I C S B A C
13 –12 11 –2 –11 18 –6
K
8
S
–7
L A M A
–9 –14 15 20
H E R T
I N Y C H I
L D
2 –20 12 19 –10 –16 6 –13 –1 14 –15 9
R E N
10 –17 –5
Maths Quest 9
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number and algebra
Activities
4.1 Overview
video
• The story of mathematics: The mighty Roman armies
(eles-1691)
ONLINE PAGE PROOFS
4.2 Solving linear equations
digital docs
• SkillSHEET (doc-6150): Solving one-step equations
• SkillSHEET (doc-6151): Checking solutions to equations
• SkillSHEET (doc-6152): Solving equations
• SkillSHEET (doc-10826): Writing equations from
worded statements
• WorkSHEET 4.1 (doc-6156): Solving linear equations
4.3 Solving linear equations with brackets
digital doc
• SkillSHEET (doc-10827): Expanding brackets
4.4 Solving linear equations with pronumerals
on both sides
digital doc
• SkillSHEET (doc-10828): Simplifying like terms
To access ebookPluS activities, log on to
Interactivity
• Solving equations (int-2764)
4.5 Solving problems with linear equations
digital docs
• SkillSHEET (doc-10826): Writing equations from
worded statements
• WorkSHEET 4.2 (doc-6159): Solving equations with
pronumerals on both sides
4.6 rearranging formulas
digital doc
• SkillSHEET (doc-10829): Transposing and
substituting into a formula
elesson
• Formulas in the real world (eles-0113)
4.7 review
Interactivities
• Word search (int-0686)
• Crossword (int-0700)
• Sudoku (int-3204)
www.jacplus.com.au
Topic 4 • Linear equations 131
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number and algebra
Answers
topic 4 Linear equations
4.2 Solving linear equations
1 a Nob
Yesc
No
d Yes
e Yesf
Nog
Noh
No
Nok
Yesl
No
i Yesj
2 a x = 210b
x = 52c
x = 230d
x = 23
x = −96g
x = 37h
x = 20
e x = 142f
x = 138k
x = 442l
x = 243
i x = 50j
ONLINE PAGE PROOFS
my = −15n
y = 1.8o
y = 12.8p
y = 2 13
1
q y = 24r
y = −18
s
y = 21.5t
y = 172.5
u y = −8.32
3 a y = 5b
y = −2c
y = 0.2d
y=1
e y = 1f
y = 2g
y = 5 13h
y = −1 23
i y = 24.5j
y = 1.2k
y = 22.25l
y = 383
600
4 a x = 1b
x = −2c
x = 3d
x = −6
e x = −1f
x = 1 23h
x = 12g
x = −2.32
1
i x = 3j
x = −56k
x = 1 14l
x = 3 13
5 a x = −1b
x = 1c
x = 0d
x=5
x = −0.9g
x = 8.9h
x = 19
e x = −8.6f
i x = −26j
x = −12
x = −1k
x = −1l
6 a x = 8b
x = 3c
x = 4d
x = −15
e x = 26f
x = −42g
x = 9h
x = −1 15
9 13j
x = −2 14k
x=
−7l
x = 23
i x =
7 a z = 16b
z = 31c
z = −4d
z=6
z = −6g
z = −1.9h
z = 6.88
e z = −9f
z = 0.6k
z = −35.4l
z=8
i z = 140j
8 a x = 1b
x = 13c
x = −2d
x = −8
7
e x = −4f
x = 30 12g
x = −6h
x = −10
i x = 10.35j
x = 0.326k
x = 22l
x = −5
3
1
2
9 a x = 4b
x = 7c
x = −1 7
d x = −6
3
e x =
4
f
x = 8g
x = −6h
x = 7.5
45
13
5.1j
x = −6k
x = −5 15l
x = −61 37
i x =
10 a Db
Dc
C
11 a a = −1b
b = 10c
c = 12.1d
d=4
e e = 5f
f = 12g
g = 15 29
h
h = 31
30
13
i i = 226
j
j = 16k
k = −8l
l = 10.3
12 a −1b
6c
303
13 $91
14 No. 46.7°C ≈ 116.1°F.
15 Answers will vary.
16 The mistake is in the second line: the 1 should have been
multiplied by 5.
17 60 lollies
18 Old machine: 6640 cans; new machine: 9360 cans
19 a$34
b Yes, a saving of $7
2017
Challenge 4.1
x = −8, −2, 0, 1, 2, 3, 5, 6, 7, 8, 10, 16
4.3 Solving linear equations with brackets
1 a x = 6b
x = −3c
x = 0d
x = 56
x = 0g
x = −1h
x = −2
e x = 1f
x = 0k
x = −0.8l
x=6
i x = 6j
1
6
1
1
2
2
1
3
2
p
=
m
=
2
d m = −1 e
f
3
5
3
g a = −1 12h
p = 1 12i
a = 3 13
j m = −2 12k
a = −16l
m = −14
1
3 a x = 16 b
x = 2c
a = 5d
b=7
9
1
e y = 7f
x = −33g
m = 4h
y=1
2 a b = 1 b
m = 4 c
a = −1
p = −2k
m = 3l
p=1
i a = 5j
4 a y = −7.5b
y = −4.667cy = −26.25 d y = 8.571
e y = −2.9f
y = 3.243g
y = 3.667h
y = 2.75
i y = 1.976j
y = −2k
y = −3.167ly = 1.98
5 a Db
C
6 1990
7 a 2(2x + 7)b
Width 10 m, length 17 m
8 Answers will vary; x = 3.
9 Adding 7 to both sides is the least preferred option, as it does not
resolve the subtraction of 7 within the brackets.
103
11 Kyle: $90, Noah: $35
12$10
4.4 Solving linear equations with pronumerals on both sides
1 a y = −1b
y = 1c
y = −3d
y = −5
8
g
y = −2h
y = 20
27
1
1j
y = 4l
y = −2
y = −15k
e y = −45f
y=
i y =
2 a Cb
A
3 a x = 1b
x = −2c
x = 3d
x=4
e x = 9f
x = 3g
x = −2h
x=4
x = 12k
x = −34
i x = −4j
x = 0l
4 a x = −6b
x = 7c
x = 2d
x = −2
2
x = −7g
x = 16h
x = −17
e x = 13f
i x = 1
5 a Db
C
6 a x = 5b
x = −3c
x = −8d
x = −712
e x = 4f
x = 10
g
x = −18h
x = 323
11
i x = 23j
x = −137k
x = 0
l x = −15
7 −3
8 24
9 3(n − 36) − 98 = −11n + 200
10 You cannot easily divide the left-hand side by 6 or the right-hand
side by 4.
11Daughter = 10 years, Tom = 40 years
12 The unknown number is −3.
13 a C = 40h + 250
b 18 hours, 45 minutes
c 18 750
d The printing is cheaper by $1375.
14 a 65 games
b$25
4.5 Solving problems with linear equations
x
5
1 a x + 3 = 5b
x − 9 = 7c
7x = 24d
= 11
x
= −9
f 5x − 3 = −7g 2(14 − x) = −3
2
3x + 4
h 3x + 5 = 8i2x − 12 = 15 j
=5
2
2 a Cb
Dc
Bd
B
3 0
e
132 Maths Quest 9
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number and algebra
36 years
$66
20 × 15; 30 × 10; Area = 300
7m
12 jackets
15 days
947 CDs
a Post Quick (cost = $700)
b The cost is nearly the same for 333 flyers ($366.50 and
$366.40).
12 2
13 $42.50
14 a Afenced = (5x + 20) m2
b Apath = (3x + 16) m2
c l = 8 m, w = 2 m
ONLINE PAGE PROOFS
4
5
6
7
8
9
10
11
d
b2 − d2
4c
9 a = 1 and b = 5
10 a A = πR2 − x2
b A = 160.71 cm2
c Answers will vary.
8 a=
Challenge 4.2
r = 10.608 cm
Investigation — Rich task
1 Table for males
8
25
y−b
y+b
c x=
a
2a
8 − 6y
1
e x=
f x=
y−2
5
4y + 1
h x=
i x = −5y − 12
5
5a − 9
14a + 13
j x = 12a
k x=
l x=
19
93
g+3
5f
5(f − 32)
a P=
b c=
c c=
4
9
9
b2 − d
V
v−u
d I=
e t=
f c=
a
R
4a
g y = hm + k h y = m(x − b) + a
y − a + mb
i a = y − m(x − b)
j x=
m
f − by
2(s − ut)
2π
k r=
l x=
m a=
a
C
t2
2
Fr
n G=
Mm
500 km
a 24 000 cm3 b t = 80 min = 1 h 20 min
Answers will vary.
B
=x
3(1 − 2y)
Answers will vary.
y
a
y+7
d x=
2
y+1
g x=
y−2
2
3
4
5
6
7
20
25
30
35
40
Body height h (cm)
132
147
163
178
194
Length of humerus l (cm)
20
25
30
35
40
Body height h (cm)
125
142
159
176
192
Table for females
4.6 Rearranging formulas
1 a x=
Length of humerus l (cm)
b x=
2 and 3
4
5
6
7
Linear
Answers will vary.
(44.6, 207.8)
James King
Code puzzle
The Atacama Desert in Chile
Topic 4 • Linear equations 133
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