NUMBER AND ALGEBRA
TOPIC 13
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Linear equations
13.1 Overview
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Many everyday situations can be represented by equations. You will
almost certainly have been solving equations without even realising it.
If you pay for an item which costs $4.25 and you hand $5.00 to the
shop assistant, you will expect 75c in change. This answer is obtained
using an equation. To work out which mobile phone plan is better value
requires solving equations. If you are travelling to a country which uses
different currency, you can use the exchange rate and equations to
work out the equivalent cost of an item in Australian dollars.
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Why learn this?
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What do you know?
Learning sequence
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Overview
Solving equations using trial and error
Using inverse operations
Building up expressions
Solving equations using backtracking
Checking solutions
Keeping equations balanced
Review ONLINE ONLY
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13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
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1 THINK List what you know about 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 that shows your class’s knowledge of equations.
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number and algebra
13.2 Solving equations using trial and error
Writing equations
Digital docs
SkillSHEET
Completing number
sentences
doc‐6571
SkillSHEET
Writing number
sentences from
written information
doc‐6572
SkillSHEET
Applying the four
operations
WOrKed eXamPle 1
doc‐6573
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Write an equation to represent each of these puzzles.
I am thinking of a number.
a When I multiply the number by 8, the answer is 24.
b When I divide the number by 5, the answer is 7.
c When I divide 60 by the number, the answer is 10.
d When I subtract 7 from the number, the answer is 25.
e When I subtract the number from 72, the answer is 52.
f When I square the number, the result is 36.
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• By writing an expression equal to a constant or another expression, we create an
equation; for example, 2x + 5 = 7, 3y − 4 = y + 6 are equations.
• We can use equations to describe a problem that we want to solve.
• To write a problem as an equation, use a pronumeral to stand for the unknown
number.
• To solve an equation means to find the value of the pronumeral.
THInK
Multiply the number by 8.
3
Write the equation.
1
Use a pronumeral to describe the number.
2
Divide the number by 5.
3
Write the equation.
1
Use a pronumeral to describe the number.
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e
476
a
Let m be the number.
m × 8 = 8m
8m = 24
b
c
Let t be the number.
t
t÷5=
5
t
=7
5
Let s be the number.
60
60 ÷ s =
s
60
= 10
s
Let l be the number.
2
3
Write the equation.
1
Use a pronumeral to describe the number.
2
Subtract 7 from the number (that is,
take 7 away from the number).
l−7
3
Write the equation.
l − 7 = 25
1
Use a pronumeral to describe the number.
2
Subtract the number from 72 (that is,
take the number away from 72).
72 − a
3
Write the equation.
72 − a = 52
Divide 60 by the number.
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c
Use a pronumeral to describe the number.
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b
1
E
a
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d
e
Let a be the number.
Maths Quest 7
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number and algebra
f
1
Use a pronumeral to describe the number.
2
Square the number (that is, multiply
the number by itself).
3
Write the equation.
f
Let z be the number.
z × z = z2
z2 = 36
• Sometimes the solution to an equation is obvious after close inspection.
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WOrKed eXamPle 2
Solve the following equations by inspection.
w
a
=4
b h − 9 = 10
3
Write the equation.
2
Think of a number which when
divided by 3 gives 4. Try 12.
3
So w must be 12.
1
Write the equation.
2
Think of a number which equals 10
when 9 is subtracted from it. Try 19.
3
So h must be 19.
w
=4
3
12 ÷ 3 = 4
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b
1
w = 12
b
19 − 9 = 10
h = 19
EV
Guess, check and improve
h − 9 = 10
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a
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O
THInK
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WOrKed eXamPle 3
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• We can try to guess the solution to an equation.
• By noting the result when we substitute each guess, we can improve the guess and come closer to the
solution.
Use guess, check and improve to solve the equation 2x + 21 = 4x − 1.
THInK
Set up a table with four columns displaying the
value of x, the value of the left‐hand and right‐
hand side equations (after substitution) and a
comment on how these two values compare.
SA
1
2
Substitute the first guess, say x = 1, into the LHS
and RHS equations and comment on results.
3
Repeat step 2 for x = 6, x = 10 until the correct
answer is obtained.
4
State the solution.
WrITe
Check
Guess
x
2x + 21 4x − 1
1
23
3
6
33
23
10
41
39
11
43
43
Comment
4x − 1 is too small
This is closer.
Very close
That’s it!
The solution is x = 11.
• This method also works when we need to find two variables.
Topic 13 • Linear equations 477
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number and algebra
WOrKed eXamPle 4
Find two numbers whose sum is 31 and whose product is 150.
WrITe
The numbers add up to 31 so guess two numbers
that do this. Then check by finding their product.
2
Guess 1 and 30.
3
Guess 10 and 21.
Try a number between 1 and 10 for the first
number.
Try a number between 5 and 10 for the first
number.
Try a number between 5 and 8 for the first number.
State the solution.
6, 25
Comment
P is too low.
P is too high.
P is too low.
P is too high.
6 × 25 = 150
That’s it!
The numbers are 6 and 25.
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Guess
Check
Sum (small
number first) Product (P)
1, 30
1 × 30 = 30
10, 21
10 × 21 = 210
5, 26
5 × 26 = 130
8, 23
8 × 23 = 184
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1
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THInK
IndIVIdual PaTHWaYS
⬛ PraCTISe
Questions:
1, 2, 3, 4, 5, 6, 7, 12
COnSOlIdaTe
Questions:
1, 2, 3, 4, 6e, h, 5, 6 column 1,
7, 10, 12
⬛
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reFleCTIOn
Why is it a good idea to write
our guesses down when using
guess, check and improve to
solve equations?
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Exercise 13.2 Solving equations using trial and error
FluenCY
1 WE1 Write
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⬛ ⬛ ⬛ Individual pathway interactivity
⬛ maSTer
Questions:
1, 2, 3, 4e, f, i, 5, 6 column 2,
6–13
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E
an equation to represent each of these puzzles.
I am thinking of a number.
a When I add 7, the answer is 11.
b When I add 3, the answer is 5.
c When I add 12, the answer is 12.
d When I add 5, the answer is 56.
e When I subtract 7, the answer is 1.
f When I subtract 11, the answer is 11.
g When I subtract 4, the answer is 7.
h When I subtract 8, the answer is 0.
i When I multiply by 2, the answer is 12.
j When I multiply by 6, the answer is 30.
k When I multiply by 5, the answer is 30.
l When I multiply by 6, the answer is 12.
m When I divide by 7, the answer is 1.
n When I divide by 3, the answer is 100.
o When I divide by 5, the answer is 2.
p When I divide by 7, the answer is 0.
q When I subtract the number from 15, the answer is 2.
r When I subtract the number from 52, the answer is 8.
s When I divide 21 by the number, the answer is 7.
t When I square the number, the answer is 100.
478
Maths Quest 7
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number and algebra
2
Solve the following equations by inspection.
x + 7 = 18
b y−8=1
c 3m = 15
WE2
a
m
=3
10
k
=0
5
h − 14 = 11
m − 1 = 273
g 4w = 28
h
w
=6
i b + 15 = 22
j
k 5k = 20
l
3
c
= 1 .4
m b − 2.1 = 6.7
n
o 5x = 14
3
3 WE3 Use guess, check and improve to solve the equation 3x + 11 = 5x − 1. The process
has been started for you.
Check
3x + 11 5x − 1
14
4
41
49
Comment
5x − 1 is too small.
5x − 1 is too big.
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Guess
x
1
10
5
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f
N
e
w + 25 = 26
d
Use guess, check and improve to solve the following equations. Note: There are
2 solutions for parts f and g.
a 5x + 15 = x + 27
b 2x + 12 = 3x − 2
c x + 20 = 3x
d 12x − 18 = 10x
e 10(x + 1) = 5x + 25
f x(x + 1) = 21x
g x(x + 7) = 12x
h 6(x − 2) = 4x
i 3(x + 4) = 5x + 4
5 WE4 Find two numbers whose sum is 21 and whose product is 98. The process has been
started for you.
Guess
Check
Comment
1 × 20 = 20
P is too low.
10 × 11 = 110
P is too high.
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10, 11
Product (P)
EV
Sum (small number first)
1, 20
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4
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Use guess, check and improve to find two numbers whose sum and product are given:
a sum = 26, product = 165
b sum = 27, product = 162
c sum = 54, product = 329
d sum = 45, product = 296
e sum = 178, product = 5712
f sum = 104, product = 2703
g sum = 153, product = 4662
h sum = 242, product = 14 065
i sum = 6.1, product = 8.58
j sum = 8, product = 14.79
k sum = 978, product = 218 957
l sum = 35, product = 274.89
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underSTandIng
7 Copy and complete this table by substituting each x‐value
into x + 4 and 4x + 1. The first row has been completed
for you. Use the table to find a solution to the equation
x2 + 4 = 4x + 1. (Remember, x2 means x × x.)
2
x
0
1
2
3
4
x2 + 4
4
Digital doc
Spreadsheet
Solving equations
doc‐1966
4x + 1
1
9
13
Topic 13 • Linear equations 479
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number and algebra
reaSOnIng
8 A football
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team won 4 more games than
it lost. The team played 16 games. How
many did it win?
9 Lily is half the age of Pedro. Ross is
6 years older than Lily and 6 years
younger than Pedro. How old is Pedro?
10 A plumber cut a 20‐metre pipe into
two pieces. One of the pieces is three times
as long as the other. What are the lengths
of the two pieces of pipe?
11 Julie has the same number of sisters as brothers. Her brother Todd has twice as many
sisters as brothers. How many children are in the family?
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PrOblem SOlVIng
12 Angus is the youngest
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in his family and today he and his Dad share a birthday. Both
their ages are prime numbers. Angus’s age has the same two digits as his Dad’s but in
reverse order. In 10 years’ time, Dad will be three times as old as Angus. How old will
each person be when this happens?
13 Sam accidentally divided two numbers on his calculator and got 0.6, when he should
have added the two numbers and got 16. What are the two numbers?
13.3 Using inverse operations
SkillSHEET
Inverse operations
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We can use flowcharts to help us organise harder equations.
The first number is called the input number.
The last number is called the output number.
Inverse operations mean ‘doing the opposite operation’.
For example, + is the inverse of −.
− is the inverse of +.
× is the inverse of ÷.
÷ is the inverse of ×.
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•
•
•
Digital doc
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WOrKed eXamPle 5
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Complete the flowchart to find the output number.
THInK
1
Follow the steps
and fill in the boxes.
8 × 2 = 16
16 + 4 = 20
20 ÷ 5 = 4
4−3=1
×2
+4
÷5
−3
×2
+4
÷5
–3
8
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8
16
20
4
1
• It is possible to travel in either direction through a flowchart. Working backwards,
against the arrows, is called backtracking.
• To work backwards, carry out inverse operations, or undo each step.
480
Maths Quest 7
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number and algebra
WOrKed eXamPle 6
Use backtracking and inverse operations to find the input number in this flowchart.
+5
×2
÷7
4
Fill in the numbers as you backtrack.
The inverse of ÷7 is ×7 (4 × 7 = 28).
2
The inverse of ×2 is ÷2 (28 ÷ 2 = 14).
3
The inverse of +5 is −5 (14 − 5 = 9).
+5
9
×2
14
−5
IndIVIdual PaTHWaYS
⬛ PraCTISe
Questions:
1–4, 7
⬛ COnSOlIdaTe
Questions:
1–5, 7, 8
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15
c i
−5
×2
SA
7
+ 10
+6
e i
−9
÷2
d i
ii
ii
ii
÷5
×2
−5
ii
+6
÷2
−9
÷3
÷3
+6
4
ii
30
+6
0
+ 10
7
30
f i
×3
15
4
÷3
+1
5
E
÷5
reFleCTIOn
When you’re backtracking,
does the order of the
operations matter?
int-4373
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+1
5
b i
×7
the following flowcharts to find the output number.
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×3
a i
4
÷2
⬛ maSTer
Questions:
1–8
⬛ ⬛ ⬛ Individual pathway interactivity
FluenCY
1 WE5 Complete
28
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Exercise 13.3 Using inverse operations
÷7
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WrITe
N
THInK
÷3
ii
0
Topic 13 • Linear equations 481
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number and algebra
+3
g i ×5
7
+3
× 10
–5
7
–5
h i × 10
ii
15
15
Use backtracking and inverse operations to find the input number in each of these
flowcharts.
WE6
a
+5
×2
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2
×5
ii
–3
b
÷2
2
÷5
–3
×7
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0
e
–3
f
32
+3
×2
÷7
4
–8
–7
×8
24
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g
×2
d
N
c
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12
6
SA
j
k
+2
EV
+4
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i
÷3
E
h
×8
÷ 2.17
÷9
× 10
–6
44
×8
–7
1
+6
–4
× 10
20
– 3.41
3.25
l
× —2
3
×
3
—
4
2
482 Maths Quest 7
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number and algebra
m
× 28
– 56
× 15
420
n
×9
– 152
× 19
+ 53
72
÷ 1.4
+ 2.31
× 6.5
– 0.04
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o
27
× 78
+ 2268
÷ 12
– 2605
O
p
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underSTandIng
3 a In question 1, what did you notice about each pair of flowcharts?
b Does changing the order of operations affect the end result?
4 Complete the two statements below.
a Adding and _____ are inverse operations.
b _____ and dividing are inverse operations.
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reaSOnIng
5 a Explain with
Digital doc
Spreadsheet
Backtracking
doc‐1967
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the use of examples why addition and subtraction are inverse operations.
Start with x + 3 = 7 by trying to cancel down the +3.
b Explain with the use of examples why multiplication and division are inverse
operations. Start with 2 x = −8 by trying to cancel down the 2.
6 a Which of the following statements is correct and which one would you use to
calculate the value of x?
i If x + 6 = −11, then x = −11 − 6.
ii If x + 6 = −11, then 6 = −11 − x.
b Which of the following statements is correct and which one would you use to
calculate the value of x?
56
56
i If −7x = 56, then −7 =
.
ii If −7x = 56, then x =
.
x
−7
PrOblem SOlVIng
7 Form an equation from
the following statements.
I thought of a number and added 5. Three times the result was equal to 27.
Use your equation to find the number that I first thought of.
8 In an amusement park, 0.3 of the children were girls. If there were 80 more boys than
girls, how many children were there in the park?
Topic 13 • Linear equations 483
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number and algebra
13.4 Building up expressions
• Flowcharts can be used to construct algebraic expressions.
WOrKed eXamPle 7
Complete the flowcharts below to find the output number.
×3
WrITe
m × 3 = 3m
Adding 5 to 3m gives 3m + 5.
a
Adding 5 to m results in m + 5.
To multiply all of m + 5 by 3,
m + 5 will need to be inside brackets.
b
×3
3m
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+5
×3
m+5
m
3m + 5
3(m + 5)
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WOrKed eXamPle 8
+5
O
THInK
b
×3
m
m
a
+5
b
+5
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a
EV
Draw a flowchart whose input number is m and whose output number is given by
the expressions:
m
m+9
+2 .
a 2m − 11
b
c4
3
5
THInK
1
The first step is to obtain 2m;
that is, multiply m by 2.
b
1
2
484
This is followed by subtracting 11.
The expression m + 9 is
grouped as in a pair of
brackets, so we must obtain
this part first. Therefore
add 9 to m.
Then the whole expression
is divided by 5.
2m
m
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SA
2
×2
a
E
a
WrITe
×2
– 11
2m – 11
2m
m
+9
b
m+9
m
+9
m
÷5
m+9
m+9
–––––
5
Maths Quest 7
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number and algebra
1
The pair of brackets indicates
we must first work from
within the brackets:
(a) divide m by 3
(b) then add 2 to this result.
÷3
c
m
—
3
m
÷3
m
—
3
m
2
Multiply the final result
obtained in step 1 by 4.
+2
÷3
+2
m
—
3
m
m
—+2
3
×4
m
— +2
3
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c
+2)
4(—
3
m
O
• Using flowcharts, we can backtrack to our input number using inverse operations.
THInK
x
Copy the flowchart and look at the
operations that have been performed.
+2
5x + 2
5x
x
–2
The inverse operation of ×5 is ÷5.
Show this on the flowchart.
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5x + 2
5x
×5
E
EV
The inverse operation of +2 is −2.
Show this on the flowchart.
+2
×5
×5
+2
5x + 2
5x
x
÷5
–2
SA
3
5x + 2
WrITe
x
2
+2
5x
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1
×5
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Complete the flowchart at right by
writing in the operations which must be
carried out in order to backtrack to x.
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WOrKed eXamPle 9
Exercise 13.4 Building up expressions
IndIVIdual PaTHWaYS
⬛ PraCTISe
Questions:
1, 2, 3, 4, 5, 6, 7, 9
⬛ COnSOlIdaTe
Questions:
1a, c, e, g, 2a, c, e, g, 3a, c, e,
g, i, k, 4, 5a, c, 6, 7, 8, 9, 11
⬛ ⬛ ⬛ Individual pathway interactivity
⬛ maSTer
Questions:
1d, f, h, 2b, d, f, h, 3b, f, l, 4, 5b,
d, 6, 7, 8, 10, 11
reFleCTIOn
Do the order of operation
laws apply when constructing
flowcharts?
int-4374
Topic 13 • Linear equations 485
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number and algebra
FLUENCY
1 WE7 Build up
a
×5
an expression by following the instructions on the flowcharts.
5x
x
×2
+7
+ 11
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÷8
–2
x
+9
× 3.1
h
+ 1.8
N
÷7
AT
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x
Build up an expression by following the instructions on the flowcharts. Use a grouping
x−5
device, such as a pair of brackets or a vinculum; for example, 2(x + 3) or
.
4
+5
x
x
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x
g
×3
– 1–
2
+ 10
÷3
+3
×9
+ 4.9
÷5
– 3.1
÷ 1.8
x
d
x
f
E
‒ 2.1
e
÷7
EV
‒2
c
b
×4
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a
x
×7
h
x
x
SA
÷5
f
–3
x
3
3
d
x
2
x–
x
× 1–2
g
+1
x
x
e
÷3
O
c
b
–2
Copy and complete the following flowcharts by filling in the missing expressions.
a
+2
×6
b
–8
x
–2
×4
+1
x
x
c
×3
÷ 12
d
x
486 Maths Quest 7
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number and algebra
÷7
+3
f
x
×6
–2
h
x
i
×3
+4
÷7
÷3
+4
×6
×3
÷4
+2
x
–8
÷5
j
+9
x
x
k
÷5
x
+5
g
–3
× 11
l
×4
–2
x
O
x
N
LY
e
4
Draw a flowchart whose input number is x and whose output is given by the
following expressions.
x−8
x
a 5x + 9
b 2(x + 1)
c + 4
d
7
6
x+6
x
−2
e 12(x − 7)
f
g 7x − 12
h
3
5
3x + 7
x
+6
i 3(x + 7) − 5
j
k 4(3x + 1)
l 3
2
5
5
Complete the following flowcharts by writing in the operations that must be carried
out in order to backtrack to x.
WE9
×7
7x + 3
x
÷2
c
+1
‒5
E
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UNDERSTANDING
6 Add the operations
a SA
5(x ‒ 2)
÷4
x
x‒5
–––––
4
x
–x ‒ 3
6
x
4(x + 8)
x
x—
‒—
7
—
5
x
–x + 2.1
9
to complete these flowcharts.
b
2(x + 7)
c
x
d
×5
x
–x + 1
2
x
x
‒2
b
+3
EV
a
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AT
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N
WE8
d
x—
+—
8
—
3
e
f
x
8x ‒ 3
g
h
x
3(x + 55)
Topic 13 • Linear equations 487
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number and algebra
i
j
7x − 5
x
4(x + 8) + 5
x
reaSOnIng
7 I think of a
eles‐0010
PrOblem SOlVIng
9 Rachel and Jackson
O
eLesson
Backtracking
N
LY
number, and add 8 to it. I multiply the
result by 5 and then divide the result by 4. The
answer is 30. Build up an expression to write the
above in the form of an equation. Show all
your working.
8 I think of a number, multiply it by 5 and add 15. The
result is 3 less than 4 times the original number. If
the original number is n, write down an equation to
show the relation. Show all your working.
AL
U
AT
IO
N
are 7 years apart in age. Jackson is older than Rachel. The sum of
their ages is 51. Find Rachel’s age.
10 Marcus and Melanie pooled their funds and purchased shares on the Stock Exchange.
Melanie invested $350 more than Marcus. Together they invested $2780. Find how
much Marcus invested.
11 The sum of two numbers is 32 and their product is 247. What are the two numbers?
E
EV
CHallenge 13.1
Digital doc
SkillSHEET
Combining like terms
SA
doc‐6575
M
PL
13.5 Solving equations using backtracking
• Backtracking an algebraic expression can be used to find the input number.
WOrKed eXamPle 10
Draw a flowchart to represent the following puzzle and then solve it by
backtracking. I am thinking of a number. When I multiply it by 4 and then add 2
the answer is 14.
THInK
1
Build an expression using x to
represent the number.
Start with x, multiply by 4 and add 2.
The output number is 14.
WrITe
×4
x
+2
4x
4x + 2
14
488
Maths Quest 7
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number and algebra
2
Backtrack to find the value of x. The inverse
operation of +2 is −2 (14 − 2 = 12).
The inverse operation of ×4 is ÷4 (12 ÷ 4 = 3).
×4
+2
x
4x
4x + 2
3
12
14
÷4
So x = 3. The number is 3.
State the answer.
N
LY
3
‒2
WOrKed eXamPle 11
Backtrack to find x.
The inverse operation of ×3
is ÷3 (24 ÷ 3 = 8).
The inverse operation of +7
is −7 (8 − 7 = 1).
a
E
1
Build the expression on the left‐hand
side of the equation.
Start with x, divide by 3 and then
add 5. The output number is 6.
M
PL
State the answer.
2
Backtrack to solve for x.
The inverse operation of +5
is −5 (6 − 5 = 1).
The inverse operation of ÷3
is ×3 (1 × 3 = 3).
State the answer.
3(x + 7)
24
×3
x
x+7
3(x + 7)
1
8
24
‒7
÷3
÷3
+5
x=1
b
x
—
3
x
x
—
3+5
6
÷3
+5
x
x
—
3
x
—
+5
3
3
1
6
×3
3
×3
x+7
+7
SA
b
3
+7
x
EV
2
Build the expression on the left‐hand
side of the equation.
Start with x, add 7 and then multiply by 3.
The output number is 24.
N
1
AL
U
a
WrITe
AT
IO
THInK
O
Solve the following equations by backtracking.
x
a 3( x + 7) = 24
+5=6
b
3
‒5
x=3
Topic 13 • Linear equations 489
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number and algebra
WOrKed eXamPle 12
Simplify and then solve the following equation by backtracking.
5x + 13 + 2x − 4 = 23
THInK
WrITe
1
Simplify by adding the like terms together on the left‐
hand side of the equation. 5x + 2x = 7x, 13 − 4 = 9
5x + 13 + 2x − 4 = 23
7x + 9 = 23
2
Draw a flowchart and build the expression 7x + 9.
Start with x, multiply by 7 and add 9.
The output number is 23.
x
+9
N
LY
×7
7x + 9
7x
State the answer.
N
×7
AT
IO
4
Backtrack to solve for x.
The inverse operation of +9 is −9
(23 − 9 = 14).
The inverse operation of ×7 is ÷7
(14 ÷ 7 = 2).
AL
U
3
O
23
+9
x
7x
7x + 9
2
14
23
÷7
–9
x=2
EV
Exercise 13.5 Solving equations
using backtracking
IndIVIdual PaTHWaYS
M
PL
E
⬛ PraCTISe
Questions:
1–7, 9, 14, 21
SA
reFleCTIOn
Can you think of some
situations where you might
need to be able to solve
equations?
490
COnSOlIdaTe
Questions:
1–3, 4b, c, e, f, h, i, 5, 6 column
1, 7b, d, h, 8–22
⬛
⬛ ⬛ ⬛ Individual pathway interactivity
⬛ maSTer
Questions:
1–3, 4c, f, i, 5, 6 column 2, 7a,
g, h, 8–23
int-4375
FluenCY
1 WE10 Draw
a flowchart to represent each of the following puzzles and then solve them
by backtracking.
I am thinking of a number.
a When I multiply it by 2 and then add 7 the answer is 11.
b When I add 3 to it and then multiply by 5 the answer is 35.
c When I divide it by 4 and then add 12 the answer is 14.
d When I add 5 to it and then divide by 3 the answer is 6.
e When I subtract 7 from it and then multiply by 6 the answer is 18.
f When I multiply it by 8 and then subtract 11 the answer is 45.
g When I subtract 4 from it and then divide by 9 the answer is 7.
h When I divide it by 11 and then subtract 8 the answer is 0.
i When I add 5 to it and then multiply by 2 the answer is 12.
j When I multiply it by 6 and then add 4 the answer is 34.
Maths Quest 7
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number and algebra
Use backtracking to find the solution to the following equations.
3x − 7 = 23
d
x
−2=8
5
b
4(x + 7) = 40
c
e
5(x − 3) = 15
f
E
a
M
PL
6
EV
AL
U
AT
IO
N
O
I multiply it by 5 and then subtract 10 the answer is 30.
I subtract 3.1 from it and then multiply by 6 the answer is 13.2.
I divide it by 8 and then subtract 0.26 the answer is 0.99.
I divide it by 3.7 and then add 1.93 the answer is 7.62.
I add 25 to it and then divide by 6 the answer is 45.
I subtract 34 from it and then divide by 23 the answer is 16.
2 Draw a flowchart and use backtracking to find the solution to the following
equations.
a 5x + 7 = 22
b 9y − 8 = 1
c 3m − 7 = 11
d 4x + 12 = 32
e 8w + 2 = 26
f 11m − 1 = 274
5
2
1
3
g 4w + 5.2 = 28
h 6b − =
i 2a + =
9
3
3
5
3 WE11a Solve the following equations by backtracking.
a 3(x + 7) = 24
b 2(x − 7) = 22
c 5(x − 15) = 40
d 11(x + 5) = 99
e 6(x + 9) = 72
f 3(x − 11) = 3
5
2
1
4
g 4(w + 5.2) = 26
h 6 (b − ) =
i 8 (x − ) =
9
3
4
5
4 WE11b Solve the following equations by backtracking.
x
x
x
a + 5 = 6
b − 2 = 3
c + 7 = 10
3
9
4
x
x
x
d − 11 = 6
e − 5 = 6
f
+ 1=1
3
2
7
3
x
x
x 1 4
4
g + 2.3 = 4.9
h −
=
i
+ =
4 11 11
2 9 9
5
5 Solve the following equations by backtracking.
x−8
x−8
x+4
a
=6
b
=3
c
= 10
3
7
7
x−5
x + 100
x + 11
d
=6
e
=0
f
= 23
2
7
17
x + 2.21
x−1 4
x+1 3
g
= 4.9
h
=
i
=
1.4
7
4
8
5
N
LY
k When
l When
mWhen
n When
o When
p When
x+6
=6
9
x+3
=3
8
x
− 1.7 = 3.6
2.1
x
+ 10 = 12
i
3
x+5
3
j 4x + = 1
k
−3=7
l 3(2x + 5) = 21
5
3
x
m4(x − 2) + 5 = 21
n 3
+ 1 = 15
o 2(3x + 4) − 5 = 15
2
7 WE12 Simplify and then solve the following equations by backtracking.
a 2x + 7 + 3x + 5 = 27
b x + x + 1 + x + 2 = 18
c 3x + 9 + x − 4 = 17
d 3x + 5x + 2x = 40
e 6x + 6 − x − 4 = 37
f 3x − 11 + 4x = 17
g 5x − 2x + 5 − x = 19
h 2x + 3x + 4x + 5 = 7
i 7x − 4x + 8 − x = 10
6(x − 4) = 18
h
SA
g
Topic 13 • Linear equations 491
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number and algebra
UNDERSTANDING
8 MC The correct flowchart
A
×6
+5
required to solve the equation 6x + 5 = 8 is:
B
+5
×6
x
x
8
8
×6
×6
+5
x
8
÷6
–5
–5
AL
U
4x + 2 = 10
1
X
1
1
X
X
X
X
1
1
1
1
1
1
1
1
1
1
E
EV
8
÷6
Given the following balance, what operations do you
need to do to find the value of x? What is the value of x?
10 Chloë wrote the following explanation to solve the
equation 3(x − 6) + 5 = 8 by the backtracking method.
To solve this equation, work backwards and do things
in the reverse order. First add 5, then subtract 6, then
finally multiply by 3.
Explain why her instructions are not correct.
9
+5
O
÷6
×6
N
–5
÷6
x
8
x
E
D
+5
–5
AT
IO
C
–5
N
LY
÷6
REASONING
11 Imraan is 5
SA
M
PL
years older than his brother Gareth and the sum of their ages is 31 years.
How old is Gareth? (Let x represent Gareth’s age.) Show your working.
12 In three basketball games Karina has averaged 12 points each game. In the first game
she scored 11 points, in the second she scored 17 points, and in the third game she
scored x points.
a From the given information, what is the average of 11, 17 and x?
b Write an equation using the answer to part a.
c Solve the equation.
d How many points did Karina score in the
third game?
13 Melanie and Callie went tenpin bowling
together. Melanie scored 15 more pins than
Callie, and their total score was 207. What did
Callie score? Show your working.
14 The sum of three consecutive whole numbers is
51. Find the numbers. (Hint: Let the smallest
number equal x.)
492 Maths Quest 7
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number and algebra
sum of three consecutive odd numbers is 27. What are the
numbers?
16 Three consecutive multiples of 5 add up to 90. What are the
3 numbers?
17 David is 5 years younger than his twin brothers. If the sum of
their ages is 52, then how old is David?
18 In the high jump event Chris leapt 12 centimetres higher than Tim,
but their two jumps made a total of 3 metres. How high did
Chris jump?
19 Daniel and Travis are twins, but their sister Phillipa is 3 years
older. If the sum of their three ages is 36, how old are the
twins? Show your working.
20 Mel loves playing ‘Think of a number’ with friends. Here’s an
example of one of her puzzles.
• Think of a number.
• Double it.
• Add 10.
• Divide by 2.
• Take away the number you first thought of.
• Your answer is … 5!
Let’s investigate to see why the answer is always 5, whatever
number you first thought of. We can form expressions for each
of the steps, using a variable as the starting value.
• Think of a number.............................................. n
• Double it............................................................ n × 2 = 2n
• Add 10............................................................... 2n + 10
• Divide by 2........................................................ (2n + 10) ÷ 2 = n + 5
• Take away the number you first thought of....... n + 5 − n = 5
• Your answer is................................................... 5
So, your answer will always be 5, for any starting number.
a Write expressions for each step in the following, showing that you can determine the
answer in each case.
Puzzle 1
• Take the year in which you were born.
• Subtract 500.
• Multiply by 2.
• Add 1000.
• Divide by 2.
• Your answer is . . . your birth year!
Puzzle 2
• Take your age (in years).
• Add 4.
• Multiply by 10.
• Subtract 10.
• Divide by 5.
• Subtract your age.
• Take away 6.
• Your answer will be . . . your age!
SA
M
PL
E
EV
AL
U
AT
IO
N
O
N
LY
15 The
Topic 13 • Linear equations 493
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number and algebra
N
LY
Puzzle 3
• Think of a number.
• Divide it by 2.
• Add 2.
• Multiply by 4.
• Take away your original number.
• Subtract your original number again.
• Your answer should be . . . 8.
b Write some ‘Think of a number’ puzzles yourself. Try them out on friends. They
will marvel at your mystical powers!
PrOblem SOlVIng
21 Using the six consecutive
O
numbers from 4 to 9, complete the magic
square at right so that each row, column and diagonal totals 15.
22 The sum of 3 consecutive odd numbers is 39. What are the 3 numbers?
23 Form an equation for the following statement.
One‐third of a certain number is 12.
Solve the equation to find the number.
Digital doc
WorkSHEET 13.1
3
2
AT
IO
N
doc‐1946
1
13.6 Checking solutions
• It is always a good idea to check if your solution to an equation is correct.
• The solution is the answer obtained when solving an equation.
Digital doc
AL
U
SkillSHEET
Checking solutions by
substitution
WOrKed eXamPle 13
doc‐6576
EV
For each of the following, determine whether x = 7 is the solution to the equation.
x + 5
a
=4
b 2 x − 8 = 10
3
THInK
Write the equation.
2
Write the left‐hand side
(LHS) of the equation
and substitute x = 7.
a
E
1
SA
b
494
x+5
=4
3
If x = 7, LHS =
M
PL
a
WrITe
x+5
3
= 7 +3 5
= 123
=4
3
Perform the calculation.
4
Compare this with the right‐
hand side (RHS) of the equation.
RHS = 4
5
Comment on the result.
The solution is x = 7,
since LHS = RHS.
1
Write the equation.
2
Write the left‐hand side of the
equation and substitute x = 7.
b
2x − 8 = 10
If x = 7, LHS = 2x − 8
= 2(7) − 8
Maths Quest 7
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number and algebra
Perform the calculation.
= 14 − 8
=6
4
Compare this with the right‐
hand side of the equation.
RHS = 10
5
Comment on the result.
Therefore, x = 7 is not the solution
since LHS ≠ RHS.
WOrKed eXamPle 14
Write the equation.
2
Write the left‐hand side of the
equation and substitute x = 10.
Perform the calculation.
4
5
Write the right‐hand side of the
equation and substitute x = 10.
Perform the calculation.
6
Comment on the results.
1
Write the equation.
2
Write the left‐hand side of the
equation and substitute x = 10.
Perform the calculation.
3
x+2
3
= 10 3+ 2
= 123
=4
RHS = 2x − 12
= 2(10) − 12
= 20 − 12
=8
EV
5
Write the right‐hand side of the
equation and substitute x = 10.
Perform the calculation.
6
Comment on the results.
SA
4
LHS =
AL
U
3
x+2
= 2x − 12
3
If x = 10,
AT
IO
a
E
b
1
M
PL
a
WrITe
N
THInK
O
For each equation below there is a solution given. Is the solution correct?
x+2
a
= 2x − 12
b 3 x − 7 = 2 x + 3, x = 10
3
N
LY
3
Therefore x = 10 is not the solution,
since LHS ≠ RHS.
b
3x − 7 = 2x + 3
If x = 10, LHS = 3x − 7
= 3(10) − 7
= 30 − 7
= 23
RHS = 2x + 3
= 2(10) + 3
= 20 + 3
= 23
The solution is x = 10,
since LHS = RHS.
Topic 13 • Linear equations 495
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number and algebra
Exercise 13.6 Checking solutions
IndIVIdual PaTHWaYS
⬛ PraCTISe
Questions:
1–4, 9
COnSOlIdaTe
Questions:
1–5, 7, 9
maSTer
Questions:
1a, c, e, g, i, k, 2a, c, e, h, j,
3–8, 10
⬛
⬛ ⬛ ⬛ Individual pathway interactivity
⬛
int-4376
solution correct?
M
PL
E
EV
AL
U
AT
IO
N
O
FluenCY
1 WE13 For each of the following determine whether:
a x = 3 is the solution to the equation x + 2 = 6
b x = 3 is the solution to the equation 2x − 1 = 5
c x = 5 is the solution to the equation 2x + 3 = 7
d x = 4 is the solution to the equation 6x − 6 = 24
e x = 10 is the solution to the equation 3x + 5 = 20
f x = 5 is the solution to the equation 4(x − 3) = 8
g x = 7 is the solution to the equation 3(x − 2) = 25
h x = 8 is the solution to the equation 5(x + 1) = 90
i x = 12 is the solution to the equation 6(x − 5) = 42
j x = 81 is the solution to the equation 3x − 53 = 80
k x = 2.36 is the solution to the equation 5x − 7 = 4.8
l x = 4.4 is the solution to the equation 7x − 2.15 = 18.64.
2 WE14 For each equation below there is a solution given. Is the
a 2x + 1 = 3x − 5
x=6
b 5x + 1 = 2x − 7
x=8
c 3x − 5 = x + 8
x = 10
d 5x = 2x + 12
x=4
e 4x = 3x + 8
x=8
f 3x = x + 20
x = 15
g 3x − 1.2 = x + 2.9
x = 1.9
h 6x + 1.5 = 2x + 41.5
x = 10
i 2.4(x + 1) = 9.6
x=3
j 1.2(x + 1.65) = 0.2(x + 9.85)
x = 0.45
N
LY
reFleCTIOn
Is it possible for an equation to
have a decimal solution?
SA
underSTandIng
3 Complete the table
x
1
2
3
4
2x + 3
a
b
c
496
0
below to find the value of 2x + 3 when x = 0, 1, 2, 3, 4.
For what value of x does 2x + 3 = 11?
What is the solution (that is, the value of x) for 2x + 3 = 11?
What is the solution (that is, the value of x) for 2x + 3 = 5?
Maths Quest 7
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number and algebra
4
Complete the table below to find the value of 5(x − 2) when x = 2, 3, 4, 5, 6.
x
2
3
4
5
6
5(x − 2)
What is the solution (that is, the value of x) to 5(x − 2) = 10?
What is the solution (that is, the value of x) to 5(x − 2) = 20?
What do you guess is the solution (that is, the value of x) to 5(x − 2) = 30?
Check your guess.
5 a Copy and complete the table below.
2x + 1
3
7
3x − 5
O
x
4
6
7
x
x + 3
2
3
3
b
M
PL
11
8
E
7
9
2x − 6
EV
5
5?
AL
U
b What is the solution to 2x + 1 = 3x −
6 a Copy and complete the table below.
N
10
AT
IO
5
N
LY
a
b
c
What is the solution to
SA
REASONING
7 Consider the
x + 3
= 2x − 6?
2
diagram shown.
38.7 m
15.4 m
3.9 m
a
b
c
ym
Do you need to know both dimensions to be able to calculate the value of y?
Calculate the value of y.
Explain how you would check the solution.
Topic 13 • Linear equations 497
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number and algebra
Carol is making a quilt for her granddaughter. The quilt pattern requires that 17 of the
quilt is made of a pink fabric. Carol has 0.5 m2 of pink fabric. She intends to use all the
pink material in the quilt. How much more fabric does she need for the entire quilt?
a Write an equation for this problem.
b Calculate the value of the unknown.
c Check the solution.
AL
U
AT
IO
N
O
N
LY
8
PrOblem SOlVIng
9 $125 is shared between
James and Alison, but Alison is to receive $19 more than
James. Set up an equation to describe this situation. How much money do James and
Alison each receive?
10 Six consecutive numbers add to 393. Find the value of the smallest of these numbers.
Digital doc
WorkSHEET 13.2
EV
doc‐1947
E
13.7 Keeping equations balanced
SA
M
PL
• Equations are mathematical statements that show two
equal expressions; that is, the left‐hand side and the
right‐hand side of an equation are equal. For example,
2 + 5 = 7.
• A pan balance scale can be used to show whether
things have equal mass. If the contents of the two
pans have the same mass, the pans balance at the
same level and the arms of the scale are parallel to the
ground. If the contents of the two pans are different
in mass, the arms tip so that the side with the greater
mass is lower.
2
7
5
Equal
498
3
7
6
Not equal
Maths Quest 7
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number and algebra
• Any change that is made to one side must also be made to the other side if balance is to
be maintained.
2 2
7
5 5
7
7
5
2 3
3
Equal
Equal
WOrKed eXamPle 15
a
b
EV
THInK
On the LHS, there are 2 bags with h
weights in each, and 2 weights outside
the bags. On the RHS, there are
8 weights. Write this as an equation.
h
Equal
WrITe
a
1
1 1 1
1 1 1 1
1
1
2h + 2 = 8
1
Remove 2 weights from both sides so
that the balance will be maintained
(−2). This leaves the 2 bags on the
LHS and 6 weights on the RHS.
b
h
h
SA
b
M
PL
E
a
h
AL
U
For the following pan balance scale:
write the equation represented by the scale
calculate the value of the variable.
AT
IO
N
O
N
LY
If we double what is on the left‐hand side,
If we add 3 to the left‐hand side, we must
we must double the amount on the right‐
also add 3 to the right‐hand side.
hand side.
• Pronumerals in expressions can be represented on a pan
x
1 1 1
1
balance scale using bags. Each bag contains a particular
1 1 1 1
1 1
number of weights that is equal to the value of the variable
shown on the front of the bag. For the example, the scale
on the right shows the equation x + 3 = 7.
Equal
• Making changes to both sides of the scale allows you to
work out how many weights are in the bag. This is the same as working out the value of
the pronumeral. On the scale above, if we remove 3 weights from each pan, the bag with
x weights in it weighs the same as 4 weights, so x = 4.
2
Remove half of the contents
of each pan (÷2).
1 1
1 1 1 1
Equal
2h = 6
h
1
1 1
Equal
h=3
Each bag contains 3 weights.
Therefore, h = 3.
Topic 13 • Linear equations 499
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number and algebra
• Both backtracking and the balance method of solving equations use inverse operations to
find the value of a pronumeral. For example, the equation 2q + 3 = 11 can be solved as
follows.
Backtracking
4
2q
+3
2q + 3
8
q
q
1 1
1 1 1 1
1 1 1 1 1
1
11
1 1
N
LY
The equivalent equations made when
backtracking are:
2q + 3 = 11
2q = 8
q=4
Equal
2q + 3 = 11
Remove three counters from both sides (−3).
O
×2
q
Balance method
q
1
1 1 1
1 1 1 1
AT
IO
N
q
Equal
2q = 8
Halve the contents of each pan (÷2).
AL
U
q
1
1 1 1
Equal
EV
q=4
SA
M
PL
E
• In both methods:
– the operations and the order in which they are performed to solve the equation are the
same
– the equivalent equations created on the way to the solution are the same
– the aim is to get the pronumeral by itself on one side of the equation
– the last operation performed on the pronumeral when building the equation is the first
operation undone.
WOrKed eXamPle 16
Use inverse operations to solve the following equations.
h
a 2 y + 3 = 11
b
+1=3
5
THInK
a
500
c
2(k − 4) = 4
WrITe
1
The last operation performed when
building 2y + 3 = 11 was +3. Subtract 3
from both sides and then simplify.
2
To get y by itself, divide both
sides by 2 and then simplify.
a
2y + 3 = 11
2y + 3 − 3 = 11 − 3
2y = 8
2 1 y 84
= 1
21
2
y=4
Maths Quest 7
c13LinearEquations.indd 500
18/05/16 12:19 PM
number and algebra
The last operation performed when
h
building + 1 = 3 was +1. Subtract 1
5
from both sides and then simplify.
To get h by itself, multiply both
sides by 5 and then simplify.
1
The last operation performed when
building 2(k − 4) = 4 was ×2. Divide
both sides by 2 and then simplify.
2
To get k by itself, add 4 to both sides
and then simplify.
h
+1−1=3−1
5
h
=2
5
h
× 51 = 2 × 5
51
h
= 10
5
2(k − 4) = 4
21 (k − 4) 42
= 1
21
2
k−4=2
c
O
2
h
+1=3
5
b
N
c
1
k−4+4=2+4
k=6
AT
IO
b
WrITe
N
LY
THInK
Equations with rational number solutions
AL
U
• When solving equations, sometimes the solution is a rational number.
WOrKed eXamPle 17
Solve the following equations.
5d = 4
b 3c + 1 = 6
a
To get d by itself, divide both sides
by 5 and then simplify.
WrITe
a
1
The last operation performed when
building 3c + 1 = 6 was +1. Subtract 1
from both sides and simplify.
2
To get c by itself, divide both sides by 3
and then simplify.
SA
b
M
PL
E
a
EV
THInK
b
5d = 4
51 d 4
=
5
51
4
d=
5
3c + 1 = 6
3c + 1 − 1 = 6 − 1
3c = 5
31 c 5
=
3
31
5
c=
3
2
=1
3
Negative integers
• When the pronumeral is being subtracted, it is difficult to represent the equation using
backtracking or the balance method.
• By adding the pronumeral to both sides, the equation can be solved as normal.
Topic 13 • Linear equations 501
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18/05/16 12:19 PM
number and algebra
WOrKed eXamPle 18
Solve the following equations.
5−v=2
b 17 − 2 a = 11
a
THInK
Undo subtracting v by adding
v and then simplify.
2
To get v by itself, subtract 2 from
both sides and simplify.
1
Undo subtracting 2a by adding
2a and then simplify.
2
The last operation performed when
building 17 = 11 + 2a was +11. Subtract
11 from both sides and simplify.
3
To get a by itself, divide both
sides by 2 and then simplify.
5−v=2
5−v+v=2+v
5=2+v
a
N
LY
5−2=2+v−2
3=v
v=3
17 − 2a = 11
17 − 2a + 2a = 11 + 2a
17 = 11 + 2a
N
O
b
AT
IO
b
1
17 − 11 = 11 + 2a − 11
6 = 2a
6 3 21 a
= 1
21
2
3=a
AL
U
a
WrITe
Exercise 13.7 Keeping equations balanced
IndIVIdual PaTHWaYS
reFleCTIOn
How will you decide the order
to undo the operations?
PraCTISe
Questions:
1–9
EV
⬛
⬛ COnSOlIdaTe
Questions:
1–10
maSTer
Questions:
1–8, 10, 11, 12
⬛
⬛ ⬛ ⬛ Individual pathway interactivity
E
int-4377
M
PL
FluenCY
1 WE15 For each of the following pan balance scales:
i write the equation represented by the scale
ii calculate the value of the pronumeral.
a
b
SA
r
r r r
1 1
1 1 1 1 1
1 1 1 1 1
n
n
Equal
Equal
c
t
t
d
t
1 1 1 1
1 1 1 1
1 1
1 1 1
Equal
502
1 1 1
1 1 1 1
1 1 1
p
p p p
1
1 1 1
1 1 1 1 1
1 1 1 1 1
Equal
Maths Quest 7
c13LinearEquations.indd 502
18/05/16 12:19 PM
number and algebra
2
a
Use inverse operations to solve the following.
3g + 7 = 10
b 4m − 6 = 6
d
2(n − 5) = 8
WE16
e
g−2
=3
4
Solve the following equations.
3h = 7
b 2k = 5
3t − 4 = 9
e 6h − 3 = 10
WE18
4
Solve the following equations.
a 6−m=2
b 4−d=1
d 13 − 2s = 7
e 19 − 3g = 4
3
c
q
+ 8 = 11
3
f
4(y + 1) = 16
c
f
2w + 1 = 8
3l + 4 = 8
c
f
12 − 3v = 6
30 − 5k = 20
Interactivity
Balancing equations
int‐0077
a
d
underSTandIng
5 Solve the following
a 2(x + 3) = 11
equations.
N
LY
WE17
AL
U
reaSOnIng
7 A taxi company
AT
IO
N
O
b 3(5 − y) = 12
1
c p+3=6
d 5(8 − 2h) = 15
4
5(F − 32)
6 The formula C =
is used to convert degrees Fahrenheit to degrees Celsius.
9
Use the formula to find:
a 45° Fahrenheit in degrees Celsius
b 45° Celsius in degrees Fahrenheit.
charges a $2.70 flag fall. An additional $1.87 per km is charged for the
SA
M
PL
E
EV
journey.
a Calculate the cost of a 4.6-km journey.
b If the journey cost $111.16, how far was the journey? Show your working.
8
While shopping for music online, Olivia found an album she liked. She could purchase
the entire album for $17.95 or songs from the album for $1.69 each.
How many songs would Olivia want to buy for it to be cheaper to buy the entire album?
Explain how you calculated the answer.
Topic 13 • Linear equations 503
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18/05/16 12:19 PM
number and algebra
PrOblem SOlVIng
9 A class of 25 students
x+2
AT
IO
x
N
O
N
LY
has 7 more boys
than girls. How many boys are there?
10 Form an equation to represent the
following statement. When 12 is
subtracted from 7 times
a number, the result is 9. Solve your
equation to find the number.
11 A number, x, becomes the fifth member
of the set of numbers {3, 7, 9, 13} to
make the mean of the five numbers
equal to their median. Find the possible
values for x.
12 Given that the perimeter of the following triangle is 22 cm, find the value of x.
AL
U
3x
SA
M
PL
E
EV
CHallenge 13.2
504
Maths Quest 7
c13LinearEquations.indd 504
18/05/16 12:19 PM
number and algebra
13.8 Review
www.jacplus.com.au
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.
Download the Review
questions document
from the links found in
your eBookPLUS.
AT
IO
N
A summary of the key points covered and a concept map
summary of this chapter are available as digital documents.
Review
questions
N
LY
The Maths Quest Review is available in a customisable format
for students to demonstrate their knowledge of this topic.
O
ONLINE ONLY
Language
solution
solve
SA
M
PL
int-3173
inverse operation
operations
output number
pronumeral
simplify
E
EV
int-2606
backtracking
equation
flowchart
input number
inverse
AL
U
int-2605
Link to assessON for
questions to test your
readiness FOr learning,
your progress aS you learn and your
levels OF achievement.
Link to SpyClass, an exciting
online game combining a
comic book–style story with
problem-based learning in
an immersive environment.
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.
Join Jesse, Toby and Dan and help
them to tackle some of the world’s
most dangerous criminals by using the
knowledge you’ve gained through
your study of mathematics.
www.assesson.com.au
www.spyclass.com.au
Topic 13 • Linear equations 505
c13LinearEquations.indd 505
18/05/16 12:20 PM
InVeSTIgaTIOn
number and algebra
<InVeSTIgaTIOn>
FOr rICH TaSK Or <number and algebra> FOr PuZZle
rICH TaSK
SA
M
PL
E
EV
AL
U
AT
IO
N
O
N
LY
Equations at the Olympic games
506
Maths Quest 7
c13LinearEquations.indd 506
18/05/16 12:20 PM
number
number and
and algebra
algebra
1 Use the equations provided to calculate the men’s and women’s running times for the 100‐metre
event in the year:
• 1928
• 1968
• 1988
• 2008.
AT
IO
2 How well do these actual results compare with the times
N
O
N
LY
The table below shows the winners and their running times for the 100-metre event final at
four Olympic games.
you calculated using the equations? What does this
say about the equations?
3 Predict the times for both men and women at the 2016
Olympic games.
AL
U
When making predictions about the future, we must
remember that these predictions are based on the
assumption that the patterns we observe now will
continue into the future.
SA
M
PL
E
EV
Your answers in question 1 show you that men run
faster times than women in the 100-metre event. However, closer inspection of the times shows that
women are making greater improvements in their times over the years.
4 If the running times continue to follow these patterns in future Olympic games, is it possible that women’s
times will become equal to men’s times? Discuss your answer.
5 Why do you think the running times are coming down? Will they follow this pattern forever?
Topic 13 • Linear equations 507
c13LinearEquations.indd 507
18/05/16 12:20 PM
<InVeSTIgaTIOn>
number
and algebra
FOr rICH TaSK Or <number and algebra> FOr PuZZle
COde PuZZle
Solve the equations below to find the puzzle’s answer code.
What tools do we use in arithmetic?
17
19
12
19
8
6
G – 12 = 3
G=
12 + B = 15 B =
H–7=0
H=
27 = C + 9
C=
16 = I – 3
I=
5=4+D
D=
11 = K – 2
K=
E + 38 = 46 E =
L – 13 = 4
L=
10
AT
IO
A=
17
AL
U
A + 7 = 18
16
O
14
N
9
N
LY
Joke time
Why was Christopher Columbus a crook?
1
5
14
8
1
16
7
8
M=
17
11
16
M
PL
3M = 27
3
N=
12O = 60
O=
7P = 84
P=
9R = 54
R=
SA
5N = 20
8
18
6
5
10
10
17
11
4
16
19
18
EV
8
E
7
S÷5=2
S=
T
—=4
4
U
—= 7
2
T=
U=
W÷4=5
W=
Y
—= 1
2
Y=
What walks on its head all day?
508
11
1
6
11
20
19
4
15
12
19
4
10
16
14
18
13
19
4
2
5
14
6
10
7
5
8
Maths Quest 7
c13LinearEquations.indd 508
18/05/16 12:20 PM
NUMBER AND ALGEBRA
Activities
N
LY
O
13.7 Keeping equations balanced
Interactivities
• Balancing equations (int‐0077)
• IP interactivity 13.7 (int-4377) Keeping
equations balanced
13.8 Review
Interactivities
• Word search (int‐2605)
• Crossword (int‐2606)
• Sudoku (int‐3173)
Digital docs
• Topic summary (doc‐10740)
• Concept map (doc‐10741)
AL
U
13.4 Building up expressions
eLesson
• Backtracking (eles‐0010)
Interactivity
• IP interactivity 13.4 (int-4374) Building up expressions
13.6 Checking solutions
Digital docs
• SkillSHEET (doc‐6576) Checking solutions
by substitution
• WorkSHEET 13.2 (doc‐1947)
Interactivity
• IP interactivity 13.6 (int-4376) Checking
solutions
N
13.3 Using inverse operations
Digital docs
• SkillSHEET (doc‐6574) Inverse operations
• Spreadsheet (doc‐1967) Backtracking
Interactivity
• IP interactivity 13.3 (int-4373) Using inverse operations
Interactivity
• IP interactivity 13.5 (int-4375) Solving
equations using backtracking
AT
IO
13.2 Solving equations using trial and error
Digital docs
• SkillSHEET (doc‐6571) Completing number sentences
• SkillSHEET (doc‐6572) Writing number sentences
from written information
• SkillSHEET (doc‐6573) Applying the four operations
• Spreadsheet (doc‐1966) Solving equations
Interactivity
• IP interactivity 13.2 (int-4372) Solving
equations using trial and error
EV
13.5 Solving equations using backtracking
Digital docs
• SkillSHEET (doc‐6575) Combining like terms
• WorkSHEET 13.1 (doc‐1946)
www.jacplus.com.au
SA
M
PL
E
To access eBookPLUS activities, log on to
Topic 13 • Linear equations 509
c13LinearEquations.indd 509
18/05/16 1:36 PM
number and algebra
Answers
TOPIC 13 Linear equations
b
f
j
n
y=9
m = 274
w = 18
c = 4.2
5
2
8
3
13
4
20
+1
ii
5
b i ×3
6
+5
15
18
+ 10
3
13
EV
5
÷2
10
13
10
30
+6
+6
+3
ii
10
7
35
15
2 a 1
c 15
e 5
g 4
i 5
k 14.4522
m 3
o 2.59
38
× 10
10
× 10
15
50
+3
–5
ii
6
×5
×5
h i 2
0
7
17
÷3
÷3
g i 7
6
0
1
÷3
21
0
5
–9
–9
ii
9
E
8
÷3
f i 1
16
+6
+6
ii
1
15
9
2
30
4
5
4
e i 0
+1
–5
14
4
4x + 1
×3
4
÷2
d i ii
x +4
13.3 Using inverse operations
2
7
d m = 30
h k = 0
l h = 25
×2
×2
ii
x
x = 1 and x = 3 are both solutions.
10
24 years old
5 metres, 15 metres
7
Angus will be 27 and his father will be 81.
6 and 10
1 a i 7
b 9 and 18
d 8 and 37
f 51 and 53
h 97 and 145
j 2.9 and 5.1
l
l 11.9 and 23.1
2
–5
5
N
LY
x = 14
x=9
x = 20 or x = 0
x=6
t
SA
8
9
10
11
12
13
b
d
f
h
r
M
PL
7
c
g
k
o
p
c i 25
O
=2
q 15 − x = 2
21
s
=7
x
2 a x = 11
e w = 1
i b = 7
m b = 8.8
3 x = 6
4 a x = 3
c x = 10
e x = 3
g x = 5 or x = 0
i x = 4
5 7 and 14
6 a 11 and 15
c 7 and 47
e 42 and 136
g 42 and 111
i 2.2 and 3.9
k 347 and 631
o
15
÷5
AT
IO
7
x
5
+ 10
ii
x+3=5
x + 5 = 56
x − 11 = 11
x−8=0
6x = 30
6x = 12
x
= 100
3
x
=0
7
52 − x = 8
x2 = 100
m=5
w=7
k=4
x = 2.8
b
d
f
h
j
l
n
AL
U
1 a x + 7 = 11
c x + 12 = 12
e x − 7 = 1
g x − 4 = 7
i 2x = 12
k 5x = 30
x
m = 1
N
13.2 Solving equations using trial and error
100
–5
150
145
b
d
f
h
j
l
n
p
7
14
10
9
0
4
17
532
510 Maths Quest 7
c13LinearEquations.indd 510
18/05/16 12:20 PM
number and algebra
d
x+3
e
+1
+7
+ 11
1
×–
2
M
PL
÷8
x
–2
SA
× 3.1
2 a +5
x
+ 1.8
1
––
2
g
+ 10
b
x
x + 10
÷3
x + 10
3
+3
–3
÷5
+5
x–3
—
5
×6
x+5
x
×3
+9
5
5
×6
+4
3
7
x–8
x–8
—
—
+9
x–8
–x
3x + 4
—
3x + 4
÷5
÷3
6(x + 5) – 2
÷7
+4
–8
x
–2
6(x + 5)
3x
x
x
x
–+3
7
x–3
x
j
4x + 1
x
–
7
x
4(x + 5)
12
+1
÷7
i
x–8
—
4x
x
g
3x – 2
÷ 12
×4
f
–2
x–8
x
e
6(x + 2)
3x
–8
1.8
×6
x+2
x
d
x – 3.1
—
x – 3.1
×3
h
×4
x+5
÷ 1.8
+2
c
1
7(x – – )
2
1
x ––
2
– 3.1
h
x + 4.9
5
×7
x
3.1x + 1.8
3.1x
x
÷5
x + 4.9
x
x
–+ 9
7
x
–
7
x
h
+9
3(x – 2.1)
+ 4.9
f
x
––2
8
x
–
8
÷7
g
x
––3
2
x
–
2
E
–3
x
f
x
– + 11
5
x
–
5
x
EV
÷5
e
2x + 7
2x
x
d
AL
U
×2
c
×3
x – 2.1
x
b
x
–+ 1
3
x
–
3
x
9(x + 3)
– 2.1
x
÷3
b
×9
x
3 a 5x – 2
5x
x
+3
x
–2
x–2
7
x–2
AT
IO
×5
÷7
x
13.4 Building up expressions
1 a –2
N
LY
c
O
reverse order.
b The results are quite different.
4 a Adding and subtracting are inverse operations.
b Multiplying and dividing are inverse operations.
5 a x + 3 = 7
x+3−3=7−3
x+0=4
x=4
Addition and subtraction are inverse operations because
+ a − a = 0 and −a + a = 0; they undo each other.
b 2x ÷ 2 = −8 ÷ 2
x = −4
Multiplication and division are inverse operations because
1
a ÷ a = 1 and × a = 1; they undo each other.
a
6 a B
oth i and ii are correct. i should be used because the unknown
x is isolated.
b Both i and ii are correct. ii should be used because the
unknown x is isolated.
7 3(x + 5) = 27; 4
8 200
N
3 a E
ach pair of flowcharts has the same operations, but in the
–x + 4
3
x
6(– + 4)
3
Topic 13 • Linear equations 511
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number and algebra
×3
×5
4
5x + 9
6 a ×2
x+1
÷6
2(x + 1)
+4
12(x – 7)
e
– 12
7x – 12
7x
×3
3(x + 7)
+7
×3
3x + 7
SA
÷5
–x
x
5 a 5
×7
b
3x + 7
—
i
4(3x + 1)
x
3(– + 6)
5
×3
×4
4(x + 8)
÷4
×8
–3
8x – 3
8x
+3
÷5
x–7
—
—–
x–7
5
×5
×3
x + 55
3(x + 55)
÷3
+ 2.1
x
–
9
x
– + 2.1
9
×9
– 2.1
×7
–5
7x – 5
7x
÷7
+8
j
+5
×4
x+8
–8
3
+5
4(x + 8)
÷4
4(x + 8) + 5
–5
5(x + 8)
= 30
4
8 5n + 15 = 4n − 3
9 22 years old
10 $1215
11 The two numbers are 19 and 13.
7
5(x – 2)
÷5
x+8
—
——
–8
x
–3
×5
+2
+3
÷3
x+8
x
–x + 6
x
– –3
6
x+8
x
2
7x + 3
x–2
x
x
–
6
– 55
÷9
h
+3
÷7
–2
–3
+7
+ 55
×3
5
7x
x
÷2
÷6
x
3(x + 7) – 5
3x + 1
+6
2(x + 7)
–7
x
×4
+1
3x
x
x+7
÷8
–7
g
÷2
3x
x
k
–5
M
PL
×3
j
3
x+7
x
EV
+7
i
x+6
—
x+6
x
l
×3
E
+6
×2
x
f
h
+7
x
x
––2
5
x
–
5
x
×4
–8
+8
d
–2
×7
+5
x
7
x–7
x
g
x–8
—
× 12
÷5
f
4
AT
IO
–7
x
x–5
—
–—
x–5
×6
+8
c
AL
U
e
÷7
x–8
x
÷4
x
x
–+ 4
6
–8
d
–1
–5
x
x
–
6
x
×2
x
b
c
x
–+1
2
+9
+1
x
d
3x
+2
—
4
5x
x
b
3x
—
x
–
2
x
+2
3x
x
4 a ÷4
+1
N
LY
l
4(11x – 2)
11x – 2
11x
÷2
c
N
x
×4
–2
O
× 11
k
512 Maths Quest 7
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number and algebra
Challenge 13.1
13.6 Checking solutions
Teacher to check. Some examples are:
12 + 3 − 4 + 5 + 67 + 8 + 9 = 100
123 + 4 − 5 + 67 − 89 = 100
123 − 45 − 67 + 89 = 100.
1 a No
e No
i Yes
2 a Yes
e Yes
i Yes
13.5 Solving equations using backtracking
3 a x = 1
d x = 4
g w = 1.3
b
e
h
4 a x = 3
d x = 34
g x = 13
b
e
h
5 a x = 14
d x = 1
g x = 4.65
b
e
h
6 a x = 10
c x = 48
e x = 6
g x = 7
i x = 11.13
21.053
y=1
w=3
b = 11
54
x = 18
x=3
b = 23
x = 45
x = 77
x = 28
11
x = 29
x=5
x = 27
7
o
c
f
i
c
f
i
c
f
i
c
f
i
b
d
f
h
j
4 25
m=6
m = 25
2
a = 15
x = 23
x = 12
7
x = 20
x = 12
x=0
x = 23
x = 78
x = 291
x = 12
x=3
x = 50
x = 21
x=6
1
x = 10
x=1
x=8
p
3
31
36
0
1
2
3
4
2x + 3
3
5
7
9
11
b x = 4
2
3
4
5
6
5(x − 2)
0
5
10
15
20
a x = 4
b x = 6
8
1
6
3
5
7
4
9
22 11, 13, 15
23 36
2
AL
U
c x = 8
2x + 1
3x − 5
3
7
4
4
9
7
5
11
10
6
13
13
7
15
16
x
x+3
2
2x − 6
3
3
0
5
4
4
7
5
8
9
6
12
11
7
16
x
5 a b x = 6
6 a EV
E
M
PL
SA
21
c x = 1
x
4
The forward order is:
First subtract 6, then multiply by 3 and finally add 5.
The reverse order is:
Subtract 5, divide by 3, then add 6.
11 Gareth is 13 years old.
x + 28
12 a 12
b
= 12
3
c x = 8 d Karina scored 8 points in the third game.
13 Callie scored 96 pins.
14 16, 17 and 18
15 7, 9 and 11
16 25, 30 and 35
17 David is 14 years old.
18 Chris jumped 156 centimetres.
19 The twins are 11 years old.
20 a Puzzle 1 [(n − 500) × 2 + 1000] ÷2 = n
Puzzle 2 [(n + 4) × 10 − 10] ÷5 − n − 6 = n
Puzzle 3 (n ÷ 2 + 2) × 4 − n − n = 8
b Ask your teacher.
No
No
No
Yes
Yes
x
a x = 4
k x = 25
l
m x = 6
n
o x = 2
7 a x = 3
b x = 5
c x = 3
d x = 4
e x = 7
f x = 4
2
g x = 7
h x =
i x = 1
9
8 A
9 First subtract 2 from both sides, then divide both side by 4; x = 2.
10 Inverse operations must be performed in the reverse order.
d
h
l
d
h
N
LY
n
b
e
h
d 13
h 88
l 5.3
No
No
Yes
No
No
O
m 10
2 a x = 3
d x = 5
g w = 5.7
c 8
g 67
k 8
c
g
k
c
g
N
b 4
f 7
j 5
Yes
Yes
No
No
No
No
AT
IO
1 a 2
e 10
i 1
b
f
j
b
f
j
b x = 5
7 a No. Length is enough.
b 34.8 m
c Adding 3.9 and the value of y. The answer should be 38.7 m.
1
7
b x = 3.5
9 James $53 and Alison $72
10 63
8 a x = 0.5
13.7 Keeping equations balanced
1 a i 4r = 12
b i 2n + 3 = 7
c i 3t + 5 = 8
d i 4p + 1 = 13
2 a g = 1
b m = 3
d n = 9
e g = 14
1
1
3 a h = 2
b k = 2
d t = 4
3
1
3
4 a m = 4
d s = 3
1
5 a x = 2
2
e h = 2
b d = 3
e g = 5
b y = 1
2
1
6
r=3
n=2
t=1
p=3
c q = 9
f y = 3
1
c w = 3
2
ii
ii
ii
ii
f l = 1
1
3
c v = 2
f k = 2
1
c p = 3
4
d h = 2
1
2
Topic 13 • Linear equations 513
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number and algebra
9
7
8
9
10
11
12
a $11.30
b 113 °F
2 All calculated times are well within 0.4 seconds of the actual
b 58 km
3 Using the formula: Men’s time = 9.817 s;
running times.
11 songs
16 boys
3
3, 8, 13
4 cm
Women’s time = 10.502 s.
4 Women’s times are coming down faster than men’s. It is possible
only if these patterns continue.
5 Different training programs, fitter athletes, better shoes, etc.
There must be some levelling out, as the times cannot keep
coming down forever; it will be impossible to run 100 metres in
0 seconds.
Challenge 13.2
1 13 kilograms
Investigation — Rich task
1 Men’s: 1928, 10.607 s; 1968, 10.231 s; 1988, 10.043 s; 2008,
Multipliers. He double crossed the Atlantic. A drawing pin stuck in
your shoe
SA
M
PL
E
EV
AL
U
AT
IO
N
O
9.855 s
Women’s: 1928, 11.956 s; 1968, 11.264 s; 1988, 10.918 s; 2008,
10.572 s
Code puzzle
N
LY
6 a 7 2 °C
514 Maths Quest 7
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N
LY
O
N
AT
IO
AL
U
EV
E
M
PL
SA
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