nuMber AnD AlgebrA • Linear and non-Linear reLationships 11 … · 2014-06-18 · Chapter 11 linear equations 293 nuMber AnD AlgebrA • Linear and non-Linear reLationships Combining

11A Identifying patterns 11B Backtracking and inverse operations 11C Keeping equations balanced 11d Using algebra to solve problems 11E Equations with the unknown on

both sides

WhAT Do you knoW?

1 List what you know about equations. Create a concept map to show your list.

2 Share what you know with a partner and then with a small group.

3 As a class, create a large concept map that shows your class’s knowledge equations.

opening QuesTion

A plane uses an average of 40 L of fuel an hour. Write an equation that will enable a pilot to input the speed and output the number of hours of fl ight.

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11nuMber AnD AlgebrA • Linear and non-Linear reLationships

linear equations

ContentsLinear equations 291Are you ready? 292Identifying patterns 294Identifying patterns 297Backtracking and inverse operations 298Backtracking and inverse operations 300Keeping equations balanced 301Keeping equations balanced 302Using algebra to solve problems 303Using algebra to solve problems 307Equations with the unknown on both sides 309Equations with the unknown on both sides 312Summary 314Chapter review 315

Activities 318

292 Maths Quest 8 for the Australian Curriculum

nuMber AnD AlgebrA • Linear and non-Linear reLationships

Are you ready?Try the questions below. If you have diffi culty with any of them, extra help can be obtained by completing the matching SkillSHEET located on your eBookPLUS.

Number patterns 1 For each of the following sequences of numbers, describe the pattern in words and then write

down the next three numbers in the pattern.a 7, 9, 11, 13, … b 28, 24, 20, 16, …c 3, 6, 12, 24, … d 100 000, 10 000, 1000, …

Using tables to show number patterns 2 a Complete the table shown, using

the diagrams at right as a guide.

Number of squares 1 2 3 4 5 6

Number of sides 4 8

b How many sides are there for 10 squares?

Describing a number pattern from a table 3 Describe each number pattern shown in the tables.

a First number 1 2 3 4 5 b First number 1 3 5 7 9

Second number 7 8 9 10 11 Second number 3 9 15 21 27

Flowcharts 4 Complete these fl owcharts to fi nd the output number.

a -9ì5

4

b ó4+3

5

Inverse operations 5 Write the inverse operation for each of the following.

a ì 2 b + 8 c - 17 d ó -5

Solving equations by backtracking 6 Solve each of the following equations by fi rst completing the fl owcharts below. Remember to

show the operations needed to backtrack to x.

a 7(x - 4) = 35 b x + =95

3

x

35

x - 4 7(x - 4)

x x + 9 x + 9

3

5

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1 square 2 squares 3 squares

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12 16 20 24

40

Each fi rst number is one more than the previous fi rst number and each second number is one more than the previous second number. Each second number is always 6 more than the matching fi rst number.

11 2

ó 2 - 8 + 17 ì -5

- 4 ì 7

+ 4 ó 7

9

x

5 35

x - 4 7(x - 4)

Solution is x = 9.

+ 9 ó 5

- 9 ì 5

6

x

15 3

5x + 9 x + 9

Solution is x = 6.

1 a Add 2 to obtain the next number in the sequence. The next three numbers are 15, 17, 19.

b Subtract 4 to obtain the next number in the sequence. The next three numbers are 12, 8, 4.

Each fi rst number is two more than the previous fi rst number and each second number is six more than the previous second number. Each second number is three times the matching fi rst number.

293Chapter 11 linear equations


Combining like terms 7 Simplify each of the following expressions by combining like terms.

a 7v + 3 + 3v + 4 b 6c + 15 - 5c - 8

Expanding expressions containing brackets 8 Expand each of the following expressions containing brackets.

a 2(3x + 5) b -7(m - 1)

Checking solutions by substitution 9 For each equation below there is a solution given. Is the solution correct?

a 5x - 7 = 2x + 2 x = 3 b x

x+ = −92

2 7 x = 5

Writing equations from worded statements 10 Write an equation for each of the following statements, using x to represent the unknown

number.a When 2 is added to a certain number, the result is 9.b Eight times a certain number is 40.c When 11 is subtracted from a certain number, the result is 3.d Dividing a certain number by 6 gives a result of 2.

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10v + 7 c + 7

6x + 10 -7m + 7

Yes No

x + 2 = 9 8x = 40

x - 11 = 3

x6

= 2



Identifying patterns ■ Mathematics is used to describe relationships in the world around us. ■ Mathematicians study patterns in numbers and shapes found in nature to discover rules. ■ These rules can then be applied to other, more general situations. ■ Looking at the number pattern 1, 4, 7, 10, . . . we can see that by adding 3 to any of these numbers, we obtain the next number.

■ This number pattern is called a sequence. ■ Each number in the sequence is called a term. ■ Each sequence has a rule that describes the pattern. For the sequence above, the rule is ‘add 3’.

WorkeD exAMple 1

Describe the following number patterns in words then write down the next three numbers in the pattern.a 4, 8, 12, . . . b 4, 8, 16, . . .

Think WriTe

a 1 The next number is found by adding 4 to the previous number.

a Next number = previous number + 4

2 Add 4 each time to get the next three numbers.

3 Write down the next three numbers. Next three numbers are 16, 20, 24.

b 1 The next number is found by multiplying the previous number by 2.

b Next number = previous number ì 2

2 Multiply by 2 each time to get the next three numbers.

3 Write the next three numbers. Next three numbers are 32, 64, 128.

WorkeD exAMple 2

Using the following rules, write down the first five terms of the number pattern.a Start with 32 and divide by 2 each time.b Start with 2, multiply by 4 and subtract 3 each time.

Think WriTe

a 1 Start with 32 and divide by 2. a 32 ó 2 = 16

2 Keep dividing the previous answer by 2 until five numbers have been calculated.

16 ó 2 = 88 ó 2 = 44 ó 2 = 22 ó 2 = 1

3 Write the answer. The first five numbers are 16, 8, 4, 2 and 1.

b 1 Start with 2 then multiply by 4 and subtract 3. b 2 ì 4 - 3 = 55 ì 4 - 3 = 17

2 Continue to apply this rule to the answer until five numbers have been calculated.

17 ì 4 - 3 = 6565 ì 4 - 3 = 257

257 ì 4 - 3 = 1025

3 Write the five numbers. The first five numbers are 5, 17, 65, 257, 1025.

11A



WorkeD exAMple 3

Describe the pattern that occurs in the final digit of the number set represented by:71, 72, 73, …

Think WriTe

1 Calculate a few of these powers. 71 = 7

2 Continue until a pattern in the last digit is noticed. When the number becomes large, be concerned with only the last digit.

72 = 4973 = 49 ì 7 = 34374 = 343 ì 7 = 240175 = 2401 ì 7 = 16 80776 = . . . 7 ì 7 = . . . 9

3 Write the pattern. The pattern in the last digit is 7, 9, 3, 1 repeated.

geometric patterns ■ Patterns can be found in geometric shapes. ■ If we examine the three shapes below, we can see patterns by investigating the changes from one shape to the next. For example, look at the number of matchsticks in each set of triangles.

By using a table of values we can see a number pattern developing:

Number of triangles Number of matchsticks

1 3

2 6

3 9

4 12

5 15

6 18

The pattern in the bottom row is 3, 6, 9, . . .; we can see that the rule here is ‘add 3’.We can also look for a relationship between the number of triangles and the number of

matchsticks in each shape. If you examine the table, you will see that a relationship can be found. In words, the relationship is ‘the number of matchsticks equals 3 times the number of triangles’.

WorkeD exAMple 4

Consider a set of hexagons constructed according to the pattern shown below.

a Using matches, pencils or similar objects, construct the above figures. Draw the next two figures in the series.

b Draw up a table showing the relationship between the number of hexagons in the figure and the number of matches used to construct the figure.



c Devise a rule to describe the number of matches required for each figure in terms of the number of hexagons in the figure.

d Use your rule to determine the number of matches required to make a figure consisting of20 hexagons.

Think WriTe

a 1 Construct the given figures with matches. Note the number of additional matches it takes to progress from one figure to the next — 5 in this case.

a The next two figures are

2 Draw the next two figures, adding another 5 matches each time.

b Draw up a table showing the number of matches needed for each figure in terms of the number of hexagons. Fill it in by looking at the figures.

bNumber of hexagons 1 2 3 4 5

Number of matches 6 11 16 21 26

c 1 Look at the pattern in the number of matches going from one figure to the next. It is increasing by 5 each time.

c The number of matches increased by 5 in going from one figure to the next.

2 If we take the number of hexagons and multiply this number by 5, it does not give us the number of matches. However, if we add 1 to this number, it does give us the number of matches in each shape.

Number of matches = number of hexagons ì 5 + 1

d 1 Use the rule to find the number of matches to make a figure with 20 hexagons.

d Number of matches for 20 hexagons = 20 ì 5 + 1 = 101

2 Work out the answer and write it down.

So 101 matches would be required to construct a figure consisting of 20 hexagons.

reMeMber

1. A number pattern has a rule that can describe the pattern.2. Geometric patterns have a rule that can describe the pattern.3. The rule can be used to make predictions.



Identifying patternsfluenCy

1 We1 Copy the patterns below, describe the pattern in words and then write down the next three numbers in the pattern.a 2, 4, 6, 8, … b 3, 8, 13, 18, … c 27, 24, 21, 18, …d 1, 3, 9, 27, … e 128, 64, 32, 16, … f 1, 4, 9, 16, …

2 Fill in the missing numbers in the following number patterns.a 3, …, 9, 12, …, … b 8, …, …, 14, … c 4, 8, …, 32, …d …, …, 13, 15, … e 66, 77, …, 99, … 121 f 100, …, …, 85, 80, …

3 We2 Using the following rules, write down the fi rst fi ve terms of the number patterns.a Start with 1 and add 4 each time.b Start with 5 and multiply by 3 each time.c Start with 50 and take away 8 each time.d Start with 64 and divide by 2 each time.e Start with 1, multiply by 2 and add 2 each time.f Start with 1, add 2 then multiply by 2 each time.

unDersTAnDing

4 Each of the following represents a special number set. What is common to the numbers in the set?a 2, 4, 6, 8, 10 b 1, 8, 27, 64 c 2, 3, 5, 7, 11d 1, 1, 2, 3, 5, 8 e 1, 2, 3, 4, 6, 12 f 3, 9, 27, 81

5 We3 Investigate the pattern that occurs in the fi nal digit of the following sets. Describe the pattern in each case.a 21, 22, 23, … b 31, 32, 33, … c 41, 42, 43, …d 51, 52, 53, … e 81, 82, 83, …

6 We4 For each of the sets of shapes below: i Construct the shapes using matches. Draw the next two shapes in the series. ii Construct a table to show the relationship between the number of shapes in each fi gure

and the number of matchsticks used to construct it.iii Devise a rule in words that describes the pattern relating the number of shapes in each

fi gure and the number of matchsticks used to construct it.iv Use your rule to work out the number of matchsticks required to construct a fi gure made

up of 20 such shapes. a

b

c

d

exerCise

11A

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Activity 11-A-1Identifying patterns

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Activity 11-A-2More patterns

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Activity 11-A-3Advanced patterns

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1 a Add 2; 10, 12, 14 b Add 5; 23, 28, 33 c Subtract 3; 15, 12, 9 d Multiply by 3; 81, 243, 729 e Divide by 2; 8, 4, 2 f Squares of numbers 1; 25, 36, 49

1, 5, 9, 13, 17 5, 15, 45, 135, 405

50, 42, 34, 26, 18 64, 32, 16, 8, 4

1, 4, 10, 22, 46 1, 6, 16, 36, 76

2 a 6, 15, 18 b 10, 12, 16 c 16, 64 d 9, 11, 17 e 88, 110 f 95, 90, 75 4 a Even numbers/

multiples of 2 b Cubes c Prime numbers d Fibonacci

sequence e Factors of 12 f Multiples/

powers of 3

6

a i

i

iN

umbe

r of

squ

ares

12

3 4

5

Num

ber

of m

atch

es4

710

1316

iii N

umbe

r of

mat

ches

= N

umbe

r of

squ

ares

ì 3

+ 1

iv 6

1

2, 4, 6, 8 3, 9, 7, 1 4, 6

5 8, 4, 2, 6

6 b i

iii Number of matches = Number of triangles ì 2 + 1 iv 41

c i

iii Number of matches = Number of houses ì 5 + 1 iv 101

d i

iii Number of matches = Number of fence units ì 3 + 1

iv 61



reAsoning

7 Consider the triangular pattern of even numbers shown below.

2

4 6

8 10 12

a Complete the next three lines of the triangle using this pattern.b Complete the triangle as far as necessary to find the position of the number 60.c Explain how, without completing any more of the triangle, you could find the position of

the number 100.d Study the triangle you have created in part 2 and write down as many patterns as you can

find. Illustrate each pattern with numbers from the triangle.e Create a similar triangle using odd numbers.

Look for the patterns in this triangle. Are they the same as or different from those for the triangle of even numbers? Justify your answers by illustrations from the triangle.

backtracking and inverse operations ■ An equation links two expressions with an equals sign. ■ Adding and subtracting are inverse operations. ■ Multiplying and dividing are inverse operations. ■ A flowchart can be used to represent a series of operations. ■ In a flowchart, the starting number is called the input number and the final number is called the output number.

■ By using inverse operations, it is possible to reverse the flowchart and work from the output number to the input number.

WorkeD exAMple 5

Find the input number for this flowchart.

- 7 ó -2 + 3

7

Think WriTe

1 Copy the flowchart. - 7 ó -2 + 3

7

2 Backtrack to find the input number.The inverse operation of +3 is -3(7 - 3 = 4).The inverse operation of + -2 is ì -2(4 ì -2 = -8).The inverse operation of -7 is +7 (-8 + 7 = -1).Fill in the missing numbers.

- 7 ó -2 + 3

+ 7 ì -2 - 3

7-1 -8 4

refleCTion

What should you look for when trying to determine number patterns?

11b

Check with your teacher.

Check with your teacher.

Che

ck w

ith y

our

teac

her.

24 6

8 10 1214 16 18 20

22 24 26 28 3032 34 36 38 40 42

44 46 48 50 52 54 5658 60

7 a, b



WorkeD exAMple 6

Find the output expression for this flowchart.

ì 3 + 2 ó 4

x

Think WriTe

1 Copy the flowchart and look at the operations that have been performed.

ì 3 + 2 ó 4

x

2 Multiplying x by 3 gives 3x. ì 3 + 2 ó 4

x 3x

3 Adding 2 gives 3x + 2. ì 3 + 2 ó 4

x 3x 3x + 2

4 Now place a line beneath all of 3x + 2 and divide by 4.

ì 3 + 2 ó 4

4x 3x 3x + 2 3x + 2

WorkeD exAMple 7

Starting with x, draw the flowchart whose output number is given by the expressions:a 6 - 2xb -2(x + 6).

Think WriTe

a 1 Rearrange the expression.Note: 6 - 2x is the same as -2x + 6.

a ì -2 ó 6

x -2x -2x + 62 Multiply x by -2, and then add 6.

b 1 The expression x + 6 is grouped in a pair of brackets, so we must obtain this part first. Therefore, add 6 to x.

b ó 6 ì - 2

x x + 6 -2(x + 6)

2 Multiply the whole expression by –2.

reMeMber

1. To work backwards through a flowchart we use inverse operations.2. Adding and subtracting are inverse operations.3. Multiplying and dividing are inverse operations.



backtracking and inverse operationsfluenCy

1 We5 Find the input number for each of the following fl owcharts.

a+ 6 ì 2

28b

ó 5 + 3

7

cì –3 + 2

14d

- 5 ó -4

6

eì -2 - 6 ì -3

12f

ó -8

-1

+ 5 ì 2

g+ 11 ó -3 - 2

-5h

ì -3

12

ó 4 + 7

i- 8 ó 6 ì -5

0j

- 5

-11

- 7 ì 2

k+ 0.5 ì 4 - 5.1

1.2l

ì 5

4

− 2 ó 3

2 We6 Find the output expression for each of the following fl owcharts.

aì 2 - 7

xb

ì 2− 7

w

cì -5 + 3

sd

ì -5+ 3

n

eó 2 + 7

mf

ó 2+ 7

y

gì 6 - 3 ó 2

zh

+ 5 ì -3 ó 4

d

iì 2 ó 5 + 1

ej

ì -1 + 3 ì 4

x

kì -2- 5 ó 7

wl

+ 6 ì -3 - 11

z

exerCise

11b

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Activity 11-B-1Sudoku challenge A

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Activity 11-B-2Sudoku challenge B

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Activity 11-B-3Sudoku challenge C

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8 20

-4 -19

-1 -1

-2 -44

8 4

1.075 4.4

2x - 7 2(w - 7)

-5s + 3 -5(n + 3)

m27+

y + 72

6 32

z − − +3 54( )d

251

e + 4(3 - x)

− −2 57( )w -3(z + 6) - 11



m- 8- 3 ó 6

v n- 4ì 8 ì -7

m

oì -5 + 2ó 6

kp

ì -5 - 7 ó 3

p

3 We7 Starting with x, draw the flowchart whose output number is:a 2(x + 7) b -2(x - 8) c 3m - 6

d -3m - 6 e x − 58

f x85−

g -5x + 11 h -x + 11 i -x - 13

j 5 - 2x k 3 74

x − l − −3 24( )x

m x + −58

3 n − −

752

x

o 3274

x +

p 146113

x −

keeping equations balanced ■ As an equation can be thought of as two expressions with an equals sign between them, an equation can be thought of as a balanced scale. The diagram at right represents the simple equation x = 3.

■ If the amount of the left-hand side (LHS) is doubled, the scale will stay balanced provided that the amount on the right-hand side (RHS) is doubled.

■ Similarly, the scale will stay balanced if we add a quantity to both sides.

■ The scales will remain balanced as long as we do the same to both sides.

refleCTion

What do you need to be careful of when you are backtracking equations?

11C

x 1 1 1

x = 3

xx1 1 1

1 1 1

2x = 6

xx1 1 1

11

1

1 1

1

1

2x + 2 = 8

-7(8m - 4)

− −5 73p

v − −36

8

− +56

2k

3 a + 7 ì 2

x x + 7 2(x + 7)

b x x - 8 -2(x - 8)

- 8 ì -2

c - 6ì 3

m 3m 3m - 6

d - 6 ì -3

m -3m -3m - 6

e 8x x - 5 x - 5

- 5 ó 8

––––

f - 5

- 58

ó 8

x x –8x –

g x -5x -5x + 11

ì -5 + 11

h x -x -x + 11

ì -1 + 11

i ì -1

x -x -x - 13

- 13

j ì -2

x -2x -2x + 5

+ 5

k –––––

ì 3

x 3x 3x - 7 3x - 7

- 7 ó 4

4

l ì -3

x x - 2 -3(x - 2) -3(x - 2)

- 2 ó 4

4–––––––

m ––––

+ 5

x x + 5x + 5

- 3ó 8

- 38 ––––x + 58

n –

ó 5

x

ì -7

-7

- 2

- 2x5

–x5

- 2–x5( )



WorkeD exAMple 8

Starting with the equation x = 4, write the new equation when we: a multiply both sides by 4 b take 6 from both sides c divide both sides by 2

5.

Think WriTe

a 1 Write the equation. a x = 42 Multiply both sides by 4. x ì 4 = 4 ì 4

3 Simplify by removing the multiplication signs. Write numbers before variables.

4x = 16

b 1 Write the equation. b x = 4

2 Subtract 6 from both sides. x - 6 = 4 - 6

3 Simplify. x - 6 = -2

c 1 Write the equation. c x = 4

2 Dividing by a fraction is the same as multiplying by its reciprocal.Multiply both sides by 5

2.

x ó 25 = 4 ó 2

5

x ì 52 = 4 ì 5

2

3 Simplify. 52x

= 202

52x

= 10

reMeMber

1. An equation links two expressions with an equals sign.2. An equation is like a pair of balanced scales (or a seesaw). The scales (or seesaw) will

remain balanced as long as we do the same to both sides.

keeping equations balancedfluenCy

1 We8 Starting with the equation x = 6, write the new equation when we:a add 5 to both sides b multiply both sides by 7c take 4 from both sides d divide both sides by 3e multiply both sides by -4 f multiply both sides by -1g divide both sides by -1 h take 9 from both sidesi multiply both sides by 2

3j divide both sides by 2

3

k take 23 from both sides.

unDersTAnDing

2 a Write the equation that is represented by the diagram at right.

b Show what happens when you halve the amount on both sides. Write the new equation.

exerCise

11C

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Activity 11-C-1Riddle A

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Activity 11-C-2Riddle B

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Activity 11-C-3Riddle C

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xx 1 1 1 1

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InteractivityBalancing equations

int-0077

x + 5 = 11 7x = 42 x - 4 = 2

x32=

-4x = -24 -x = -6 -x = -6 x - 9 = -3

234

x = 329

x =

x − =23513

2x = 4

x = 2




b Show what happens when you take three from both sides. Write the new equation.


b Show what happens when you add three to both sides. Write the new equation.


b Show what happens when you double the amount on each side. Write the new equation.

6 MC If we start with x = 5, which of these equations is not true?A x + 2 = 7 B 3x = 8 C -2x = -10

d x51= E x - 2 = 3

7 MC If we start with x = 3, which of these equations is not true?

A 23

2x

= B -2x = -6 C 2x - 6 = 0

d x535

= E x - 5 = 2

8 MC If we start with x = -6, which of these equations is not true?A -x = 6 B 2x = -12 C x - 6 = 0d x + 4 = -2 E x - 2 = -8

9 MC If we start with 2x = 12, which of these equations is not true?

A 23

4x

= B -2x = -12 C 2x - 6 = 2

d 4x = 24 E 2x + 5 = 17

using algebra to solve problems ■ A linear equation is an equation where the variable has an index (power) of 1. This means that it never contains terms like x2 or x .

■ To solve a linear equation, perform the same operations on both sides until the variable or unknown is left by itself.

■ Sometimes the variable or unknown is called a pronumeral. ■ A fl owchart is useful to show you the order of operations applied to x, so that the reverse order and inverse operation can be used to solve the equation. As you become confi dent with solving equations algebraically, you can leave out the fl owchart steps.

x1 1 1

11

11

1

xxx 1 1 1

11

1

1

1

1 1 1

1 1xx 1

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Digital docWorkSHEET 11.1

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refleCTion

How does using inverse operations keep the equations balanced?

11D

x + 3 = 5

x = 2

3x + 1 = 7

3x + 4 = 10

2x + 1 = 5

4x + 2 = 10

✔

✔

✔

✔



WorkeD exAMple 9

Solve these one-step equations by doing the same to both sides.

a p - 5 = 11 b x16

= -2

Think WriTe

a 1 Write the equation. a p - 5 = 11

2 Draw a flowchart and fill in the arrow to show what has been done to p.

p

- 5

p -5

3 Backtrack from 11.

p

- 5

16 11

p -5

+ 54 Add 5 to both sides. p - 5 + 5 = 11 + 5

5 Give the solution. p = 16

b 1 Write the equation. b x16

= -2

2 Draw a flowchart and fill in the arrow to show what has been done to x.

x

16

x16

3 Backtrack from -2.

x

ó 16

ì 16

x16

-32 -2

4 Multiply both sides by 16. x16

ì 16 = -2 ì 16

5 Give the solution. x = -32

■ The equations in Worked example 9 are called one-step equations because only one operation needs to be undone to obtain the value of the unknown.

WorkeD exAMple 10

Solve these two-step equations by doing the same to both sides.

a 2(x + 5) = 18 b x3

+ 1 = 7

Think WriTe

a 1 Write the equation. a 2(x + 5) = 18



2 Draw a flowchart and fill in the arrow to show what has been done to x.

ì 2+ 5

x x + 5 2(x + 5)

3 Backtrack from 18. ì 2

4 9 18

+ 5

- 5 ó 2

x x + 5 2(x + 5)

4 Divide both sides by 2.2 52

182

( )x + =

x + 5 = 9

5 Subtract 5 from both sides. x + 5 - 5 = 9 - 5

6 Give the solution. x = 4

b 1 Write the equation. b x31 7+ =

2 Draw a flowchart and fill in the arrows to show what has been done to x.

ó 3 + 1

+ 1x 3x–

3x–

3 Backtrack from 7.

ì 3

ó 3 + 1

+ 1

- 1

18

x

6 7

3x–

3x–

4 Subtract 1 from both sides.x

x31 1 7 1

36

+ − = −

=

5 Multiply both sides by 3.x33 6 3× = ×

6 Give the solution. x = 18

■ The equations in Worked example 10 are called two-step equations because two operations need to be undone to obtain the value of the unknown.

WorkeD exAMple 11

Solve the following equations by doing the same to both sides.

a 3(m - 4) + 8 = 5 b 62

5 18x +

= −

Think WriTe

a 1 Write the equation. a 3(m - 4) + 8 = 5



2 Draw a flowchart and fill in the arrows to show what has been done to m.

m m -4 3(m -4) 3(m - 4)+8

- 4 ì 3 + 8

3 Backtrack from 5.

m m - 4 3(m - 4) 3(m - 4)+8

- 4 + 8

+ 4 ó3 - 8

53 - 1 - 3

ì 3

4 Subtract 8 from both sides. 3(m - 4) + 8 - 8 = 5 - 83(m - 4) = -3

5 Divide both sides by 3.3 43

33

4 1

( )m

m

− = −

− = −6 Add 4 to both sides. m - 4 + 4 = -1 + 4

7 Give the solution. m = 3

b 1 Write the equation. b 625 18

x +

= −

2 Draw a flowchart and fill in the arrows to show what has been done to x.

x + 5 6( + 5)

ó 2 + 5 ì 6

x2

x2

x2

3 Backtrack from -18.

x + 5 6( + 5)

ó 2 + 5

- 5 ó 6

-18-16 - 8 -3

x2

x2

x2

ì 6

4 Divide both sides by 6.625

6186

25 3

x

x

+

= −

+ = −

5 Subtract 5 from both sides.x

x25 5 3 5

28

+ − = − −

= −

6 Multiply both sides by 2.x22 8 2× = − ×

7 Give the solution. x = -16



reMeMber

1. A linear equation is an equation where the variable has an index (power) of 1.2. When solving linear equations, perform the same operations on both sides of the

equation until the unknown is left by itself.3. You can draw a fl owchart to help you to decide what to do next.

using algebra to solve problemsfluenCy

1 We9a Solve these one-step equations by doing the same to both sides.a x + 8 = 7 b 12 + r = 7 c 31 = t + 7 d w + 4.2 = 6.9e 58 = m + 1

8f 27 = j + 3 g q - 8 = 11 h -16 + r = -7

i 21 = t - 11 j y - 5.7 = 8.8 k - 117

= z - 23

l - 913

= f - 1

2 We9b Solve these one-step equations by doing the same to both sides.a 11d = 88 b 7p = -98 c 5u = 4 d 2.5g = 12.5

e 8m = 14

f - 35 = 9j g

t8

= 3 h k5

= -12

i -5.3 = l4

j v6

= 23

k c9

= - 527

l - 712

= h5

3 We 10a Solve these two-step equations by doing the same to both sides.a 3m + 5 = 14 b -2w + 6 = 16 c -5k - 12 = 8 d 4t - 3 = -15e 2(m - 4) = -6 f -3(n + 12) = 18 g 5(k + 6) = -15 h -6(s + 11) = -24i 2m + 3 = 10 j 40 = -5(p + 6) k 5 - 3g = 14 l 11 - 4f = -9m 2q - 4.9 = 13.2 n 7.6 + 5r = -8.4 o 13.6 = 4t - 0.8 p -6k + 7.3 = 8.5

q -4g - 15 = 45

r - 38 = 2f - 18

8

4 We 10b Solve these two-step equations by doing the same to both sides.

a x32 9+ = b x − =5

41 c m + = −3

27 d h

−+ =31 5

e − − =m53 1 f 2

54

w = − g − = −37

1m h c − = −7

32

i − =54

10m j t + = −2

75 k c − = −21

94 5. l x

83 2 5 8− = −. .

5 We 11 Solve these equations by doing the same to both sides. They will need more than two steps.

a 2(m + 3) + 7 = 3 b − + =2 55

6( )x c 5 6

34

m + = d 3 27

1x − =

e 4 23

6− =x f − + = −x 3

24 g 3

72 1

x − = h 453 5

b − =

i 792 5

f + = − j 643

2− = −z k 8652− =m l − − = −9

511

4u

m 3 52

7m −−

= n -7(5w + 3) = 35 o 526 10

x −

= − p d − + =72

10 8

q 3 14

5 2n + − = r 3 5

79 6

( )t − + =

exerCise

11D

eBookpluseBookplus

Activity 11-D-1Using algebra to solve problems

doc-2345

Activity 11-D-2More problems

using algebradoc-2346

Activity 11-D-3Advanced problems

using algebradoc-2347

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Digital docSpreadsheet

2-step equationsdoc-2353

x = -1 r = -5 t = 24 w = 2.7

m = 12

f

j = –

19 7 o

r – 25 7

q = 19 r = 9

t = 32y = 14.5

z = - 1921

f = 413

d = 8 p = -14 u = 45 or 0.8 g = 5

m = 132

j = – 115

t = 24 k = -60

l = -21.2 v = 4c = - 5

3 or -12

3

h = - 3512

or -21112

x = 21 x = 9 m = -17 h = -12

m = -20 w = -10 m = 213

c = 1

m = -8 t = -37c = -19.5

x = -20.8

m = -5 x = -20m = 11

5x = 3

x = -7 x = 11 x = 7 b = 10

f = -9 z = 6 m = 5 u = -11

m = -3w = -13

5x = 8

d = 3

n = 9 t = -2

3 a m = 3 b w = -5 c k = -4 d t = -3 e m = 1 f n = -18 g k = -9 h s = -7 i m = 3.5 j p = -14 k g = -3 l f = 5 m q = 9.05 n r = -3.2 o t = 3.6 p k = -0.2 q g = - 1

4 or -0.25

r f = 1516



unDersTAnDing

6 Below is Alex’s working to solve the equation 2x + 3 = 14.

2x + 3 = 14

223142

x + =

x + 3 = 7

x + 3 - 3 = 7 - 3

x = 4a Is the solution correct?b If not, can you fi nd where the error is and correct it?

7 Simplify the left-hand side of the following equations by collecting like terms, and then solve.a 3x + 5 + 2x + 4 = 19 b 13v - 4v + 2v = -22c -3m + 6 - 5m + 1 = 15 d -3y + 7 + 4y - 2 = 9e -3y - 7y + 4 = 64 f 5t + 4 - 8t = 19g 5w + 3w - 7 + w = 13 h w + 7 + w - 15 + w + 1 = -5i 7 - 3u + 4 + 2u = 15 j 7c - 4 - 11 + 3c - 7c + 5 = 8

reAsoning

8 A repair person calculates his service fee using the equation F = 40t + 55, where F is the service fee in dollars and t is the number of hours spent on the job.a How long did a particular job take if the service fee was $155?b Explain what the numbers 40 and 55 could represent as costs in the service fee equation.

9 Lyn and Peta together raised $517 from their cake stalls at the school fete. If Lyn raised l dollars and Peta raised $286, write an equation that represents the situation and determine the amount Lyn raised.

10 a Write an equation that represents the perimeter of the fi gure at right and then solve for x.

b Write an equation that represents the perimeter of the fi gure at right and then solve for x.

11 Two positive integers have a difference of 5 and a sum of 13. Find the two numbers. 12 If four times a certain number equals nine minus a half of the number, fi nd the number. 13 Tom is 5 years old and his dad is 10 times his age, being 50 years old. Is it possible, at any

stage, for Tom’s dad to be twice the age of his son? Explain your answer. 14 Laurie earns the same amount for mowing the four of the neighbour’s lawns every month.

Each month he saves all his pay except $30, which he spends on his mobile phone. If he has $600 at the end of the year, how much did he earn each month? (Write an equation to solve for this situation fi rst.)

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3-step equationsdoc-2354

2x + 19

5x + 18

3x + 17

Perimeter = 184 cm

3x - 21

2x + 16

2x + 30

4x - 13

Perimeter = 287 cm

The solution is not correct.

x = 2 v = -2m = -1 y = 4

y = -6 t = -5w = 22

9w = 2

3u = -4 c = 6

212 hours

l + 286 = 517, l = $231

10x + 54 = 184, x = 13 cm

11x + 12 = 287, x = 25 cm

9 and 4

2

$80 per month

40 could be the charge per hour; 55 could be the fl at fee covering travel and other expenses.

Yes, when Tom is 45 and his dad is 90.

Alex should have also divided 3 by 2 in the second line or subtracted 3 from both sides fi rst before dividing both sides by 2. The solution should be x = 51

2.



15 The linear relationship between two variables x and y is displayed in this table.

x18

14

38

12

y712

56

1312

43

Write the linear relationship as an equation.

equations with the unknown on both sides

■ Some equations have unknowns on both sides of the equation. ■ If an equation has unknowns on both sides, eliminate the unknowns from one side and then solve as usual.

■ Consider the equation 4x + 1 = 2x + 5. • Drawing the equation on a pair of scales, looks

like this:

• The scales remain balanced if 2x is eliminated from both sides:

• Writing this algebraically, we have 4x + 1 = 2x + 5 4x + 1 - 2x = 2x + 5 - 2x 2x + 1 = 5 2x + 1 - 1 = 5 - 1 2x = 4

2242

x =

x = 2

WorkeD exAMple 12

Solve the equation 5t - 8 = 3t + 12 and check your solution.

Think WriTe

1 Write the equation. 5t - 8 = 3t + 12

2 Subtract the smaller unknown (that is, 3t) from both sides and simplify.

5t - 8 - 3t = 3t + 12 - 3t2t - 8 = 12

refleCTion

How do you decide on the order to undo the operations?

11e

4x + 1 = 2x + 5

xxxx 1

1 1 1

1 1xx

2x + 1 = 5

1xx 1 1 1

1 1

y = 2x + 13



3 Add 8 to both sides and simplify. 2t - 8 + 8 = 12 + 82t = 20

4 Divide both sides by 2 and simplify. 22202

t =

t = 10

5 Check the solution by substituting t = 10 into the left-hand side and then the right-hand side of the equation.

If t = 10, LHS = 5t - 8 = 50 - 8 = 42If t = 10, RHS = 3t + 12 = 30 + 12 = 42

6 Comment on the answers obtained. Since the LHS and RHS are equal, the equation is true when t = 10.

WorkeD exAMple 13

Solve the equation 3n + 11 = 6 - 2n and check your solution.

Think WriTe

1 Write the equation. 3n + 11 = 6 - 2n

2 The inverse of -2n is +2n.Therefore, add 2n to both sides and simplify.

3n + 11 + 2n = 6 - 2n + 2n5n + 11 = 6

3 Subtract 11 from both sides and simplify. 5n + 11 - 11 = 6 - 115n = -5


55

n = −

n = -1

5 Check the solution by substitutingn = -1 into the left-hand side and then the right-hand side of the equation.

If n = -1,LHS = 3n + 11

= -3 + 11= 8

If n = -1,RHS = 6 - 2n

= 6 - -2= 6 + 2= 8

6 Comment on the answers obtained. Since the LHS and RHS are equal, the equation is true when n = -1.

WorkeD exAMple 14

Expand the brackets and then solve the following equation, checking your solution.a 3(s + 2) = 2(s + 7) + 4b 4(d + 3) - 2(d + 7) + 4 = 5(d + 2) + 7



Think WriTe

a 1 Write the equation. a 3(s + 2) = 2(s + 7) + 42 Expand the brackets on each side of the

equation first and then simplify.3s + 6 = 2s + 14 + 43s + 6 = 2s + 18

3 Subtract the smaller unknown term (that is, 2s) from both sides and simplify.

3s + 6 - 2s = 2s + 18 - 2ss + 6 = 18

4 Subtract 6 from both sides and simplify. s + 6 - 6 = 18 - 6s = 12

5 Check the solution by substituting s = 12 into the left-hand side and then the right-hand side of the equation.

If s = 12,LHS = 3(s + 2)

= 3(12 + 2)= 3(14)= 42

If s = 12,RHS = 2(s + 7) + 4

= 2(12 + 7) + 4= 2(19) + 4= 38 + 4= 42

6 Comment on the answers obtained. Since the LHS and RHS are equal, the equation is true when s = 12.

b 1 Write the equation. b 4(d + 3) - 2(d + 7) + 4 = 5(d + 2) + 72 Expand the brackets on each side of the

equation first, and then simplify.4d + 12 - 2d - 14 + 4 = 5d + 10 + 7

2d + 2 = 5d + 173 Subtract the smaller unknown term (that is,

2d) from both sides and simplify.2d - 2d + 2 = 5d - 2d + 17

2 = 3d + 174 Rearrange the equation so that the unknown is

on the left-hand side of the equation.3d + 17 = 2

5 Subtract 17 from both sides and simplify. 3d + 17 - 17 = 2 - 173d = -15


153

d = −

d = -57 Check the solution by substituting

d = -5 into the left-hand side and then the right-hand side of the equation.

If d = -5,LHS = 4(d + 3) - 2(d + 7) + 4

= 4(-5 + 3) - 2(-5 + 7) + 4= 4(-2) - 2(2) + 4= -8 - 4 + 4= -8

If d = -5,RHS = 5(-5 + 2) + 7

= 5(-3) + 7= -15 + 7= -8

8 Comment on the answers obtained. Since the LHS and RHS are equal, the equation is true when d = -5.

■ Note: When solving equations with the unknown on both sides, it is good practice to remove the unknown with the smaller coefficient from the relevant side.



reMeMber

1. When a unknown appears on both sides of an equation, remove the unknown term from one side. It is good practice to remove the smaller unknown from the relevant side.

2. For a positive term we can remove by subtraction. For example, 7x + 7 = 5x - 3 (subtract 5x from both sides).

3. For a negative term we can remove by addition. For example, 3x + 11 = 7 - 2x (add 2x to both sides).

equations with the unknown on both sidesfluenCy

1 We 12 Solve the following equations and check your solutions.a 8x + 5 = 6x + 11 b 5y - 5 = 2y + 7 c 11n - 1 = 6n + 19d 6t + 5 = 3t + 17 e 2w + 6 = w + 11 f 4y - 2 = y + 9g 3z - 15 = 2z - 11 h 5a + 2 = 2a - 10 i 2s + 9 = 5s + 3j k + 5 = 7k - 19 k 4w + 9 = 2w + 3 l 7v + 5 = 3v - 11

2 We 13 Solve the following equations and check your solutions.a 3w + 1 = 11 - 2w b 2b + 7 = 13 - b c 4n - 3 = 17 - 6nd 3s + 1 = 16 - 2s e 5a + 12 = -6 - a f 7m + 2 = -3m + 22g p + 7 = -p + 15 h 3 + 2d = 15 - 2d i 5 + m = 5 - mj 7s + 3 = 15 - 5s k 3t - 7 = -17 - 2t l 16 - 2x = x + 4

3 We 14 Expand the brackets and then solve the following equations, checking your solutions.a 3(2x + 1) + 3x = 30 b 2(4m - 7) + m = 76c 3(2n - 1) = 4(n + 5) + 1 d t + 4 = 3(2t - 7)e 3d - 5 = 3(4 - d) f 4(3 - w) = 5w + 1g 2(k + 5) - 3(k - 1) = k - 7 h 4(2 - s) = -2(3s - 1)i -3(z + 3) = 2(4 - z) j 5(v + 2) = 7(v + 1)k 2m + 3(2m - 7) = 4 + 5(m + 2) l 3d + 2(d + 1) = 5(3d - 7)m 4(d + 3) – 2(d + 7) + 5 = 5(d + 12) n 5(k + 11) + 2(k – 3) – 7 = 2(k – 4)o 7(v – 3) – 2(5 – v) + 25 = 4(v + 3) – 8 p 3(l – 7) + 4(8 – 2l) – 7 = –4(l + 2) – 6

unDersTAnDing

4 Solve the equation:− + − − = −3 47

5 1 33

3 4( ) ( )x x

x

5 Solve the equation: ( )x

x− + =23

5 2

6 Find the value of x given the perimeter of the rectangle is 48 cm.

7 The two shapes below have the same area. a Write an equation to show that the

parallelogram and the trapezium have the same area.

b Solve the equation for x.c State the dimensions of the

shapes.

exerCise

11e

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Activity 11-E-1Rocket A

doc-2348

Activity 11-E-2Rocket B

doc-2349

Activity 11-E-3Rocket C

doc-2350

inDiviDuAl pAThWAys

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Unknowns on both sides

doc-2355

3x + 1

x - 2

(x + 3) cm

6 cm

(x + 5) cm

6 cm

2 cm

x = 3 y = 4 n = 4t = 4 w = 5 y = 3

23

z = 4 a = -4 s = 2k = 4 w = -3 v = -4

w = 2 b = 2 n = 2s = 3 a = -3 m = 2

p = 4 d = 3 m = 0s = 1 t = -2 x = 4

x = 3 m = 10n = 12

d = 256

t = 5w = 1

29

k = 10 s = -3z = -17 v = 11

2m = 112

3d = 3 7

10d = –19 k = –10

v = 2 l = 18

x = - 1333

x = 135

or 235

6.25 cm

6(x + 3) = 12 ì 6 ì (x + 5 + 2)

x = 1

Parallelogram: base 4 cm, height 6 cm, Trapezium: base 6 cm, top 2 cm, height 6 cm.



reAsoning

8 A maths class has equal numbers of boys and girls. Eight of the girls left early to play in a netball match. This left 3 times as many boys in the class as girls. How many students are in the class?

9 Judy is thinking of a number. First she doubles it and adds 2. She realizes that if she multiplies it by 3 and subtracts 1, she gets the same result. Find the number.

10 Mick’s father was 28 years old when Mick was born. If his father is now 3 times as old as Mick is, how old are they both now?

11 Given the following number line for a line segment PR, determine the length of PR?

12 In 8 years’ time, Tess will be 5 times as old as her age 8 years ago. How old is Tess now?

P Q R2x

3x + 8

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InteractivitySolving

equationsint-2373

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Digital docWorkSHEET 11.2

doc-2352

eBookpluseBookplus

WeblinkSolving

equations

refleCTion

Does it matter to which side of the equation the unknowns are moved?

24

3

Mick is 14, his father is 42.

16

12



summaryIdentifying patterns

■ A number pattern has a rule that can describe the pattern. ■ Geometric patterns have a rule that can describe the pattern. ■ The rule can be used to make predictions.

Backtracking and inverse operations ■ To work backwards through a fl owchart we use inverse operations. ■ Adding and subtracting are inverse operations. ■ Multiplying and dividing are inverse operations.

Keeping equations balanced ■ An equation links two expressions with an equals sign. ■ An equation is like a pair of balanced scales (or a seesaw). The scales (or seesaw) will remain balanced as long as we do the same to both sides.

Using algebra to solve problems ■ A linear equation is an equation where the variable has an index (power) of 1. ■ When solving linear equations, perform the same operations on both sides of the equation until the unknown is left by itself.

■ You can draw a fl owchart to help you to decide what to do next.

Equations with the unknown on both sides ■ When an unknown appears on both sides of an equation, remove the unknown term from one side. It is good practice to remove the smaller unknown from the relevant side.

■ For a positive term we can remove by subtraction. For example, 7x + 7 = 5x - 3 (subtract 5x from both sides).

■ For a negative term we can remove by addition. For example, 3x + 11 = 7 - 2x (add 2x to both sides).

MAPPING YOUR UNdERSTANdING

Using terms from the summary, and other terms if you wish, construct a concept map that illustrates your understanding of the key concepts covered in this chapter. Compare this concept map with the one that you created in What do you know? on page 291.Have you completed the two Homework sheets, the Rich task and the two Code puzzles in your Maths Quest 8 Homework Book?Homework

Book



Chapter reviewfluenCy

1 Find the output number for each of these flowcharts.

a - 1 ì 5

x

b ó 8 ì 3

x

c + 8 ó 5

x

+ 3

d ì 3 - 7

x

ó 2

2 Draw the flowchart whose output number is given by the following expressions.a -3(m + 4)

b n35+

c m − −75

4

d 7 - 15w

3 a Write an equation that is represented by the diagram below.

represents an unknown amount

represents 1

Key

b Show what happens when you take 2 from both sides, and write the new equation.

4 MC If we start with x = 5, which of these equations is not true?A x + 2 = 7 B 3x = 12

C -2x = -10 d x51=

E x - 2 = 3

5 MC If we start with x = 3, which of these equations is not true?

A 232

x =

B -2x = -6

C 2x - 6 = 0

d x535

=

E x - 5 = 2

6 Solve these equations by doing the same to both sides.a z + 7 = 18b -25 + b = -18

c - 89 = z - 4

3

d 9t = 13

e − =8 75

.l

f - 613

= h8

7 Solve these equations by doing the same to both sides.a 5v + 3 = 18

b 5(s + 11) = 35

c d − =74

10

d -2(r + 5) - 3 = 5

e 2 37

9y − =

f x53 2− =

8 Solve the following equations and check each solution.a 5k + 7 = k + 19b 4s - 8 = 2s - 12c 3t - 11 = 5 - td 5x + 2 = -2x + 16

9 Expand the brackets first and then solve the following equations.a 5(2v + 3) - 7v = 21b 3(m - 4) + 2m = m + 8

5(x - 1)

38x

x + 85

+ 3

3 72

x −

+ 4

m m + 4 -3(m + 4)

ì - 3

ó 3 + 5

n + 5–n3

–n3

- 7 - 4

m m - 7 m - 7——–5

ó 5

m - 7——–5

- 4

ì -15

w -15w 7 -15w

+ 7

2x + 2 = 8

(See bottom of page)

3 b

2x = 6

✔

✔

11

7

49

127

-43.5

- 4813

or -3 913

v = 3

s = -4

d = 47

r = -9

y = 33

x = 25

k = 3

s = -2

t = 4

x = 2

v = 2

m = 5



probleM solving

1 Ray the electrician charges $80 for a call-out visit and then $65 per half hour. a Write an equation for his fees, where C is his

cost and t is the number of 30-minute periods he spent on the job.

b How long did a particular job take if he charged $275.00?

c Ray’s brother Roger is a plumber and uses the equation C = 54t + 86 to calculate his costs. He also charged $275 for one job. How long did Roger spend at this particular job?

d Explain what the number 54 and 86 could mean.

2 At the end of the term, Katie’s teacher gave the class their average scores. They had done four tests for the term. Katie’s average was 76%. She had a mark of 83% for Probability, 72% for Geometry and 91% for Measurement but had forgotten what she got for Algebra. Write an equation to show how Katie would work out her Algebra test score and then solve this equation.

3 My three daughters were each born 2 years apart. Their combined ages come to 63 years. What is the age of the eldest?

4 A truck carrying 50 bags of cement weighs 5.2 tonnes. After delivering 15 bags of cement, the truck weighs 5.173 tonnes. How much would an empty truck weigh?

5 You are 4 times as old as your sister. In 8 years time you will be twice as old as your sister. What are your ages now?

6 What whole number must be added to both the numerator and the denominator of a fraction of 1

4 to

obtain an answer of 23?

7 You lend three friends a total of $45. You lend the first friend x dollars. To the second friend you lend $5 more than you do for the first friend. To the third friend you lend three times as much as for the second friend. How much does each person receive?

8 What is the greatest possible perimeter of a triangle with sides 5x + 20, 3x + 76, x + 196, given that the triangle is isosceles? All sides are in mm.

9 Penny was looking at the following phone plans to see which one would be the most economical for her phone habits. Penny’s average phone conversation was approximately 2 minutes long.1. 15 cents connection fee and 30 cents

per minute

2. No connection fee and 40 cents per minute 3. 40 cents connection fee and 19 cents

per minute a Help Penny by writing an algebraic equation

for each phone plan. b Use the equations to find the most

economical phone plan for Penny if her conversations average: 1 minute, 2 minutes, 3 minutes

c At what average talking time should you switch to Plan 3?

10 A rectangular vegetable patch is (3x + 4) metres long and (2x − 5) metres wide. Its perimeter is 58 metres. What are the dimensions of the vegetable plot?

11 Knitting involves following patterns, often with outstanding designs resulting. Consider a simple pattern to knit a triangular scarf. The pattern might read like this. • Start with 3 stitches. • Increase by 1 stitch at both ends of each row. • Continue in this manner until the scarf is as

long as you would like.a If r represents the row number, and n the

number of stitches in that row, write an equation showing the relationship between n and r.

b Use your equation to calculate the number of rows you would have to knit to have 65 stitches.

12 Three buckets (one black, one white and one red) each hold a maximum of 5 L. Different volumes of water (in whole numbers of litres) are poured into each one. Twice the amount of water in the black bucket is equal to half the amount of water in the red bucket. If the water in the black bucket was poured into the white bucket, the total amount would be the same as the amount of water in the red bucket.How much water is in each bucket?Explain the reasoning you use to arrive at your answer.

13 While on holidays Amy hired a bicycle for $9 an hour and $3 for the use of a helmet. Her brother, Ben, found a cheaper hire place, which charged $6 per hour; but the hire of the helmet was $5 and he had to pay $5 for insurance. Both places measured the time in 20-minute increments. They both ended up paying the same amount and were gone for the same number of hours. Construct a table of values to determine how long they were gone and how much it cost?

C = 65t + 80

112 hours

Katie scored 58% for her Algebra test.

Eldest is 23 years old.

Truck weighs 5.11 tonnes.

You are 16 and your sister is 4.

5

$5, $10, $30

832 mm

y = 0.15 + 0.3x

y = 0.4x

1

c 1

hour

45

min

utes

d $5

4: c

harg

e pe

r ha

lf h

our,

$86:

flat

cal

l-ou

t fee

.y = 0.4 + 0.19x where x in the time of the call and y is the cost of the call.

22 m ì 7 m

n = 2r + 1

32 rows

1 L

in the black bucket, 3 L in the

white bucket, 4 L

in the red bucket.

(See next page)



14 Michael checked his bank balance before going shopping. He had $450. While shopping he paid with his bank card. He bought two suits, which cost the same, and three pairs of shoes, each of which cost half the price of a suit. He also had lunch for $12. When he checked his balance again he was $33 overdrawn. What did one suit cost?

15 Shannon is saving to buy a new computer, which costs $3299. So far he has $449 in the bank and he wants to make regular deposits each month until he reaches his target of $3299. If he wants to buy the computer in 8 months time, how much does he need to save as a monthly deposit?

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InteractivitiesTest yourself

Chapter 11int-2374

Word searchChapter 11

int-2633

CrosswordChapter 11

int-2634

13 2 hours 20 minutes Time Amy Ben

0 $3 $10 20 $6 $12 40 $9 $14 60 $12 $16 80 $15 $18100 $18 $20120 $21 $22140 $24 $24

$134.57

$356.25


eBookpluseBookplus ACTiviTies

Chapter opener

Digital doc (page 291)• Hungry brain activity Chapter 11 (doc-6977)

Are you ready?

Digital docs (pages 292–93)• SkillSHEET 11.1 (doc-6978) Number patterns• SkillSHEET 11.2 (doc-6979) Using tables to show

number patterns• SkillSHEET 11.3 (doc-6980) Describing a number

pattern from a table• SkillSHEET 11.4 (doc-6981) Flowcharts• SkillSHEET 11.5 (doc-6982) Inverse operations• SkillSHEET 11.6 (doc-6983) Solving equations by

backtracking• SkillSHEET 11.7 (doc-6984) Combining like terms• SkillSHEET 11.8 (doc-6985) Expanding expressions

containing brackets• SkillSHEET 11.9 (doc-6986) Checking solutions by

substitution• SkillSHEET 11.10 (doc-6987) Writing equations

from worded statements

11A Identifying patterns

Digital docs (page 297)• Activity 11-A-1 (doc-2336) Identifying patterns• Activity 11-A-2 (doc-2337) More patterns• Activity 11-A-3 (doc-2338) Advanced patterns

11B Backtracking and inverse operations

Digital docs (page 300)• Activity 11-B-1 (doc-2339) Sudoku challenge A• Activity 11-B-2 (doc-2340) Sudoku challenge B• Activity 11-B-3 (doc-2341) Sudoku challenge C

11C Keeping equations balanced

Digital docs (page 302)• Activity 11-C-1 (doc-2342) Riddle A• Activity 11-C-2 (doc-2343) Riddle B• Activity 11-C-3 (doc-2344) Riddle C• WorkSHEET 11.1 (doc-2351) (page 303)Interactivity (page 302)• Balancing equations (int-0077)

11d Using algebra to solve problems

Digital docs (page 307)• Activity 11-D-1 (doc-2345) Using algebra to solve

problems• Activity 11-D-2 (doc-2346) More problems using

algebra• Activity 11-D-3 (doc-2347) Advanced problems

using algebra• Spreadsheet (doc-2353) 2-step equations (page 307)• Spreadsheet (doc-2354) 3-step equations (page 308)

11E Equations with the unknown on both sides

Digital docs (page 312)• Activity 11-E-1 (doc-2348) Rocket A• Activity 11-E-2 (doc-2349) Rocket B• Activity 11-E-3 (doc-2350) Rocket C• Spreadsheet (doc-2355) Unknowns on both sides• WorkSHEET 11.2 (doc-2352) (page 313)Weblink (page 313)• Solving equationsInteractivity (page 313)• Solving equations (int-2373)

Chapter review

Interactivities (page 317)• Test yourself Chapter 11 (int-2374) Take the end-of-

chapter test to test your progress. • Word search Chapter 11 (int-2633)• Crossword Chapter 11 (int-2634)

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nuMber AnD AlgebrA • Linear and non-Linear reLationships 11 … · 2014-06-18 · Chapter 11 linear equations 293 nuMber AnD AlgebrA • Linear and non-Linear reLationships Combining

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