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10Number and algebra

EquationsOne of the most common ways to solve complex practicalproblems is to use equations. By relating the various aspectsof a problem using variables, we can often find the bestsolution, or set of solutions. Equations are used in numerousprofessions and trades, from accountancy to zoology. In thischapter, we will learn the techniques involved in solvingequations.

n Chapter outlineProficiency strands

10-01 One-step equations U F R10-02 Two-step equations U F R10-03 Equations with

variables on both sidesU F R

10-04 Equations with brackets U F R10-05 Simple quadratic

equations x2 ¼ cU F R C

10-06 Equation problems U F PS R C

n Wordbankbacktracking method A method of solving equations by‘undoing’, or performing inverse (opposite) operations inreverse order

balancing method A method of solving equations by usingthe same operations on both sides of the equation

equation A mathematical statement that two quantities areequal, involving algebraic expressions and an equals sign (¼)

inverse operation An opposite used in solving anequation, for example, the inverse operation of multiplyingis adding

solve (an equation) To find the value of an unknownvariable in an equation

solution The answer to an equation or problem, thecorrect value(s) of the variable that makes an equation true

substitute To replace a variable with a number

variable A quantity that can take on different values,represented by a symbol such as a letter of the alphabet

9780170189538

NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l um8

n In this chapter you will:• solve linear equations using algebraic techniques• verify solutions by substitution• solve real-life problems by using pronumerals to represent unknowns• solve simple quadratic equations of the form x2 ¼ c

SkillCheck

1 Find the number represented by h each time.

a h þ 3 ¼ 10 b 4 3 h ¼ 28 c 12 � h ¼ 4d h 4 3 ¼ 9 e 9 þ h ¼ 15 f h � 7 ¼ 16g h 3 6 ¼ 30 h 24 4 h ¼ 8 i 10 þ h ¼ 20

j h � 5 ¼ 0 k 12

3 h ¼ 4 l h 4 5 ¼ 4

2 Evaluate each expression.

a 5 � 8 b �8 þ 3 c 3 3 (�4)d �6 4 3 e �2 þ 5 f �2 � 9g �4 3 (�9) h �12 4 (�4) i �3 � 7j 80 4 (�10) k 6 � 15 l �8 3 9

3 If d ¼ 7, evaluate:

a 2d þ 3 b 3d � 4 c d þ 12

d 18 � 2d

4 If x ¼ �3, evaluate:

a 5x þ 9 b 4 � x c x� 3�1

d 2x � 8

5 Simplify each expression.

a 3n � 2n b 8x þ 3x c 5r � r d 6t þ 2t

6 Expand each expression.

a 2(m þ 3) b 3(x � 2) c 4(k þ 7) d 5(d � 1)e 6(a � 3) f 4(b þ 8) g 5(2q � 6) h 9(5j þ 1)

7 What is the opposite operation to:

a adding? b dividing? c multiplying? d subtracting?

10-01 One-step equationsAn equation is a statement involving a variable (such as x), numbers and an equals (¼) sign, forexample, 2x � 5 ¼ 11. In Year 7 we learnt how to solve equations, that is, to find the value of thevariable that makes the equation true. This value is called the solution to the equation.There are two algebraic methods for solving equations.The balancing method involves representing both sides of an equation as balance scales, andsolving the equation by performing the same operation on both sides to keep it ‘balanced’.

Video tutorial

Solving equations

MAT08NAVT00012

Worksheet

StartUp assignment 10

MAT08NAWK10090

Worksheet

Equations 1

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Puzzle sheet

Equations dominoes

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Equations

The backtracking method involves using a flow chart to show what operations are performed onthe variable (say, x) to create the equation, then using a reverse flow chart to undo each operationby performing the inverse (opposite) operation in reverse order.

Summary

To solve an equation, aim to have the variable (such as x) on one side of the equation and anumber on the other side, in the form:

x ¼ ____

Check your solution by substituting it back into the equation.

Operation Inverse operationþ �� þ3 4

4 3

Example 1

Solve each equation.

a m þ 4 ¼ 12 b 3y ¼ 18 c k � 3 ¼ 5 d t8¼ 6

Solutiona Method 1: The balancing method

m

m þ 4 ¼ 12m þ 4 � 4 ¼ 12 � 4 Subtracting 4 from both sides.

m

m ¼ 8

Check: 8 þ 4 ¼ 12.

Method 2: The backtracking method+ 4m m + 4

− 48 12

= =

m þ 4 ¼ 12m ¼ 12 � 4 Undo ‘þ 4’ by subtracting 4.

m ¼ 8 Check: 8 þ 4 ¼ 12.

Puzzle sheet

One-step equations

MAT08NAPS00046

Worksheet

One-step equations

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b Method 1: The balancing method

y

y

yPlace the 18 balls in

three equal rows

3y ¼ 183y3¼ 18

3Dividing both sides by 3.

y

y ¼ 6

Check: 3 3 6 ¼ 18

Method 2: The backtracking method× 3y 3y

÷ 36 18

= =

3y ¼ 18

y ¼ 183

Undo ‘3 3’ in 3y by dividing by 3.

y ¼ 6 Check: 3 3 6 ¼ 18

These are all one-step equations because they require only one step or operation to solve.

c Method 1: The balancing method

k � 3 ¼ 5

k � 3 þ 3 ¼ 5 þ 3 Adding 3 to both sides.

k ¼ 8 Check: 8 � 3 ¼ 5

Method 2: The backtracking method

k � 3 ¼ 5

k ¼ 5 þ 3 Undo ‘� 3’ by adding 3.

k ¼ 8 Check: 8 � 3 ¼ 5

d Method 1: The balancing methodt8¼ 6

t8

3 8 ¼ 6 3 8 Multiplying both sides by 8.

t ¼ 48 Check: 488¼ 6

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Equations

Method 2: The backtracking methodt8¼ 6

t ¼ 6 3 8 Undo ‘4 8’ by multiplying by 8.

t ¼ 48 Check: 488¼ 6

Exercise 10-01 One-step equations1 Solve each equation, showing the working.

a x þ 5 ¼ 12 b y þ 13 ¼ 36 c m þ 9 ¼ 21 d k þ 27 ¼ 54e k þ 6 ¼ 2 f s þ 3 ¼ 1 g n þ 9 ¼ �4 h p þ 17 ¼ �3i m þ 8 ¼ 0 j g þ 4.5 ¼ 20 k aþ 1 3

4 ¼ 12 l k þ 2.7 ¼ �5

2 Solve each equation, showing the working.

a t � 11 ¼ 40 b w � 7 ¼ 4 c a � 5 ¼ 2 d j � 23 ¼ 51e q � 12 ¼ 17 f f � 42 ¼ 68 g x � 9 ¼ 24 h g � 10 ¼ 1i j � 3 ¼ �5 j w � 9 ¼ �2 k m � 12 ¼ �27 l d � 1 ¼ �1

m y � 17 ¼ 3.9 n n � 2.1 ¼ 5.9 o b� 45¼ 1

4p s� 2 1

3 ¼ 4 12


a 6x ¼ 24 b 4g ¼ 16 c 10y ¼ 90 d 5b ¼ 45e 12d ¼ 108 f 9c ¼ 81 g 23p ¼ 115 h 2m ¼ 26i 5x ¼ �25 j 7q ¼ �42 k �3c ¼ 54 l �15x ¼ �120m 4p ¼ 37 n 8r ¼ 18 o 3q ¼ �10 p 9w ¼ �12


a s3¼ 5 b m

7¼ 6 c d

10¼ 12 d w

2¼ 29

e a3¼ �4 r f

�2¼ 9 g r

5¼ �3 h m

�7¼ 1

i x�8¼ �6 j t

2¼ 3

4k n�4¼ �4 l x

3¼ � 1

6

5 Which of the following is the solution to 3x ¼ �12? Select the correct answer A, B, C or D.

A x ¼ �9 B x ¼ �36 C x ¼ �4 D x ¼ �15

6 Write an equation whose solution is:

a m ¼ 4 b x ¼ 0 c r ¼ 7 d d ¼ �5

10-02 Two-step equationsTwo-step equations require two steps or operations to solve.

See Example 1

Worked solutions

Exercise 10-01

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Skillsheet

Solving equations bybalancing

MAT08NASS10035

Skillsheet

Solving equations bybacktracking

MAT08NASS10036

Skillsheet

Solving equationsusing diagrams

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Example 2


a 3x þ 2 ¼ 17 b k5þ 4 ¼ 7


x

x

x

3x þ 2 ¼ 173x þ 2 � 2 ¼ 17 � 2

x

x

x

Step 1: Subtracting 2 from both sides.

3x ¼ 153x3¼ 15

3Step 2: Dividing both sides by 3.

x

x ¼ 5

Check: 3 3 5 þ 2 ¼ 17

Method 2: The backtracking method

3x þ 2 ¼ 17× 3x 3x

÷ 35 15

=

+ 2 3x + 2

– 2 17

=

3x ¼ 17 � 2 Step 1: Undo ‘þ 2’ by subtracting 2.

3x ¼ 15

x ¼ 153

Step 2: Undo ‘3 3’ in 3x by dividing by 3.

x ¼ 5 Check: 3 3 5 þ 2 ¼ 17

b Method 1: The balancing methodk5þ 4 ¼ 7

k5þ 4� 4 ¼ 7� 4 Step 1: Subtracting 4 from both sides.

k5¼ 3

Homework sheet

Equations 1

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Worksheet

Backtracking

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Equations

k5

3 5 ¼ 3 3 5 Step 2: Multiplying both sides by 5.

k ¼ 15 Check: 155þ 4 ¼ 3þ 4 ¼ 7

Method 2: The backtracking methodk5þ 4 ¼ 7

k5¼ 7 � 4 Step 1: Undo ‘þ 4’ by subtracting 4.

k5¼ 3

k ¼ 3 3 5 Step 2: Undo ‘4 5’ by multiplying by 5.

k ¼ 15 Check: 155þ 4 ¼ 3þ 4 ¼ 7

Example 3


a 2h3¼ 4 b xþ 1

2¼ 7


2h3¼ 4

2h3


2h ¼ 12

2h2¼ 12

2Step 2: Dividing both sides by 2.

h ¼ 6 Check: 2 3 63¼ 4

Method 2: The backtracking method2h3¼ 4

2h ¼ 4 3 3 Step 1: Undo ‘4 3’ by multiplying by 3.

2h ¼ 12

h ¼ 122

Step 2: Undo ‘3 2’ in 2h by dividing by 2.

h ¼ 6 Check: 2 3 63¼ 4

b Method 1: The balancing methodxþ 1

2¼ 7

xþ 12


x þ 1 ¼ 14

x þ 1 � 1 ¼ 14 � 1 Step 2: Subtracting 1 from both sides.

x ¼ 13 Check: 13þ 12¼ 7

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Method 2: The backtracking methodxþ 1

2¼ 7 Step 1: Undo ‘4 2’ by multiplying by 2.

x þ 1 ¼ 7 3 2

x þ 1 ¼ 14

x ¼ 14 � 1 Step 2: Undo ‘þ 1’ by subtracting 1.

x ¼ 13 Check: 13þ 12¼ 7

Exercise 10-02 Two-step equations1 Solve each equation, showing all steps. Remember to check your answers.

a 2m þ 7 ¼ 19 b 3x � 5 ¼ 13 c 5k þ 12 ¼ 52d 6w � 17 ¼ 19 e 4h þ 21 ¼ 9 f 8d � 5 ¼ �27g �2x þ 18 ¼ 4 h �3a � 9 ¼ �6 i 14c þ 2 ¼ 37j 12p � 10 ¼ 86 k 9y þ 11 ¼ 101 l 2r � 15 ¼ 16

2 Solve each equation.

a m2þ 4 ¼ 13 b c

5þ 9 ¼ 12 c d

7þ 2 ¼ 8

d k3þ 15 ¼ 6 e h

2þ 7 ¼ 17 f x

4þ 2 ¼ �4

g n5� 9 ¼ 4 h t

3� 6 ¼ �2 i a

8� 2 ¼ �6

j x6� 16 ¼ �13 k v

15� 2 ¼ 4 l b

3� 4 ¼ 3

3 The following is Liam’s incorrect solution for 7x þ 5 ¼ 13.

7x þ 5 ¼ 137x ¼ 13 � 5 Line 17x ¼ 8 Line 2

x ¼ 87

Line 3

x ¼ 1 18 Line 4

In which of the following was the error made? Select A, B, C or D.

A Line 1 B Line 2 C Line 3 D Line 4


a 3x2¼ 9 b 15x

9¼ 15 c 2x

3¼ 8 d 3N

5¼ 3

e 6N7¼ 18 f 5B

2¼ �10 g 3x

4¼ �3 h 2x

5¼ �4

i 5m2¼ 11 j �x

3¼ 8 k �4x

5¼ 1 l �2x

3¼ �10

5 Which of the following is the solution to k � 129¼ 6? Select the correct answer A, B, C or D.

A k ¼ 27 B k ¼ 42 C k ¼ 66 D k ¼ 162

See Example 2

Worked solutions

Exercise 10-02

MAT08NAWS10073

See Example 3

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Equations


a xþ 13¼ 2 b x� 3

2¼ 3 c N þ 2

5¼ 1 d N � 3

4¼ 5

e N þ 85¼ 6 f xþ 1

2¼ �2 g k � 5

2¼ �5 h mþ 2

3¼ 11

7 Write 4 equations, one of each type shown in questions 1, 2, 4 and 5, that have the solutionp ¼ 2.

10-03Equations with variables onboth sides

For equations with variables on both sides, such as 3x þ 4 ¼ 2x þ 7, we can only use thebalancing method, not the backtracking method.

Summary

For equations with variables on both sides, perform operations on both sides to move:

• all the variables onto one side of the equation• all the numbers onto the other side of the equation.

Example 4


a 3x þ 4 ¼ 2x þ 7 b 5n � 3 ¼ 2n � 15 c 2d � 8 ¼ �5d � 71

Solutiona 3x þ 4 ¼ 2x þ 7

x

x

x

x

x

3x þ 4 � 2x ¼ 2x þ 7 � 2x Subtracting 2x from both sides toremove it from the RHS.

x þ 4 ¼ 7 (RHS ¼ ‘right-hand side’)

x

Worked solutions

Exercise 10-02

MAT08NAWS10073

Technology worksheet

Excel: Solving linearequations

MAT08NACT00017

Technology

Excel: Linear equationsolver

MAT08NACT00027

Video tutorial

Equations withvariables on both sides

MAT08NAVT10018

Video tutorial

Solving equations

MAT08NAVT00012

Animated example

Formally solvingequations

MAT08NAAE00020

Worksheet

Equations withunknowns on both

sides

MAT08NAWK00076

Puzzle sheet

Equations withunknowns on both

sides

MAT08NAPS00048

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x þ 4 � 4 ¼ 7 � 4 Subtracting 4 from both sides toremove it from the LHS.

x ¼ 3 (LHS ¼ ‘left-hand side’)

x

Check:LHS ¼ 3 3 3 þ 4 ¼ 13RHS ¼ 2 3 3 þ 7 ¼ 13LHS ¼ RHS.

b 5n � 3 ¼ 2n � 15

5n � 3 � 2n ¼ 2n � 15 � 2n Subtracting 2n from both sides toremove it from the RHS.

3n � 3 ¼ �15 Now this is a two-step equation.

3n � 3 þ 3 ¼ �15 þ 3 Adding 3 to both sides to remove itfrom the LHS.

3n ¼ �12 Now this is a one-step equation.

3n3¼ �12


n ¼ �4

Check:LHS ¼ 5 3 (�4) � 3 ¼ �23RHS ¼ 2 3 (�4) � 15 ¼ �23LHS ¼ RHS

c 2d � 8 ¼ �5d � 71

2d � 8 þ 5d ¼ �5d � 71 þ 5d Adding 5d to both sides to remove itfrom the RHS.

7d � 8 ¼ �71 Now this is a two-step equation.

7d � 8 þ 8 ¼ �71 þ 8 Add 8 to both sides to remove it fromthe LHS.

7d ¼ �63 Now this is a one-step equation.

7d7¼ �63


d ¼ �9Check:LHS ¼ 2 3 (�9) � 8 ¼ �26RHS ¼ �5 3 (�9) � 71 ¼ �26LHS ¼ RHS.

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Exercise 10-03 Equations with variables on both sides1 Solve each equation, showing all steps. Remember to check your answers.

a 3a þ 6 ¼ a þ 18 b 2k þ 4 ¼ k þ 8 c 3x þ 6 ¼ 2x þ 9d 6p þ 4 ¼ p þ 19 e 6n þ 5 ¼ 2n þ 17 f 5q þ 6 ¼ 4q þ 12g 5y þ 14 ¼ 3y þ 14 h 4a þ 7 ¼ 2a þ 21 i 2r þ 9 ¼ r þ 15

2 What is the solution to the equation 4x � 5 ¼ 2x þ 7? Select the correct answer A, B, C or D.

A x ¼ 1 B x ¼ 2 C x ¼ 4 D x ¼ 6


a 5a � 3 ¼ 2a þ 6 b 6x � 2 ¼ 3x þ 6 c 3d � 6 ¼ d � 4d 8p � 15 ¼ 3p � 10 e 3m þ 5 ¼ �m þ 26 f 4x þ 6 ¼ �6x þ 56g 4x þ 3 ¼ �2x þ 7 h 2r þ 18 ¼ �5r þ 11 i 3y � 12 ¼ �y þ 6

4 Here is Bree’s solution for 3x � 4 ¼ �2x þ 8.

3x � 4 ¼ �2x þ 83x þ 2x � 4 ¼ 8 Line 1

5x � 4 ¼ 8 Line 25x ¼ 8 � 4 Line 35x ¼ 4 Line 4

x ¼ 54

Line 5

In which lines did Bree make mistakes? Select the correct answer A, B, C or D.

A Lines 1 and 3 B Lines 2 and 5 C Lines 1 and 4 D Lines 3 and 5


a 3d þ 4 ¼ d b 10k ¼ 12 � 8k c 5 � p ¼ p þ 9d 7 þ x ¼ 6x þ 22 e 6k � 11 ¼ k f 3m þ 20 ¼ 7m � 2g 4t ¼ 12 � 4t h 8j � 17 ¼ 10j i 6 � 3q ¼ 8 � q

6 Write an equation with x on both sides that has the solution x ¼ 7.

10-04 Equations with brackets

Summary

For equations with brackets (grouping symbols), expand the expressions and then solveas usual.

Example 5


a 3(x þ 5) ¼ 9 b 2(3 � y) ¼ 4y þ 4 c 5(r � 3) ¼ 2(r þ 6)

See Example 4

Worked solutions

Exercise 10-03

MAT08NAWS10074

Worked solutions

Exercise 10-03

MAT08NAWS10074

Worksheet

Equations 2

MAT08NAWK10093

Video tutorial

Equations with brackets

MAT08NAVT10019

Worksheet

Writing equations

MAT08NAWK10094

Puzzle sheet

Equations match

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Solutiona 3(x þ 5) ¼ 9

3x þ 15 ¼ 9 Expanding the expression to make it a two-step equation.

3x þ 15 � 15 ¼ 9 � 15 Subtracting 15 from both sides.

3x ¼ �63x3¼ �6


x ¼ �2 Check: 3(�2 þ 5) ¼ 3 3 3 ¼ 9

b 2(3 � y) ¼ 4y þ 4

6 � 2y ¼ 4y þ 4 Expanding the brackets.

6 � 2y � 4y ¼ 4y þ 4 � 4y Subtracting 4y from both sides to remove itfrom the RHS.

6 � 6y ¼ 4 Now this is a two-step equation.

6 � 6y � 6 ¼ 4 � 6 Subtracting 6 from both sides to remove itfrom the LHS.

�6y ¼ �2�6y�6¼ �2�6

Dividing both sides by (�6).

y ¼ 13

Check:

LHS ¼ 2 3 3� 13

� �¼ 5 1

3RHS ¼ 4 3

13þ 4 ¼ 5 1

3

LHS ¼ RHS

c 5(r � 3) ¼ 2(r þ 6)

5r � 15 ¼ 2r þ 12 Expanding both sides.

5r � 15 � 2r ¼ 2r þ 12 � 2r Subtract 2r from both sides to remove it fromthe RHS.

3r � 15 ¼ 12 Now this is a two-step equation.

3r � 15 þ 15 ¼ 12 þ 15 Add 15 to both sides to remove it from theLHS.

3r ¼ 27

3r3¼ 27


r ¼ 9

Check:LHS ¼ 5 3 (9 � 3) ¼ 5 3 6 ¼ 30RHS ¼ 2 3 (9 þ 6) ¼ 2 3 15 ¼ 30LHS ¼ RHS

Worksheet

Equations 3 (Extension)

MAT08NAWK10095

Homework sheet

Equations 2

MAT08NAHS10024

Video tutorial

Solving equations

MAT08NAVT00012

Can you think of a way to solvethis equation withoutexpanding?

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Equations

Exercise 10-04 Equations with brackets1 Solve each equation, showing all steps. Remember to check your answers.

a 2(x þ 3) ¼ 8 b 3(m þ 2) ¼ 18 c 5(k þ 1) ¼ 35 d 4(p þ 4) ¼ 32e 10(j þ 2) ¼ 50 f 7(d þ 4) ¼ 63 g 3(x � 4) ¼ 15 h 3(x � 4) ¼ 12i 5(x � 1) ¼ 10 j 8(x � 5) ¼ 24 k 4(k � 5) ¼ 28 l 2(q � 7) ¼ 26


a 2(6 � x) ¼ 6 b 3(4 � p) ¼ 24 c 2(1 � q) ¼ 8 d 3(16 � h) ¼ 15e 2(4 þ r) ¼ �8 f 3(e � 16) ¼ �15 g 5(z � 3) ¼ �10 h �7(y þ 2) ¼ 35i 9(6 � a) ¼ 0 j 5(d þ 3) ¼ �20 k �2(3 � k) ¼ 12 l 4(6 � c) ¼ �16

3 What is the solution to 2(12 � 3x) ¼ 30? Select the correct answer A, B, C or D.

A x ¼ �2 B x ¼ �1 C x ¼ 1 D x ¼ 2

4 Solve each equation.a 6(x þ 1) ¼ 3x þ 3 b 2(r þ 2) ¼ r þ 5 c 3(p þ 3) ¼ 2p þ 2d 2(x þ 1) ¼ x e 5(p � 2) ¼ 3p f 4(e � 2) ¼ 2e þ 4g 5(y þ 2) ¼ 3y þ 12 h 3(a � 5) ¼ 5 � 2a i 3(2w þ 1) ¼ 3w � 15

5 Here is Robert’s solution for 7(h � 4) ¼ 3h þ 20.

7(h � 4) ¼ 3h þ 207h � 4 ¼ 3h þ 20 Line 1

7h � 4 � 3h ¼ 3h þ 20 � 3h Line 24h � 4 ¼ 20 Line 3

4h � 4 þ 4 ¼ 20 þ 4 Line 44h ¼ 24 Line 54h4¼ 24

4Line 6

h ¼ 6 Line 7

In which lines did Robert make a mistake? Select the correct answer A, B, C or D.

A Line 1 B Line 3 C Line 5 D Line 7


a 5(x � 1) ¼ 4(x þ 2) b 6(x þ 2) ¼ 4(x þ 6) c 4(k þ 3) ¼ 3(k � 2)d 7(y þ 2) ¼ 4(y þ 5) e 3(v þ 2) ¼ 2(v þ 5) f 4(x � 2) ¼ 2(x þ 7)g 3(p þ 1) ¼ 5(p � 1) h 3(2s þ 1) ¼ 5(s þ 2) i 5(d þ 4) ¼ 2(2d � 1)

7 Write an equation involving brackets whose solution is:

a c ¼ 5 b k ¼ �1 c n ¼ 2 d q ¼ �4

Mental skills 10 Maths without calculators

Multiplying and dividing by 5, 15, 25 and 50It is easier to multiply or divide a number by 10 than by 5. So whenever we multiply ordivide a number by 5, we can double the 5 (to make 10) and then adjust the first number.1 Study each example.

a To multiply by 5, halve the number, then multiply by 10.

18 3 5 ¼ 18 312

3 10 ðor 9 3 2 3 10Þ

¼ 9 3 10

¼ 90

See Example 5

Worked solutions

Exercise 10-04

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b To multiply by 50, halve the number, then multiply by 100.

26 3 50 ¼ 26 312

3 100 ðor 13 3 2 3 100Þ

¼ 13 3 100

¼ 1300

c To multiply by 25, quarter the number, then multiply by 100.

44 3 25 ¼ 44 314

3 100 ðor 11 3 4 3 25Þ

¼ 11 3 100

¼ 1100

d To multiply by 15, halve the number, then multiply by 30.

8 3 15 ¼ 8 312

3 30 ðor 4 3 2 3 15Þ

¼ 4 3 30

¼ 120

e To divide by 5, divide by 10 and double the answer. We do this because there aretwo 5s in every 10.

140 4 5 ¼ 140 4 10 3 2

¼ 14 3 2

¼ 28

f To divide by 50, divide by 100 and double the answer. This is because there aretwo 50s in every 100.

400 4 50 ¼ 400 4 100 3 2

¼ 4 3 2

¼ 8

g To divide by 25, divide by 100 and multiply the answer by 4. This is because thereare four 25s in every 100.

600 4 25 ¼ 600 4 100 3 4

¼ 6 3 4

¼ 24

h To divide by 15, divide by 30 and double the answer. This is because there are two15s in every 30.

240 4 15 ¼ 240 4 30 3 2

¼ 8 3 2

¼ 16

2 Now evaluate each expression.

a 32 3 5 b 14 3 5 c 48 3 5 d 18 3 50e 52 3 50 f 36 3 25 g 28 3 5 h 12 3 25i 12 3 15 j 22 3 35 k 90 4 5 l 170 4 5m 230 4 5 n 1300 4 50 o 900 4 50 p 300 4 25q 1000 4 25 r 360 4 45 s 210 4 15 t 360 4 15

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Equations

10-05 Simple quadratic equations x2 ¼ cAn equation involving a pronumeral squared, such as x2 ¼ 25 or 3x2 � 4 ¼ 7, is called a quadraticequation. In this section, we will solve simple quadratic equations of the type x2 ¼ c, where c is anumber.

Example 6

Solve each quadratic equation.a x 2 ¼ 36b x 2 ¼ 121c x 2 ¼ 40, writing the solution correct to one decimal placed x 2 ¼ 83, writing the solution as a surd

Solutiona x 2 ¼ 36

x ¼ �ffiffiffiffiffi36p

¼ �6

Finding the square root of both sides.

The equation has two solutions, x ¼ 6 or x ¼ �6.

b x 2 ¼ 121

x ¼ �ffiffiffiffiffiffiffiffi121p

¼ �11

Finding the square root of both sides.

The equation has solutions x ¼ 11 or x ¼ �11.

Investigation: Solving x2 ¼ c

1 What is the inverse operation of ‘squaring’?2 The equation x2 ¼ 9 has two solutions. What are the two numbers, x, which when

squared gives 9?3 What are the solutions for each equation? What is the pattern found in the answers?

a x 2 ¼ 25 b x 2 ¼ 100 c x 2 ¼ 14 Study this example:

x 2 ¼ 49


which means x ¼ffiffiffiffiffi49p

or �ffiffiffiffiffi49p

¼ �7 which means x ¼ 7 or � 7

Check: When x ¼ 7, x 2 ¼ 72 ¼ 49When x ¼ �7, x 2 ¼ (�7)2 ¼ 49

Use the same method to solve each equation and check your answers:a x 2 ¼ 81 b x 2 ¼ 64

NSW

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c x2 ¼ 40


¼ �6:2345 . . .

� �6:3 The equation has solutions x � 6.3 or x � �6.3.

d x2 ¼ 83


Leaving the answer as a surd.x ¼

ffiffiffiffiffi83p

or �ffiffiffiffiffi83p

Summary

The simple quadratic equation x2 ¼ c (where c is a positive number) has two solutions,x ¼ � ffiffiffi

cp

(which means x ¼ ffiffifficp

or x ¼ � ffiffifficp

).

Exercise 10-05 Simple quadratic equations x2 ¼ c

1 Solve each quadratic equation.

a x 2 ¼ 81 b x 2 ¼ 144 c x 2 ¼ 1d x 2 ¼ 169 e m2 ¼ 5041 f u 2 ¼ 1849

2 Solve each quadratic equation, writing the solution correct to two decimal places.

a x 2 ¼ 13 b x 2 ¼ 54 c x 2 ¼ 88d t 2 ¼ 129 e h2 ¼ 946 f z 2 ¼ 527

3 Solve each quadratic equation, writing the solution as a surd.

a x 2 ¼ 41 b x 2 ¼ 30 c x 2 ¼ 48d a 2 ¼ 126 e p 2 ¼ 75 f b2 ¼ 509

4 Many simple quadratic equations have two solutions (one positive, one negative) but there isone simple quadratic equation that has only one solution. What is this equation and what is itssolution?

5 The equation x 2 ¼ �100 has no solutions. Explain why.

6 Write two more simple quadratic equations that have no solutions.

7 Copy and complete:a x 2 ¼ c has two solutions if c is _______________.

b x 2 ¼ c has no solutions if c is _______________.

c x 2 ¼ c has one solution if c is _______________.

See Example 6

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Equations

10-06 Equation problems

Summary

To solve word problems that require an equation to be solved:

• choose your pronumeral• translate the words into an equation• solve the equation• write a sentence that answers the problem.

Example 7

a When a number is doubled and 11 is subtracted, the result is 23. Find the number.b Five times a number is the same as six more than three times the number. What is the

number?

Solutiona Let the number be x.

Translating the words into anequation:

x 3 2� 11 ¼ 23

2x� 11 ¼ 23Solving the equation:2x � 11 þ 11 ¼ 23 þ 11 Adding 11 to both sides.

2x ¼ 34

2x2¼ 34


x ¼ 17

The number is 17.

Check: 2 3 17 � 11 ¼ 23

b Let n represent the number.

5n ¼ 3n þ 65n � 3n ¼ 3n þ 6 � 3n Subtracting 3x from both sides.

2n ¼ 6

2n2¼ 6


n ¼ 3

The number is 3.

Check:LHS ¼ 5 3 3 ¼ 15RHS ¼ 3 3 3 þ 6 ¼ 15LHS ¼ RHS

Worksheet

Equations problems

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Worksheet

Working with formulas

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Homework sheet

Equations 2

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Homework sheet

Equations revision

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Animated example

Writing equations

MAT08NAAE00019

Video tutorial

Solving equations

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Worksheet

Solving equationsreview

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Quiz

Solving equations

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Example 8

A repairman charges for fixing washing machines using the formula:

C ¼ 32h þ 45

where C is the charge in dollars and h is the number of hours the job takes.Find:a the charge for a job that takes 3 hoursb the number of hours worked if the charge is $205.

Solutiona Substitute h ¼ 3 into the formula.

C ¼ 32hþ 45

¼ 32 3 3þ 45

¼ 141The charge is $141.

b Substitute C ¼ 205 into theformula and solve the equation.

205 ¼ 32h þ 4532h þ 45 ¼ 205 Swap the sides of the equation so that ‘h’ is

on the LHS.

32hþ 45� 45 ¼ 205� 45

32h ¼ 16032h

32¼ 160

32h ¼ 5

The number of hours worked is 5.

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Equations

Exercise 10-06 Equation problems1 Solve each problem by writing an equation and

then solving it. You may use diagrams to helpyou think about the information.a Four family tickets for a theme park cost

$352. How much does each ticket cost? (Lett represent the price of a family ticket.)

b Twenty cans of drink cost $18.00. How muchdoes each can cost? (Let c represent the costof one can.)

c A number is tripled and the result is 99. Whatis the number? (Let x represent the unknownnumber.)

d A number has 7 added to it and the result is23. Find the number. (Let n represent theunknown number.)

2 Select the correct equation A, B, C or D for each problem. Then use the equation to solvethe problem.a If 23 is subtracted from a number, the answer is 79. What is the number?

A 79 þ N ¼ 23 B 23N ¼ 79 C 23 � N ¼ 79 D N � 23 ¼ 79

b The sum of a number and 18 is 67. What is the number?

A N þ 67 ¼ 18 B N þ 18 ¼ 67 C 18N ¼ 67 D N � 18 ¼ 67

c When a number is multiplied by 23, the answer is 1288. What is the number?

A 1288N ¼ 23 B N þ 23 ¼ 1288 C 1288 � N ¼ 23 D 23N ¼ 1288

d When a number is divided by 14, the answer is 42. What is the number?

A N � 14 ¼ 42 B N14¼ 42 C 14N ¼ 42 D 14 þ N ¼ 42

e Marika buys a car that costs $13 499. She pays a deposit of $4525. How much does she owe?

A 4525N ¼ 13 499 B N � 4525 ¼ 13 499C N þ 4525 ¼ 13 499 D N ¼ 13 499 þ 4525

3 For each problem, write an equation and find the unknown number.a When an unknown number is divided by 5, the result is 15.

b When 12 is subtracted from a number, the answer is 37.

c When a number is subtracted from 52, the answer is 25.

d An unknown number is multiplied by 3, then 8 is subtracted. The result is 40.

e An unknown number is halved, then 7 is added. The result is 13.

f The sum of a number and 4 is doubled. The result is 44.

g The sum of a number and 6 is divided by 3 and the result is 7.

h The product of a number and 5 is decreased by 2. The result is 28.

4 Translate each problem into an equation, then solve the equation to solve the problem.a Year 8 is holding a disco to raise money. Each ticket bought by students raises $5 but the

costs of running the disco total $130. How many tickets must be sold to make a profit of$300? (Let n stand for the number of tickets sold.)

See Example 7

Worked solutions

Exercise 10-06

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b In eight years Kelly’s age will be twice what it is now. How old is Kelly now? (Let n standfor Kelly’s age now.)

c Eight sheep have the same mass as three sheep and one cow. If the cow’s mass is 500 kg,what is the mass of one sheep? (Let s stand for the mass of one sheep.)

d If you multiply Liam’s favourite number by 3 and add 1, you get the same answer as if youmultiplied the number by 5 and took away 11. What is the number? (Let x represent the number.)

e Mr Yen says, ‘If you add 15 to my age and multiply by 7, the answer is 294.’ How old isMr Yen? (Let a stand for his age.)

f The area of a rhombus is calculated by multiplying its diagonals and dividing by 2. If arhombus has an area of 88 cm2 and one diagonal is 11 cm, what is the length of the otherdiagonal? (Let d represent the length of the diagonal.)

5 A temperature in degrees Celsius (�C) can be converted to degrees Fahrenheit (�F) using the

formula F ¼ 9C5þ 32.

a Convert 25�C to �F. b Convert 108�F to �C.

6 The charge, $C, for hiring a hall for an event is C ¼ 150 þ 2N, where N stands for the numberof people at the event. Find:a the charge when 225 people are at the event

b the number of people at the event when the charge is $394.

7 The cost, $y, of a classified ad in the local newspaper is y ¼ 0.8w þ 3.5, where w is thenumber of words in the ad. Find:a the cost of a 13-word ad

b the number of words in an ad costing $25.90

8 The perimeter of this rectangle is 47 cm. Find:a the value of x

b the length of the rectangle

c the width of the rectangle.

2x cm (x − 2)cm

9 The profit, $P, made by a DVD store is given by P ¼ 5x � 900, where x represents thenumber of DVDs sold. Find:a the profit made when 195 DVDs are sold

b the number of DVDs sold if the profit is $435.

10 Find the value of x in this isosceles triangle.

17cm

(2x + 3)cm

See Example 8

Worked solutions

Exercise 10-06

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12

Equations

Power plus

1 Find the value of x in each of the following.a 2(x þ 1) þ 2(x � 1) ¼ 12 b 2(x þ 4) � 3(x � 1) ¼ 9c 4(2x � 1) � 5(x � 2) ¼ 6 d 2 � (3x þ 5) ¼ 4(x þ 1)

e xþ 74¼ 6 x� 1ð Þ

3f 2 xþ 1ð Þ

3¼ 5 x� 2ð Þ

2g 2x

3� x

6¼ 10 h 3x

4þ 9x

10¼ 44

2 Write an equation and solve it to find the unknown values in each of the following.

2x

a b c

Perimeter = 30 cm

x

x + 2 2x

Area = 100 cm2

(x + 25)°

(x – 40)°x°

3 Each of the following equations has two solutions. Find them.a x2 ¼ 25 b x2 � 7 ¼ 9 c 2x2 ¼ 72d 3x2 þ 8 ¼ 20 e (x þ 4)2 ¼ 9 f x2 þ 2x ¼ 15

4 Diophantus was a famous mathematician who was the first to abbreviate his mathematicalthoughts using symbols. He is known as the Greek father of algebra and when he died,one of his admirers wrote the following riddle about his life:Diophantus’ youth lasted 1

6 of his life. He grew a beard after 112 more. After 1

7 more of hislife, Diophantus married; 5 years later he had a son. The son lived exactly 1

2 as long as hisfather, and Diophantus died just 4 years after his son. All this adds to the yearsDiophantus lived.Write this riddle as an equation and solve it to find how long Diophantus lived.(Hint: Let x years equal his life.)

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Chapter 10 review

n Language of mathsbacktracking

balancing

brackets

check

equation

expand

formula

inverse operation

LHS (left-hand side)

one-step equation

pronumeral

quadratic equation

RHS (right-hand side)

solution

solve

substitution

two-step equation

undoing

unknown

variable

1 What are the two algebraic methods for solving equations?

2 Which method involves ‘undoing’ operations?

3 What does the word ‘solution’ mean?

4 Which word in the list means ‘opposite’?

5 Why is the variable in an equation sometimes called an unknown?

6 What word means to rewrite an algebraic expression by removing the brackets?

7 What does RHS stand for:

a in congruent triangles? b in solving equations?

n Topic overview• Which parts of this topic did you find easy? What did you already know?• Give examples of some problems that might be solved using equations.• Are there any parts of this topic that you still don’t understand? Talk to your teacher about

them.• In what sort of careers would people use equations?• Copy and complete this mind map of the topic, adding detail to its branches and using

pictures, symbols and colour where needed. Ask your teacher to check your work.

Equations with variableson both sidesOne-and two-step

equations

Equation problemsEquations withbrackets

Simple quadraticequations x2 = c

EQUATIONS

=

Puzzle sheet

Equations crossword

MAT08NAPS10029

Worksheet

Mind map: Equations

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1 Solve each of the following equations.a k þ 6 ¼ 13 b x � 3 ¼ 8 c a þ 3 ¼ 17d a � 12 ¼ 21 e 3x ¼ 12 f 10f ¼ 120

g m4¼ 7 h x

3¼ 8 i w þ 9 ¼ 3

j k � 5 ¼ �7 k �2m ¼ �16 l x3¼ �12

2 Solve each equation.a 4p þ 3 ¼ 23 b 3m þ 17 ¼ 8 c 2x � 12 ¼ 18

d h7þ 5 ¼ 16 e n

2� 8 ¼ �12 f a

5� 8 ¼ 4

g 2y5¼ 4 h xþ 2

3¼ 4 i n� 5

2¼ 10

j 5x2¼ �15 k k þ 9

7¼ 5 l d � 1

5¼ �6

3 Solve each equation.a 3x þ 4 ¼ x þ 6 b 5u � 3 ¼ 2u þ 6c 12h � 8 ¼ 8h þ 4 d 3v � 4 ¼ 7v þ 8e 2x þ 9 ¼ 7x � 4 f 9 � 5t ¼ 3t � 15

4 Solve each equation.a 2(n þ 3) ¼ 18 b 3(x þ 1) ¼ 15 c 10(x � 3) ¼ �10d 5(x � 2) ¼ 3x þ 4 e 4(y þ 1) ¼ y þ 18 f �2(d � 2) ¼ 2d � 20g 3(r � 1) ¼ 2(r þ 9) h 7(a þ 5) ¼ 3(a þ 9) i 2(2n � 4) ¼ 2(5 � n)

5 Solve each equation:a x 2 ¼ 64b x 2 ¼ 45, correct to one decimal placec x 2 ¼ 27, as a surd

6 a If a number is doubled and has 4 added to it, the answer is 14. What is the number?b If a number has 4 added to it and is doubled, the answer is 14. What is the number?

7 a Kay is paid $45 for each jumper she knits. If n is the number of jumpers Kay knits toearn a total of $270, which equation can be used to find the value of n? Select the correctanswer A, B, C or D.

A n45¼ 270 B 45n ¼ 270 C n þ 45 ¼ 270 D 270 � n ¼ 45

b Solve the equation to find the value of n.

8 Seven times a number is the same as nine more than four times the same number. Use anequation to find the number.

9 The sum (S) of the interior angles (in degrees) of a polygon is given by S ¼ 180n � 360,where n is the number of sides. Find:a the sum of the interior angles when a polygon has 9 sidesb the number of sides in a polygon whose angle sum is 1080�.

See Exercise 10-01

See Exercise 10-02

See Exercise 10-03

See Exercise 10-04

See Exercise 10-05

See Exercise 10-06

See Exercise 10-06

See Exercise 10-06

See Exercise 10-06

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Chapter 10 revision

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