number and algebraequationsweb2.hunterspt-h.schools.nsw.edu.au/studentshared...7 number and...

7Number and algebra

EquationsEquation-solving has been recorded as far back as 1500 BCE.It was first used in ancient Babylon and Egypt and wasbrought to Europe from India by the Arabs during the 9thcentury. The word ‘algebra’ comes from the Arabic wordal-jabr, meaning restoration, the process of performing thesame operation on both sides of an equation to solve theequation.

n Chapter outlineProficiency strands

7-01 Equations with variableson both sides U F R

7-02 Equations with brackets U F R7-03 Equation problems U F PS R C7-04 Equations with algebraic

fractions U F R7-05 Simple quadratic

equations ax2 ¼ c U F R C7-06 Simple cubic equations

ax3 ¼ c* U F R C7-07 Equations and formulas U F PS R C7-08 Changing the subject of

a formula*U F R C

*STAGE 5.3

nWordbankcubic equation An equation involving a variable cubed(power of 3), such as 2x3 ¼ 250.

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

formula A rule written as an algebraic equation, usingvariables.

inverse operation An opposite used in solving an equation,for example, the inverse operation of multiplying is adding

linear equation An equation involving a variable that isnot raised to a power, such as 2x þ 9 ¼ 17.

quadratic equation An equation involving a variablesquared (power of 2), such as 3x2 � 6 ¼ 69.

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

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

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9780170193085

NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l um9

n In this chapter you will:• solve linear equations, including equations involving simple algebraic fractions• (STAGE 5.3) solve equations involving algebraic fractions• solve simple quadratic equations of the form ax2 ¼ c

• (STAGE 5.3) solve simple cubic equations of the form ax3 ¼ c

• use formulas to solve problems• (STAGE 5.3) change the subject of a formula

SkillCheck

1 Solve each equation. Use substitution to check your solutions.

a 3x � 10 ¼ 5 b 4 þ 2y ¼ 21 c 12y þ 5 ¼ 23

d m4� 6 ¼ 2 e x� 5

4¼ 7 f 5r

3¼ 10

g �2x5¼ 3 h 11 � 2a ¼ 17 i 20 þ 4d ¼ �6

j w2þ 3 ¼ 4 k a

6� 1 ¼ 7 l 6� n

3¼ 4

2 Using n to represent ‘the number’, write an expression for each of these statements.a The product of the number and 7.b The square of the number.c 5 times the sum of the number and 8.d The number decreased by 20.e The product of 6 and the number, decreased by nine.f If the number is even, the next even number.

7-01 Equations with variables on both sides

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.

Worksheet

StartUp assignment 7

MAT09NAWK10075

Puzzle sheet

Solving equations

MAT09NAPS00033

Puzzle sheet

Backtracking

MAT09NAPS00032

Skillsheet

Solving equationsby balancing

MAT09NASS10023

Skillsheet

Solving equationsby backtracking

MAT09NASS10024

Skillsheet

Solving equationsusing diagrams

MAT09NASS10025

Puzzle sheet

Equations withunknowns on

both sides

MAT09NAPS00035

Homework sheet

Equations 1

MAT09NAHS10012

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Equations

Example 1

Solve each equations.

a 7x þ 7 ¼ 2x þ 2 b 9 � 6y ¼ 10 � 2y

Solutiona 7xþ 7 ¼ 2xþ 2

7x� 2xþ 7 ¼ 2x� 2xþ 2

5xþ 7 ¼ 2

5xþ 7� 7 ¼ 2� 7

5x ¼ �55x5¼ �5

5x ¼ �1

Subtracting 2x from both sides.

Subtracting 7 from both sides.

Dividing both sides by 5.

Check:

LHS ¼ 7 3 �1ð Þ þ 7 ¼ 0

RHS ¼ 2 3 �1ð Þ þ 2 ¼ 0

LHS ¼ RHS

b 9� 6y ¼ 10� 2y

9� 6yþ 2y ¼ 10� 2yþ 2y

9� 4y ¼ 10

9� 4y� 9 ¼ 10� 9

�4y ¼ 1�4y�4¼ 1�4

y ¼ � 14

Adding 2y to both sides.


Dividing both sides by (�4).

Check:

LHS ¼ 9� 6 3 � 14

� �¼ 10 1

2

RHS ¼ 10� 2 3 � 14

� �¼ 10 1

2LHS ¼ RHS

Exercise 7-01 Equations with variables on both sides1 Solve each equation, and check your solutions.

a 5w þ 3 ¼ 2w þ 21 b 2q � 10 ¼ q � 4 c 13x þ 1 ¼ 8x þ 26d 12n þ 3 ¼ 5n � 11 e 8y � 10 ¼ 10y � 30 f 3m � 2 ¼ 10 � 3m

g 9 � 2a ¼ a � 9 h 9 � 2x ¼ 18 þ 7x i 12y þ 6 ¼ 6 þ 9y

j �12 � 10u ¼ �20 � 18u k 15 � 7x ¼ 22 � 3x l 10 � 6x ¼ �15 � 11x

See Example 1

9780170193085 241


2 For each equation, select the correct solution A, B, C or D.a 6x � 1 ¼ 2x þ 11

A x ¼ 12 B x ¼ 3 C x ¼ 0 D x ¼ 2.5

b 11 � 4p ¼ 2p þ 2

A p ¼ 6.5 B p ¼ 2 C p ¼ 1.5 D p ¼ 3

3 Solve each equation.

a 7w þ 15 ¼ w þ 3 b 10 � 3t ¼ 16 þ t c 4a þ 2 ¼ 10 � 4a

d 50 þ 7y ¼ 20 � 3y e 8y � 2 ¼ 10y þ 1 f 9y þ 3 ¼ 9 � y

g 9 � t ¼ 7t � 2 h 5y þ 2 ¼ 17 � y i 25 � 12k ¼ 15 � 6k

4 Solve �3n � 8 ¼ �7n � 12. Select A, B, C or D.

A n ¼ 5 B n ¼ 2 C n ¼ �1 D n ¼ �0.4

Just for the record Discovering planets

In 1781, British astronomer William Herschel discoveredthe planet Uranus. At that time, it was the farthest planetknown in our solar system. However, astronomers foundthat Uranus’ orbit around the Sun did not follow theexpected path.Working separately, mathematicians John Couch Adamsof England and Urbain Leverrier of France both predictedthat this different orbit was caused by an unknown planet.They calculated the position of this undiscovered planetusing a number of equations.In 1846, a German astronomer called Johann Gallelocated this planet and named it Neptune.The dwarf planet, Pluto, was discovered in a similarmanner.

How long ago was Pluto discovered?

7-02 Equations with brackets

Summary

For equations with brackets (grouping symbols), expand the expressions and then solve asusual.

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Worksheet

Equations 2

MAT09NAWK10076

Worksheet

Checking solutions

MAT09NAWK10078

Puzzle sheet

Equations

MAT09NAPS00036

Puzzle sheet

Equations order activity

MAT09NAPS10077

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Equations

Example 2

Solve each equation.

a 3(a þ 7) ¼ 6 b 9(m � 5) ¼ 7(m þ 1) c 10y � 3(2y � 5) ¼ 6(8 � 3y)

Solutiona 3 aþ 7ð Þ ¼ 6

3aþ 21 ¼ 6

3aþ 21� 21 ¼ 6� 21

3a ¼ �153a3¼ �15

3a ¼ �5

Expanding the expression to makeit a two-step equation.

Subtract 21 from both sides.

Divide both sides by 3.

Check: 3(�5 þ 7) ¼ 3 3 2 ¼ 6

b 9 m� 5ð Þ ¼ 7 mþ 1ð Þ9m� 45 ¼ 7mþ 7

9m� 7m� 45 ¼ 7m� 7mþ 7

2m� 45 ¼ 7

2m� 45þ 45 ¼ 7þ 45

2m ¼ 522m2¼ 52

2m ¼ 26

Expanding brackets on both sides.

Subtracting 7m from both sides.

Adding 45 to both sides.


Check:

LHS ¼ 9 3 26� 5ð Þ ¼ 9 3 21 ¼ 189

RHS ¼ 7 3 26þ 1ð Þ ¼ 7 3 27 ¼ 189

LHS ¼ RHS

c 10y� 3 2y� 5ð Þ ¼ 6 8� 3yð Þ10y� 6yþ 15 ¼ 48� 18y

4yþ 15 ¼ 48� 18y

4yþ 18yþ 15 ¼ 48� 18yþ 18y

22yþ 15 ¼ 48

22yþ 15� 15 ¼ 48� 15

22y ¼ 3322y22¼ 33

22

y ¼ 1 12

Expanding brackets on both sides.Collecting like terms.

Adding 18y to both sides.



Video tutorial

Equations withbrackets

MAT09NAVT10022Can you think of another wayto solve this equation withoutexpanding?

Stage 5.3

9780170193085 243


Exercise 7-02 Equations with brackets1 Solve each equation.

a 2(m þ 3) ¼ 8 b 3(x þ 1) ¼ 9 c 5(y � 2) ¼ 15y

d 35 ¼ 7(k þ 1) e 4(3 � a) ¼ 16 f 11 ¼ 9(1 þ 2p)g 3h ¼ 4(h þ 6) h 6(m � 10) ¼ �6 i 8u ¼ 11(u � 3)j 27 ¼ 7(2y þ 1) k 5(2 þ 3p) ¼ �8 l 22x ¼ 9(4x � 3)

2 In which line has an error been made in solving 5(x � 3) ¼ 25? Select A, B, C or D.

5 x� 3ð Þ ¼ 25

5x� 8 ¼ 25 Line 1

5x� 8þ 8 ¼ 25þ 8 Line 2

5x ¼ 33 Line 3

x ¼ 335

Line 4

¼ 6 35

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

3 Show that k ¼ 5 is the solution to 12(k � 1) ¼ 48.

4 Show that a ¼ 6 is the solution to 10 þ a ¼ 2(2 þ a).


a 8(m þ 2) ¼ 5(m þ 5) b 2(y � 3) ¼ 4(y � 5) c 3(2 þ x) ¼ 4(1 þ x)d 5(p þ 2) ¼ 3(6 þ p) e 5n þ 6 ¼ 2(2n þ 1) f 2(4 � 3x) ¼ 4(7 � 3x)g 4(3w � 1) ¼ 5(4 þ 3w) h �2(x þ 1) ¼ 16 � 5x i �8y � 5 ¼ 5(2y � 3)

6 Show that the solution to 5(2m � 2) ¼ 6(m þ 1) is m ¼ 4.


a 5(m þ 6) þ 10 ¼ 3(m þ 2) þ 20 b 3(y þ 2) � 10 ¼ 2(y � 1) þ 5c 7y þ 2(y þ 5) ¼ 4(y � 10) d 3x þ 4(5 þ x) ¼ 6(2 þ x) þ 20e 5y þ 2(y � 3) ¼ 4y þ 2(2y þ 10) f 11 � 2(5 þ y) ¼ 4(3 þ y) � 1g 8 � 3(1 � m) ¼ 5(m þ 3) þ 4 h 12 � 7(2y � 5) ¼ 6 � 15(2 � 5y)

7-03 Equation problemsWord problems can often be solved more easily when they are converted into equations. Followthese steps.

• Read the problem carefully and determine what needs to be found: ‘What is the question?’• Use a variable to represent the unknown quantity.• Write the problem as an equation.• Solve the equation.• Answer the problem.

See Example 2

Worked solutions


MAT09NAWS10032

Worksheet

Word problemswith equations

MAT09NAWK10079

Worksheet

Angle problemswith algebra

MAT09MGWK000065

Homework sheet

Equations 2

MAT09NAHS10013

Puzzle sheet

Writing and solvingequations

MAT09NAPS00034

9780170193085244

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Equations

Example 3

When three-quarters of a number is decreased by 8 the result is 46. What is the number?

SolutionLet the number be x.

3x4� 8 ¼ 46

3x4� 8þ 8 ¼ 46þ 8

3x4¼ 54

3x4

3 4 ¼ 54 3 4

3x ¼ 2163x3¼ 216

3x ¼ 72

Translating from words to algebra.

Adding 8 to both sides.

Multiplying both sides by 4.


Check: 3 3 724� 8 ¼ 54� 8 ¼ 46

The number is 72.

Example 4

A rectangle is three times as long as it is wide. If its perimeter is 60 cm, find its dimensions.

SolutionLet the width of the rectangle be w cm. Then the length is 3w cm.

w cm

3w cm

The perimeter is w þ 3w þ w þ 3w and this is given as 60.

wþ 3wþ wþ 3w ¼ 60

8w ¼ 60

w ¼ 7:5

[ The width of the rectangle is 7.5 cm and the length is 3 3 7.5 ¼ 22.5 cm.Check: The perimeter of a rectangle with dimensions 7.5 cm and 22.5 cmis 7.5 þ 22.5 þ 7.5 þ 22.5 ¼ 60 cm.

Technology

GeoGebra: Equationproblem

MAT09MGTC00005

9780170193085 245


Example 5

The sum of three consecutive numbers is 150.Find the numbers.

SolutionLet the first number be x.The next number is x þ 1 and the third number is x þ 2.Their sum is x þ (x þ 1) þ (x þ 2) and this is given as 150.

xþ xþ 1þ xþ 2 ¼ 150

3xþ 3 ¼ 150

3x ¼ 147

x ¼ 1473

¼ 49

[ The consecutive numbers are 49, 50 and 51.Check: 49 þ 50 þ 51 ¼ 150.

Example 6

Justin is 6 years older than his sister Chelsea.Their mother is three times Justin’s age.a If the sum of the three ages is 79, write an

equation to find Justin’s age.b Solve the equation and find each person’s age.

Solutiona Let x ¼ Justin’s age.

Chelsea’s age is x � 6.

The mother’s age is 3x.

xþ x� 6ð Þ þ 3x ¼ 79

5x� 6 ¼ 79

Chelsea is 6 yearsyounger than Justin.

b 5x� 6 ¼ 79

5x ¼ 85

x ¼ 17Justin is 17 years old.Chelsea is 17 � 6 ¼ 11 years old.Their mother is 3 3 17 ¼ 51 years old.Check: 17 þ 11 þ 51 ¼ 79

Consecutive numbers followeach other in order, such as3, 4, 5.

Animated example

Applying linearequations

MAT09NAAE00008

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9780170193085246

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Equations

Exercise 7-03 Equation problems1 When 7 is subtracted from four times a certain number, the answer is 37. What is the

number?

2 If 15 more than a number is 3 more than double the number, what is the number?

3 Two-thirds of a number is 16. What is the number?

4 When two-fifths of a number is increased by 15 the result is 27. What is the number?

5 A rectangle is four times as long as it is wide. The perimeter of the rectangle is 100 cm. Findthe dimensions of the rectangle.

6 The length of a rectangle is 7 cm longer than its width.a Let w be the width of the rectangle. Write an equation for w if the perimeter of the

rectangle is 94 cm.

b Solve the equation and find the dimensions of the rectangle.

7 Find the value of x in this triangle.88°

2(x + 11)°

8 a Find the value of y.

b What is the size of each alternate angle?

5(y + 9)°

(7y + 19)°

9 Calculate the size of each marked angle.

7(x + 3)°

2(4x – 3)°

10 The sum of two consecutive numbers is 87. Find the numbers.

11 The sum of three consecutive numbers is 87. Find the numbers.

12 The sum of three consecutive even integers is 168. What are the three integers?

13 The sum of three consecutive odd integers is 75. Find the integers.

See Example 3

See Example 4

Worked solutions

Equation problems

MAT09NAWS10033

See Example 5

9780170193085 247


14 Dean’s father, Franco, is five times Dean’s age. Dean is eight years older than his sister, Helen.The sum of all their ages is 62 years. How old is each person? (Hint: Let Dean’s age be x.)

15 William is three times as old as his daughter, Rebecca. Rebecca is six years younger than herbrother, Ben. How old is Rebecca if the sum of their three ages is 76 years?

16 In my money box I have only $1 and $2 coins. I have 240 coins in total, worth $318. Howmany $2 coins do I have in the money box?

17 Aerin bought four ice creams and received $2.80 change from his $10 note. How much dideach ice cream cost?

18 A boy is twice as tall as his little sister and 30 cm shorter than his father. Their combinedheight is 3.8 m. Find (in centimetres) the height of each person.

19 The length of a rectangle is 9.5 cm longer than its width. The perimeter of the rectangle is87 cm. Find the dimensions of the rectangle.

20 Janine is six years younger than Paul. Paul is three times the age of their son Brett. Brett isfive years older than his sister Amanda. The sum of all their ages is 125 years. How old iseach person?

7-04 Equations with algebraic fractions

Example 7


a xþ 114þ 9 ¼ �3 b a� 2

4¼ 2aþ 5

3

Solutiona xþ 11

4þ 9 ¼ �3

xþ 114þ 9 � 9 ¼ �3 � 9

xþ 114¼ �12

xþ 114

3 4 ¼ �12 3 4

xþ 11 ¼ �48

xþ 11 � 11 ¼ �48 � 11

x ¼ �59

See Example 6

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Worksheet

Equations 3

MAT09NAWS10080

Puzzle sheet

Equations code puzzle

MAT09NAPS10081

Video tutorial

Solving equationswith fractions

MAT09NAVT00008

9780170193085248

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Equations

b a� 24¼ 2aþ 5

3

For equations where all terms are fractions, multiply both sides by a common multipleof the denominators to remove the fractions.The lowest common multiple (LCM) of 3 and 4 is 12, so multiply both sides by 12.

12 3ða� 2Þ

4¼ 12 3

ð2aþ 5Þ3

12 33ða� 2Þ

4 1¼ 12 4

3ð2aþ 5Þ

3 1

3 a� 2ð Þ ¼ 4 2aþ 5ð Þ3a� 6 ¼ 8aþ 20

�5a� 6 ¼ 20

�5a ¼ 26

a ¼ 26�5

a ¼ �5 15

Example 8


a k3þ k

7¼ 4 b 4m

5� m

10¼ 4

Solutiona Multiply both sides by 21, the

LCM of 3 and 7.

21 k3þ k

7

� �¼ 21 3 4

21 3k3þ 21 3

k7¼ 84

721 3k

3 1þ 21 3

3k

7 1¼ 84

7k þ 3k ¼ 84

10k ¼ 84

k ¼ 8410

k ¼ 8 25

b Multiply both sides by 10, theLCM of 5 and 10.

10 4m5� m

10

� �¼ 10 3 4

10 34m5� 10 3

m10¼ 40

10 23

4m5 1� 10 1

3m

10 1

¼ 40

8m� m ¼ 40

7m ¼ 40

m ¼ 407

m ¼ 5 57

9780170193085 249


Example 9


a 2x� 53� xþ 7

5¼ 2 b aþ 5

6þ 3a

4¼ 5

Solutiona Multiply both sides by 15, the LCM

of 3 and 5.

15 2x� 53� xþ 7

5

� �¼ 15 3 2

515 2x� 53 1

� �� 315 xþ 7

5 1

� �¼ 30

5 2x� 5ð Þ � 3 xþ 7ð Þ ¼ 30

10x� 25� 3x� 21 ¼ 30

7x� 46 ¼ 30

7x ¼ 76

x ¼ 767

¼ 10 67

b Multiply both sides by 12, theLCM of 6 and 4.

12 aþ 56þ 3a

4

� �¼ 12 3 5

212 aþ 56 1

� �þ 312 3a

4 1

� �¼ 60

2 aþ 5ð Þ þ 9a ¼ 60

2aþ 10þ 9a ¼ 60

11aþ 10 ¼ 60

11a ¼ 50

a ¼ 5011

¼ 4 611

Exercise 7-04 Equations with algebraic fractions1 Solve each equation.

a aþ 14þ 5 ¼ 15 b xþ 4

5� 10 ¼ 2 c x� 5

5� 1 ¼ 4

d y� 25� 2 ¼ 6 e pþ 1

3� 11 ¼ 8 f 5þ 4yþ 6

9¼ 15

g 1þ 10x� 22

¼ �10 h 3yþ 54þ 9 ¼ �1 i 12� x

3� 1 ¼ 6

j 15� d7þ 10 ¼ 0 k 3� 2w

11þ 4 ¼ 5 l 6� 4p

9� 10 ¼ �6


a xþ 54¼ xþ 4

5b yþ 8

3¼ y� 2

2c 2p� 1

4¼ pþ 5

3

d 2wþ 53¼ 3wþ 1

5e 8x� 4

3¼ 4xþ 5

2f m� 6

4¼ 2m� 3

10

g 2yþ 43¼ 5y� 2

5h 4x� 2

6¼ xþ 7

4i 12� 2x

4¼ 9þ x

3

j 8� 3w5¼ 2wþ 1

4k 7� 5x

2¼ 1� 9x

4l 8� x

2¼ 2xþ 1

3

3 Solve x3þ x

2¼ 1. Select A, B, C or D.

A x ¼ 1 15

B x ¼ 1 12

C x ¼ 1 D x ¼ 2

Stage 5.3

NSW

See Example 7

See Example 8

9780170193085250

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Equations


a d4þ d

3¼ 14 b k

2� k

3¼ 3 c m

3þ m

2¼ 10

d x8þ x

3¼ 10 e p

3� p

5¼ 4 f k

4� k

5¼ 1

g 4k3� k

5¼ 34 h 3m

5þ m

2¼ 11 i 4a

5þ 2a

3¼ 10

5 What is the solution top� 1

5� pþ 3

2¼ 4? Select the correct answer A, B, C or D.

A p ¼ �12 B p ¼ �19 C p ¼ 14 D p ¼ 3


a x7þ x� 1

4¼ 0 b 2p

3� pþ 1

6¼ 2 c mþ 2

2þ mþ 1

3¼ 12

d c� 25� c� 3

2¼ 4 e 2y� 1

4þ yþ 1

2¼ 6 f x� 1

3þ x� 4

6¼ �5

g 7þ 2p5� 1� p

2¼ �1 h 6y� 1

4� yþ 2

3¼ 8 i aþ 5

4þ 3 ¼ 2a

5

j 3x� 14¼ 2xþ 5

6k 12uþ 3

7¼ 2u� uþ 4

6l m� 6

5þ 2m ¼ 3m� 1

4

Mental skills 7A Maths without calculators

Fraction of a quantityLearn these commonly-used fractions and their decimal equivalents.

Fraction 12

14

18

34

15

110

120

25

Decimal 0.5 0.25 0.125 0.75 0.2 0.1 0.05 0.4

Now we will use them to find a fraction or decimal of a quantity.

1 Study each example.

a 14

3 72 ¼ 72 4 4

¼ 18

b 25

3 40 ¼ 15

3 40� �

3 2

¼ 8 3 2

¼ 16

c 34

3 32 ¼ 14

3 32� �

3 3

¼ 8 3 3

¼ 24

d 0:5 3 66 ¼ 12

3 66

¼ 33

e 0:05 3 80 ¼ 120

3 80

¼ 4

f 0:125 3 56 ¼ 18

3 56

¼ 7

Stage 5.3

NSWSee Example 9

9780170193085 251


Technology Solving equations on a graphics

calculatorIn this activity, you will solve equations on a Casio graphics calculator.

1 Select EQUA (Equation mode) from the main menu and select Solver.2 At ‘Eq:’, enter the equation 2x þ 3 ¼ 15 by pressing 2 X,θ,T + 3 = 15 and EXE .

( = is SHIFT )

3 Select SOLV to solve the equation.

The solution is x ¼ 6. The screen also shows LHS (Lft) ¼ RHS (Rgt) so the solution is true.

4 Select REPT to repeat (solve another equation).

5 Enter the equation 4(2x þ 7) ¼ 44 using the ( ) keys.

6 Select SOLV to solve the equation. What is the solution?

7 To enter the equation xþ 52¼ x� 1

3, we need to use the ( ) and ab/c keys.

Press ( X,θ,T + 5 ) ab/c 2 = ( X,θ,T − 1 ) ab/c 3 and EXE .

8 Select SOLV to solve the equation. What is the solution?

9 Now write 5 equations of your own based on the different types studied in this chapter. Useyour graphics calculator to solve them, and write down the equations and solutions in yourbook. Swap the equations with other students in your class and try to solve their equationsusing the graphics calculator.

2 Now simplify each expression.

a 12

3 28 b 14

3 36 c 110

3 70 d 18

3 64

e 15

3 15 f 110

3 80 g 25

3 25 h 120

3 100

i 34

3 44 j 18

3 40 k 0.25 3 60 l 0.4 3 45

m 0.1 3 260 n 0.125 3 48 o 0.75 3 48 p 0.05 3 120q 0.2 3 70 r 0.5 3 320 s 0.25 3 56 t 0.125 3 16

Investigation: Solving x2 ¼ c

1 x2 ¼ 25 has two solutions. What are they?2 What are the possible solutions for each of the following?

a x2 ¼ 9 b x2 ¼ 49 c x2 ¼ 1003 What is the inverse operation of ‘squaring’?4 Study this example:

x2 ¼ 81x ¼ �

ffiffiffiffiffi81p

which meansffiffiffiffiffi81p

or �ffiffiffiffiffi81p

x ¼ �9 which means 9 or �9Check: When x ¼ 9, x2 ¼ 92 ¼ 81When x ¼ �9, x2 ¼ (�9)2 ¼ 81

9780170193085252

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Equations

7-05 Simple quadratic equations ax2 ¼ c

An equation in which the highest power of the variable is 2 is called a quadratic equation, forexample, x2 ¼ 5, 3m2 þ 7 ¼ 10, d2 � 4 ¼ 0 and 4y2 � 3y ¼ 8.

Summary

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

(which means x ¼ffiffifficp

and x ¼ �ffiffifficp

).

Example 10

Solve each quadratic equation.

a y2 ¼ 16 b p2 ¼ 65 c 5a2 ¼ 245

Solutiona y 2 ¼ 16

y ¼ �ffiffiffiffiffi16p

¼ �4Finding the square root of both sides.

b p 2 ¼ 65

p ¼ �ffiffiffiffiffi65p

65 is not a square number so leavethe answer as a surd.

Finding the square root of both sides.

c 5a 2 ¼ 245

a 2 ¼ 2455

a 2 ¼ 49

a ¼ �ffiffiffiffiffi49p

¼ �7


Now use the same method to solve each equation and check your answers.a m2 ¼ 1 b k2 ¼ 64

5 How many solutions does each quadratic equation have?a m2 ¼ 1 b k2 ¼ 64 c x2 ¼ 81

6 Do the following quadratic equations have solutions? (Give reasons for your answers.)a w2 ¼ �1 b y2 ¼ �64 c h2 ¼ �81

7 Write an example of a quadratic equation that has only one solution.

9780170193085 253


Example 11

Solve each quadratic equation, writing the solution correct to one decimal place.

a 4x2 ¼ 600 b 3h 2

5¼ 26

Solutiona 4x 2 ¼ 600

x 2 ¼ 6004

x 2 ¼ 125

x ¼ �ffiffiffiffiffiffiffiffi125p

¼ �11:1803 . . .

� �11:2

b 3h 2

5¼ 26

3h 2 ¼ 26 3 5

3h 2 ¼ 130

h 2 ¼ 1303

h 2 ¼ 43 13

h ¼ �ffiffiffiffiffiffiffiffi43 1

3

r

¼ �6:5828 . . .

� �6:6

Exercise 7-05 Simple quadratic equations ax2¼ c

1 Solve each quadratic equation, writing the solutions as surds if necessary.

a m2 ¼ 144 b x2 ¼ 400 c y2 ¼ 225

d k2 ¼ 59 e y2 ¼ 10 f w2 ¼ 36

g 8x2 ¼ 200 h 9t2 ¼ 81 i a 2

2¼ 8

j 5k2 ¼ 40 k 3w2 ¼ 30 l 2d2 ¼ 288

m k 2

2¼ 8 n w 2

10¼ 7 o 4x2 ¼ 1

p m 2

3¼ 27 q 8y2 ¼ 40 r 2p2 þ 3 ¼ 21

s 3k2 ¼ 48 t y 2

5� 2 ¼ 9 u 6x2 ¼ 42

2 Solve each equation, writing the solution correct to one decimal place.

a m2 ¼ 20 b b2 ¼ 17 c v2 ¼ 6

d 2p2 ¼ 35 e 9k2 ¼ 63 f x 2

5¼ 8

g k 2

16¼ 6 h 7u 2

10¼ 2 i 6y2 ¼ 84

j 3w 2

4¼ 20 k a2 þ 11 ¼ 28 l 2y2 � 14 ¼ 63

3 Explain why the quadratic equation k2 þ 25 ¼ 0 has no solutions.

4 State which of these quadratic equations has no solutions. Give reasons.

a x2 ¼ �9 b 2k2 þ 5 ¼ 9 c 3m2 þ 8 ¼ 4

d 9w 2

2� 1 ¼ 1 e 4þ d 2

3¼ 8 f 5a 2

2þ 3 ¼ 2

See Example 10

See Example 11

9780170193085254

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Equations

7-06 Simple cubic equations ax3 ¼ cAn equation in which the highest power of the variable is 3 is called a cubic equation, for example,x3 ¼ 12, 2m3 þ 1 ¼ 25, d3 � 14 ¼ 4 and x3 � 3x2 þ 5x þ 4 ¼ 0.

Summary

The cubic equation x3 ¼ c has one solution: x ¼ffiffiffic3p

Example 12

Solve each cubic equation.

a n3 ¼ 729 b d3 ¼ 40 c �3y3 ¼ 1029

Solutiona n 3 ¼ 729

n ¼ffiffiffiffiffiffiffiffi7293p

¼ 9

Finding the cube root of both sides.

b d 3 ¼ 40

d ¼ffiffiffiffiffi403p

Finding the cube root of both sides.40 is not a cube number so leave the answer as a surd.

Investigation: Solving x3 ¼ c

1 x3 ¼ 27 has only one solution. What is it?

2 What is the solution to each of the following equations?a x3 ¼ 125 b x3 ¼ 64 c x3 ¼ �8

3 What is the inverse operation of ‘cubing’?

4 Study this example:x3 ¼ �27x ¼

ffiffiffiffiffiffiffiffiffi�273p

x ¼ �3 The cube root of a negative number is also negative.Check: When x ¼ �3, x3 ¼ (�3)3 ¼ �27Now use the same method to solve each equation and check your answers.a r3 ¼ �1000 b u3 ¼ 216

5 How many solutions does each cubic equation have?a n3 ¼ 1 b t3 ¼ �343 c x3 ¼ �1

Stage 5.3

NSW

9780170193085 255


c �3y 3 ¼ 1029

y 3 ¼ 1029�3

y 3 ¼ �343

y ¼ffiffiffiffiffiffiffiffiffiffiffi�3433p

¼ �7

Dividing both sides by �3.

Example 13

Solve each cubic equation, writing the solution correct to one decimal place.

a 2h3 ¼ �18 b z 3

7¼ 11

Solutiona 2h 3 ¼ �18

h 3 ¼ �182

h 3 ¼ �9

h ¼ffiffiffiffiffiffiffi�93p

¼ �2:0800 . . .

� �2:1

b z 3

7¼ 11

z 3 ¼ 11 3 7

z 3 ¼ 77

z ¼ffiffiffiffiffi773p

¼ 4:2543 . . .

� 4:3

Exercise 7-06 Simple cubic equations ax3 ¼ c

1 Solve each cubic equation, writing the solutions as surds if necessary.

a r3 ¼ 1 b k3 ¼ 216 c d3 ¼ �27

d x3 ¼ �45 e w3 ¼ 100 f f 3 ¼ 64

g 4m3 ¼ 32 h 2t3 ¼ �250 i c 3

2¼ 108

j �3k2 ¼ 192 k 7a3 ¼ 105 l 2p 3

9¼ �4

m 4q3 ¼ 665.5 n n 3

5¼ �345:6 o 8s3 ¼ 150

p e 3

8¼ �1 q �6y3 ¼ 40 r 2v3 � 10 ¼ 1014

s 4b 3

5¼ 12 t z 3

6þ 4 ¼ �8 u �4x3 ¼ �144

2 Solve each equation, writing the solution correct to one decimal place.

a c3 ¼ 47 b g3 ¼ 151 c y3 ¼ �450

d �8p3 ¼ 728 e 3u3 ¼ 245 f x 3

5¼ �11

g � h 3

20¼ 25 h 11a 3

7¼ 80 i �6d3 ¼ 186

j 5v 3

8¼ �27 k a3 � 45 ¼ 220 l 4 j3 þ 72 ¼ �166

Stage 5.3

See Example 12

See Example 13

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Equations

3 a Does a cubic equation of the form x3 ¼ c always have a solution?

b When is the solution to x3 ¼ c positive?

c When does x3 ¼ c have two solutions?

7-07 Equations and formulasA formula is an algebraic equation which shows a relationship between variables. For example, theformula for the area of a circle is A ¼ pr2, where A is the area and r is the radius of the circle (p isa constant). Because the formula is for the area, A is called the subject of the formula and it is thevariable on its own on the left-hand side of the ‘¼’ sign.

Example 14

The formula for the perimeter (P) of a rectangle of length l and width w is given byP ¼ 2(l þ w). Use the formula to find:a the perimeter of a rectangle with length 20 cm and width 9 cmb the width of a rectangle if its length is 12 m and its perimeter is 70 mc the length of a rectangle if its width is 42 cm and its perimeter is 1.8 m.

Solutiona l ¼ 20, w ¼ 9:

P ¼ 2 l þ wð Þ¼ 2 20þ 9ð Þ¼ 2 3 29

¼ 58[ The perimeter is 58 cm.

b l ¼ 12, P ¼ 70:

P ¼ 2 l þ wð Þ70 ¼ 2 12þ wð Þ70 ¼ 24þ 2w

46 ¼ 2w

w ¼ 462

¼ 23[ The width is 23 m.

c w ¼ 42, P ¼ 1.8 m ¼ 180 cm (since w is given in cm).

P ¼ 2 l þ wð Þ180 ¼ 2 l þ 42ð Þ180 ¼ 2l þ 84

96 ¼ 2l

l ¼ 962

¼ 48[ The length is 48 cm.

Stage 5.3

Worksheet

Working with formulas

MAT09NAWK10082

Homework sheet

Equations revision

MAT09NAHS10014

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Example 15

The cost of hiring a portable sound systemfor a party is $80 plus $15 per hour. Thecost can be represented by the formulaC ¼ 80 þ 15h where C is the total cost(in dollars), and h is the number of hours.

a Find the cost of hiring the sound systemfor 4 hours.

b A family is willing to spend $300 forhiring the sound system. What is themaximum number of whole rentalhours that the family can afford?

Solutiona h ¼ 4:

C ¼ 80þ 15h

¼ 80þ 15 3 4

¼ 140[ The cost is $140.

b C ¼ 300:

C ¼ 80þ 15h

300 ¼ 80þ 15h

220 ¼ 15h

h ¼ 22015

¼ 14 23

[ The maximum number of whole hours is 14.

Exercise 7-07 Equations and formulas1 Given the formula y ¼ 5x þ b, find:

a y if x ¼ 5 and b ¼ 3 b y if x ¼ �1 and b ¼ 16c b if y ¼ 40 and x ¼ 3 d b if y ¼ �6 and x ¼ �1e x if y ¼ 27 and b ¼ 12 f x if y ¼ 64 and b ¼ �16

2 A temperature in degrees Celsius (�C) can be converted to degrees

Fahrenheit (�F) using the formula F ¼ 9C5þ 32. Convert each

temperature to �F.

a 35�C b �10�C c 16�C

Shut

ters

tock

.com

/Pav

elL

Phot

oan

dV

ideo

See Example 14

Shut

ters

tock

.com

/Ser

gejR

azvo

dovs

kij

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Equations

3 Use the formula in question 2 to convert each temperature to �C, correct to one decimal place.

a 100�F b �45�F c 78�F

4 The formula A ¼ 12ðaþ bÞh is used to find the area of a trapezium, where A is the area, a and b

are the lengths of the parallel sides, and h is the perpendicular height between them. Use theformula to find:a the area of a trapezium with height 6 cm and parallel sides of length 9 cm and 15 cm

b the height of a trapezium if its area is 420 cm2 and it has parallel sides of length 22 cm and20 cm

c the length of one side of a trapezium if its parallel side is 20.5 m, its area is 318 m2 and itsheight is 12 m.

5 Find the value of S in the formula M ¼ kS5

if M ¼ 12.6 and k ¼ 3.15. Select A, B, C or D.

A 60 B 7.938 C 20 D 0.8

6 The cost C (in dollars) of hiring a limousine is given by the formula C ¼ 180 þ 90h, where h isthe number of hours of hire. Find:a the cost of hiring a limousine for 4 hours

b the cost of hiring a limousine for 2 days

c the number of hours for which you could hire a limousine for $720

d the maximum number of whole hours for which a limousine could be hired at a cost of $1000.

7 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 232 DVDs are sold

b the number of DVDs sold if the profit is $1635.

8 A catering company charges $C for a function with P people using the formulaC ¼ 75 þ 12.5P

a How much does the company charge for a function with 10 guests?

b Find the cost of catering for a group of 60 people.

c Diane has $640 to spend on catering for her next party. What is the maximum number ofpeople she can invite?

9 If P ¼ 52 and l ¼ 4, find w if P ¼ 2(l þ w). Select A, B, C or D.

A 9 B 12 C 18 D 22

10 The temperature T (in �C) of a hot liquid as it cools is given by the formula T ¼ 100 � 17.5h,where h is the number of hours it has been cooling. Find:a the temperature of the liquid after 2 hours

b the temperature of the liquid after 30 minutes

c the number of hours it takes for the temperature of the liquid to reach 30�C.

11 Archeologists use the formula H ¼ 2.52t þ 75.8 to estimate the height H cm of a man whenthe tibia (shin) bone length t cm is measurable.a An intact male tibia bone measuring 42 cm long was found. Estimate the height of the male

to the nearest centimetre.

b Estimate, correct to the nearest centimetre, the length of the tibia bone of a male of height174 cm.

See Example 15

Worked solutions

Equations and formulas

MAT09NAWS10034

9780170193085 259


7-08 Changing the subject of a formulaIn the formula v ¼ u þ at, v is the subject of the formula. When the formula is rearranged so thatone of the other variables becomes the subject, the process is called changing the subject of theformula. To change the subject of a formula, use the same rules as for solving an equation. Theanswer is not a number but an algebraic expression (another formula).

Example 16

a For the formula v ¼ u þ at, change the subject to a.

b Make b the subject of the formula x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 � 4acp

.

c Given P ¼ m� 1mþ 1

, change the subject of the formula to m.

Solutiona To make a the subject of the

formula, solve it like anequation for a.

v ¼ uþ at

uþ at ¼ v

at ¼ v� u

a ¼ v� ut

Swapping sides so new subject a appears on the LHS.

Subtracting u from both sides.Dividing both sides by t.

b x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 � 4acp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 � 4acp

¼ x

b 2 � 4ac ¼ x 2

b 2 ¼ x 2 þ 4ac

b ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix 2 þ 4ac

p

Swapping sides.

Squaring both sides.

Adding 4ac to both sides.

Taking the positive and negative square rootof both sides.

c P ¼ m� 1mþ 1

m� 1mþ 1

¼ P

m� 1 ¼ P mþ 1ð Þ¼ Pmþ P

m� Pm ¼ P þ 1

mð1� PÞ ¼ P þ 1

m ¼ P þ 11� P

Swapping sides.

Multiplying both sides by m þ 1.

Expanding.

Moving the m-terms to the LHS, the 1 to the RHS.

Factorise m from the LHS.

Dividing both sides by 1 � P.

Stage 5.3

NSW

Puzzle sheet

Formulas squaresaw

MAT09NAPS10083

9780170193085260

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Equations

stage 5.3Exercise 7-08 Changing the subject of the formula1 Change the subject of each formula to x.

a d ¼ x þ c b y ¼ mx þ b c p ¼ ax � y

d y ¼ xmþ p e k ¼ mrx f v ¼ m

x

g A ¼ xþ y2

h c ¼ ax þ by i A ¼ 12

hðxþ yÞ

2 The volume of a pyramid has the formula V ¼ 13

Ah, where A is the area of the base and h isthe perpendicular height. Which of the following is the correct formula for A? Select thecorrect answer A, B, C or D.

A A ¼ 13

Vh B A ¼ 3Vh

C A ¼ 3Vh D A ¼ 3Vh

3 Make y the subject of each formula.a m ¼ an þ ay b x ¼ y 2 c k ¼ x

yd Q ¼ P þ ffiffiffi

yp e x2 þ y2 ¼ r2 f M ¼ ny2

g m ¼ rffiffiffiyp h b ¼ c2 � 2ay i t ¼ ffiffiffiffiffiffiffiffiffiffiffi

y� xp

4 Change the subject of each formula to the pronumeral shown.

a V ¼ lbh h ¼ ? b A ¼ pr2 r ¼ ? c K ¼ 12

mv 2 v ¼ ?

d F ¼ 9C5þ 32 C ¼ ? e s ¼ ut þ 1

2at 2 a ¼ ? f A ¼ h

2ðxþ yÞ y ¼ ?

5 What is the correct formula for p ¼ m(n þ x) if x is the subject? Select the correct answerA, B, C or D.

A x ¼ p � mn B x ¼ pmn

C x ¼ pm� n D x ¼ p

n� m

6 Solve each equation for a.

a a5¼ apþ q b ar ¼ 3(a þ b) c Da ¼ D � 2a

d p ¼ axaþ y

e k ¼ 1� a1þ a

f M(a þ b) ¼ N(a � b)

7 The cost ($C) of a hire car is given by C ¼ 80 þ 4.2d, where d is the distance travelled inkilometres.a Find the cost of hiring the car for a journey of 50 km.

b Make d the subject of the formula.

c Find the number of whole kilometres that can be travelled in the hire car for $402.

8 The angle sum of a shape with n sides is A�, where A ¼ 180(n � 2).a Use the formula to find the angle sum of a shape with 7 sides.

b Make n the subject of the formula.

c If the angle sum of a polygon is 1440�, how many sides does it have?

9 Make r the subject of the formula 1x¼ 1

rþ 1

s.

10 The body-mass index of an adult is given by the formula B ¼ mh 2, where m is the adult’s mass

in kilograms and h is their height in metres.a Change the subject of the formula to h.

b Hence find, correct to two decimal places, the height of a person with a body mass index of25 and a mass of 60 kg.

See Example 16

Worked solutions

Changing the subjectof a formula

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Mental skills 7B Maths without calculators

Percentage of a quantityLearn these commonly-used percentages and their fraction equivalents.

Percentage 50% 25% 12.5% 75% 20% 10% 33 13

% 66 23

%

Fraction 12

14

18

34

15

110

13

23

Now we will use them to find a percentage of a quantity.

1 Study each example.

a 20% 3 25 ¼ 15

3 25

¼ 5

b 50% 3 120 ¼ 12

3 120

¼ 60

c 12:5% 3 32 ¼ 18

3 32

¼ 4

d 75% 3 56 ¼ 34

3 60

¼ 14

3 60� �

3 3

¼ 15 3 3

¼ 45

e 33 13

% 3 27 ¼ 13

3 27

¼ 9

f 66 23

% 3 60 ¼ 23

3 60

¼ 13

3 60� �

3 2

¼ 20 3 2

¼ 40

2 Now simplify each expression.

a 25% 3 44 b 33 13

% 3 120 c 20% 3 35 d 66 23

% 3 36

e 10% 3 230 f 12 12

% 3 48 g 50% 3 86 h 20% 3 400

i 75% 3 24 j 33 13

% 3 45 k 25% 3 160 l 10% 3 650

m 12.5% 3 88 n 66 23

% 3 21 o 20% 3 60 p 75% 3 180

9780170193085262

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Equations

Power plus


a 52y¼ 14 b 7

r¼ 1

r � 3

c y 2 þ 812

¼ 4 23

d 4m2 � 3 ¼ m2 þ 72

e 3 mþ 5ð Þ4

� 2 m� 1ð Þ3

¼ m� 10þ m� 62

2 Given that W ¼ X � YX þ Y

, find:

a W when X ¼ 15 and Y ¼ 10 b W when X ¼ 6 and Y ¼ �12c X when W ¼ 25 and Y ¼ 6 d Y when W ¼ 5 and X ¼ 1

3 A farmer raises pigs and chickens. From a total of 42 animals she can count 116 legs.Write an equation and solve it to find how many chickens she has.

4 A man is twice as old as his daughter. Ten years ago he was three times as old as her.Write an equation and solve it to determine how old his daughter is now.

5 Consider x2 þ y2 ¼ 4.a Explain why the smallest value for x is �2 and the largest value for x is 2.

b Are there any restrictions on the values that y can take? Explain why.

c By making y the subject, show that y ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffi4� x 2p

.

9780170193085 263


Chapter 7 review

n Language of maths

algebraic fraction brackets consecutive cube root

cubic equation equation expand formula

inverse operation LHS lowest common multiple (LCM) linear equation

quadratic equation RHS solution’ solve

square root subject substitute surd

undoing unknown variable

1 Which method of solving equations involves using inverse operations on both sides of theequation?

2 What is the subject of the formula A ¼ 12ðaþ bÞh

3 What name is given to numbers that follow each other in order, such as 9, 10, 11?

4 Write an example of:

a a quadratic equation b a linear equation.

5 How many solutions does a linear equation have?

6 What does LHS stand for?

n Topic overview

Copy (or print) and complete this mind map of the topic, adding detail to its branches and usingpictures, symbols and colour where needed. Ask your teacher to check your work.

Equations

Simplequadratic and

cubicequations

Equations withalgebraic fractions

Equations withvariables on both sides


Equationsand

formulas

Equation problemsChanging the

subject of a formula

Puzzle sheet

Equations crossword

MAT09NAPS10084

Worksheet

Mind map: Equations(Advanced)

MAT09NAWK10086

9780170193085264

1 Solve each equation. Check your solutions.

a 2x þ 5 ¼ 3x þ 4 b 7 þ 4x ¼ x � 8 c 5 � 6w ¼ 3w � 7d 3x þ 4 ¼ 2x þ 7 e 5n � 3 ¼ 2n � 15 f 2d � 8 ¼ �5d � 71g 4t ¼ 12 � 4t h 8j � 17 ¼ 10j i 6 � 3q ¼ 8 � q

2 Write an equation with x on both sides that has the solution x ¼ 3.


a 2(w � 5) ¼ 4 b 3(1 þ 4n) ¼ 15 c 5(1 � 3p) ¼ 20d 2(3 þ x) ¼ 5(x þ 1) e 3(1 � y) ¼ 4(2 � y) f 2(3 � 4x) ¼ �(2x þ 3)

4 The length of a rectangle is 6 cm longer than it is wide. The perimeter of the rectangle is76 cm. Find the dimensions of the rectangle.

5 Find the value of each pronumeral.

4x°

56°

(2a + 15)°

107°

a b

6 If 6 more than a number is the same as 5 more than double the number, what is the number?


a d � 42þ 9 ¼ 4 b 8nþ 6

3¼ 6nþ 5

2c 2p

3� p

2¼ 4


a x� 15þ x� 4

2¼ 6 b 7nþ 3

5� 5nþ 4

3¼ 2 c 2y� 1

4þ yþ 1

2¼ 6

9 Solve each quadratic equation.

a d2 ¼ 64 b 8p2 ¼ 288 c 3z2 ¼ 105

10 Solve each cubic equation.

a x3 ¼ 1331 b �4h3 ¼ 864 c t 3

2¼ 62:5

11 The body mass index (BMI) of an adult is B ¼ Mh 2, where M is the mass in kilograms and

h is the height in metres.a Find as a whole number the BMI of Dean who is 1.85 m tall and has a mass of 72 kg.b Find the mass of a person with a BMI of 24, who is 2.1 m tall.

12 The cost, C, in dollars, of hiring a taxi is C ¼ 5 þ 2.4d, where d is the distance travelled inkilometres. Find:a the cost of a taxi trip if the distance travelled is 15 kmb the distance travelled if the cost of a taxi trip was $78.20.

13 Make w the subject of each formula.

a a ¼ kwþ v b p ¼ m t � wð Þ c x ¼ 1wþ y

See Exercise 7-01

See Exercise 7-01

See Exercise 7-02

See Exercise 7-03

See Exercise 7-03

See Exercise 7-03

See Exercise 7-04

Stage 5.3

See Exercise 7-04

See Exercise 7-05

Stage 5.3

See Exercise 7-06

See Exercise 7-07

See Exercise 7-07

Stage 5.3See Exercise 7-08

9780170193085 265

Chapter 7 revision

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