Linear equations - John Wiley & Sons Maths Quest 9 number and algebra 4.2 Solving linear equations...

4.1 OverviewWhy learn this?Looking for patterns in numbers, relationships and measurements helps us to understand the world around us. A mathematical model is a mathematical representation of a situation. If we can see a pattern in a table of values or a graph that shows ordered pairs following an approximately straight line, the model is called a linear model.

What do you know? 1 THInK List what you know about linear equations. Use a

thinking tool such as a concept map to show your list.2 PaIr Share what you know with a partner and then with a

small group.3 SHare As a class, create a thinking tool such as a large concept

map to show your class’s knowledge of linear equations.

Learning sequence4.1 Overview4.2 Solving linear equations4.3 Solving linear equations with brackets4.4 Solving linear equations with pronumerals on both sides4.5 Solving problems with linear equations4.6 Rearranging formulas4.7 Review ONLINE ONLY

Linear equations

TOPIC 4

number and algebra

c04LinearEquations.indd 102 05/07/14 5:42 PM

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WaTCH THIS vIdeOThe story of mathematicsThe mighty Roman armies

Searchlight Id: eles-1691


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104 Maths Quest 9

number and algebra

4.2 Solving linear equationsWhat is a linear equation? • An equation is a mathematical statement that contains an = sign. • For an equation, the expression on the left-hand side of the equals sign has the same

value as the expression on the right-hand side. • Solving a linear equation means fi nding a value for the pronumeral that makes the

statement true. • ‘Doing the same thing’ to both sides of the equation ensures that the two expressions

remain equal.WOrKed eXamPle 1

For each of the following equations, determine whether x = 10 is a solution.

a x + 2

3= 6 b 2x + 3 = 3x − 7 c x2 − 2x = 9x − 10

THInK WrITe

a 1 Substitute 10 for x in the left-hand side of the equation.

a LHS = x + 2

3

= 10 + 2

3

= 123

= 4

2 Write the right-hand side. RHS = 6

3 Is the equation true? That is, does the left-hand side equal the right-hand side?

LHS ≠ RHS

4 State whether x = 10 is a solution. x = 10 is not a solution.

b 1 Substitute 10 for x in the left-hand side. b LHS = 2x + 3= 2(10) + 3= 23

2 Substitute 10 for x in the right-hand side. RHS = 3x − 7= 3(10) − 7= 23

3 Is the equation true? LHS = RHS

4 State whether x = 10 is a solution. x = 10 is a solution.

c 1 Substitute 10 for x in the left-hand side. c LHS = x2 − 2x= 102 − 2(10)= 100 − 20= 80

2 Substitute 10 for x in the right-hand side. RHS = 9x − 10= 9(10) − 10= 90 − 10= 80

WOrKed eXamPle 1WOrKed eXamPle 1WOrKed eXamPle 1


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Topic 4 • Linear equations 105

3 Is the equation true? LHS = RHS

4 State whether x = 10 is a solution. x = 10 is a solution.

Solving one-step equations • If one operation has been performed on

a pronumeral, it is known as a one-step equation.

• Simple equations can be solved by applying the inverse operation.

• The inverse operation has the effect of undoing the original operation.

Solve each of the following linear equations.a x − 79 = 153 b x + 46 = 82 c 6x = 100 d

x7

= 19

THInK WrITe

a 1 79 is subtracted from x to give 153. a x − 79 = 153

2 Apply the inverse operation by adding 79 to both sides of the equation.

x = 153 + 79

3 Write the value of x. x = 232

b 1 46 is added to x to give 82. b x + 46 = 82

2 Apply the inverse operation by subtracting 46 from both sides of the equation.

x = 82 − 46


c 1 6 is multiplied by x to give 100. c 6x = 100

2 Perform the inverse operation by dividing both sides of the equation by 6.

x = 1006


d 1 x is divided by 7 to give 19. d x7

= 19

2 Perform the inverse operation by multiplying both sides of the equation by 7.

x = 19 × 7

3 Write the value of x. = 133

Note: In each case the result can be checked by substituting the value obtained for x back into the original equation and confi rming that it will make the equation a true statement.

Operation Inverse operation

+ −− +× ÷÷ ×



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106 Maths Quest 9

Solving two-step equations • If two operations have been performed on the pronumeral, it is known as a two-step

equation. • To solve two-step equations, establish the order in which the operations were performed. • Perform inverse operations in the reverse order to both sides of the equation. • Each inverse operation must be performed one step at a time. • This principle will apply to any equation with two or more steps, as shown in the

examples that follow.

Solve the following linear equations.a 2y + 4 = 12 b 6 − x = 8 c

x3

− 4 = 2 d 3x5

= 6

THInK WrITe

a 1 First subtract 4 from both sides. a 2y + 4 = 122y + 4 − 4 = 12 − 4

2 Divide both sides by 2. 2y = 8

2y2

= 82

3 Write the value of y. y = 4

b 1 6 − x is the same as −x + 6. Rewrite the equation.

b 6 − x = 8−x + 6 = 8

2 Subtract 6 from both sides. −x + 6 − 6 = 8 − 63 Divide both sides by −1. −x = 2

−x−1

= 2−1

4 Write the value of x. x = −2

c 1 Add 4 to both sides. c x3

− 4 = 2

x3

− 4 + 4 = 2 + 4

x3

= 6

2 Multiply both sides by 3. x3

× 3 = 6 × 3


d 1 Multiply both sides by 5. d 3x5

= 6

3x5

× 5 = 6 × 5

3x = 30

2 Divide both sides by 3. 3x3

= 303




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Solve the following linear equations.

a x + 1

2= 11 b

7 − x5

= −6.3

THInK WrITe

a 1 All of x + 1 has been divided by 2. a x + 12

= 11

2 Multiply both sides by 2. x + 12

× 2 = 11 × 2

x + 1 = 22

3 Subtract 1 from both sides. x = 21

b 1 All of 7 − x has been divided by 5. b 7 − x5

= −6.3

2 Multiply both sides by 5. 7 − x5

× 5 = −6.3 × 5

3 7 − x is the same as −x + 7. 7 − x = −31.5

4 Subtract 7 from both sides. 7 − x − 7 = −31.5 − 7−x = −38.5

5 Divide both sides by −1. x = 38.5

Algebraic fractions with the pronumeral in the denominator • If a pronumeral is in the denominator, there is an extra step involved in fi nding the

solution.Consider the following example:

4x

= 32

In order to solve this equation, we fi rst multiply both sides of the equations by x.

4x

× x = 32

× x

4 = 3x2

or 3x2

= 4

The pronumeral is now in the numerator, and the equation is easy to solve.

3x2

= 4

3x = 8

x = 83



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108 Maths Quest 9

Solve each of the following linear equations.

a 3a

= 45

b 5b

= −2

THInK WrITe

a 1 Multiply both sides by a. a 3a

= 45

3 = 4a5

2 Multiply both sides by 5. 15 = 4a

3 Divide both sides by 4. a = 154

or a = 334

b 1 Write the equation. b 5b

= −2

2 Multiply both sides by b. 5 = −2b

3 Divide both sides of the equation by −2. 5−2

= b

b = −212

Exercise 4.2 Solving linear equations IndIvIdual PaTHWaYS

⬛ PraCTISeQuestions:1a–f, 2a–l, 3a–h, 4, 5, 6a–f, 7a–f, 8a–f, 9a–f, 10, 11, 12, 17

⬛ COnSOlIdaTeQuestions:1d–i, 2g–r, 3d–i, 4, 5, 6d–i, 7d–i, 8d–i, 9d–l, 10–13, 17–19

⬛ maSTerQuestions:1g–l, 2i–u, 3g–l, 4, 5, 6g–l, 7g–l, 8g–l, 9g–l, 10–12, 14–20

FluenCY

1 WE1 For each of the following equations, determine whether x = 6 is a solution.a x + 3 = 7 b 2x − 5 = 7 c x2 − 2 = 38

d 6x

+ x = 7 e 2(x + 1)

7= 2 f 3 − x = 9

g x2+ 3x = 39 h 3(x + 2) = 5(x − 4) i x2 + 2x = 9x − 6

j x2 = (x + 1)2 −14 k (x − 1)2 = 4x + 1 l 5x + 2 = x2 + 4

2 WE2 Solve each of the following linear equations. Check your answers by substitution.a x − 43 = 167 b x − 17 = 35 c x + 286 = 516d 58 + x = 81 e x − 78 = 64 f 209 − x = 305g 5x = 185 h 60x = 1200 i 5x = 250

j x

23= 6 k

x17

= 26 l x9

= 27


reFleCTIOnHow are linear equations de� ned?

⬛ ⬛ ⬛ Individual pathway interactivity int-####

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m y − 16 = −31 n 5.5 + y = 7.3 o y − 7.3 = 5.5 p 6y = 14 q 0.2y = 4.8 r 0.9y = −0.05

s y5

= 4.3 t y

7.5= 23 u

y8

= −1.04

3 WE3a Solve each of the following linear equations.a 2y − 3 = 7 b 2y + 7 = 3 c 5y − 1 = 0d 6y + 2 = 8 e 7 + 3y = 10 f 8 + 2y = 12 g 15 = 3y − 1 h −6 = 3y − 1 i 6y − 7 = 140 j 4.5y + 2.3 = 7.7 k 0.4y − 2.7 = 6.2 l 600y − 240 = 143

4 WE3b Solve each of the following linear equations.a 3 − 2x = 1 b −3x − 1 = 5 c −4x − 7 = −19d 1 − 3x = 19 e −5 − 7x = 2 f −8 − 2x = −9g 9 − 6x = −1 h −5x − 4.2 = 7.4 i 2 = 11 − 3xj −3 = −6x − 8 k −1 = 4 − 4x l 35 − 13x = −5

5 Solve each of the following linear equations.a 7 − x = 8 b 8 − x = 7 c 5 − x = 5d 5 − x = 0 e 15.3 = 6.7 − x f 5.1 = 4.2 − x g 9 − x = 0.1 h 140 − x = 121 i −30 − x = −4j −5 = −6 − x k −x + 1 = 2 l −2x − 1 = 0

6 WE3c, d Solve each of the following linear equations.

a x4

+ 1 = 3 b x3

− 2 = −1 c x8

= 12

d −x3

= 5 e 5 − x2

= −8 f 4 − x6

= 11

g 2x3

= 6 h 5x2

= −3 i −3x4

= −7

j −8x3

= 6 k 2x7

= −2 l −3x10

= −15

7 WE4 Solve each of the following linear equations.

a z − 1

3= 5 b

z + 14

= 8 c z − 4

2= −4

d 6 − z

7= 0 e

3 − z2

= 6 f −z − 50

22= −2

g z − 4.4

2.1= −3 h

z + 27.4

= 1.2 i 140 − z

150= 0

j −z − 0.4

2= −0.5 k

z − 69

= −4.6 l z + 65

73= 1

8 Solve each of the following linear equations.

a 5x + 1

3= 2 b

2x − 57

= 3 c 3x + 4

2= −1

d 4x − 13

9= −5 e

4 − 3x2

= 8 f 1 − 2x

6= −10

g −5x − 3

9= 3 h

−10x − 43

= 1 i 4x + 2.6

5= 8.8

j 5x − 0.7

−0.3= −3.1 k

1 − 0.5x4

= −2.5 l −3x − 8

14= 1

2


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110 Maths Quest 9

9 WE5 Solve each of the following linear equations.

a 2x

= 12

b 3x

= 7 c −4x

= 72

d 5x

= −34

e 0.4x

= 92

f 8x

= 1 g −4x

= 23

h −6x

= −45

i 1.7x

= 13

j 6x

= −1 k 4x

= −1522

l 50x

= −3543

10 MC a The solution to the equation 82 − x = 44 is:a x = 126 b x = −126 C x = 122 d x = 38

b What is the solution to the equation 5x − 12 = −62? a x = −14.8 b x = 14.8 C x = 10 d x = −10

c What is the solution to the equation x − 12

= 5.3?

a x = 9.6 b x = 10.6 C x = 11.6 d x = 2 11 Solve each of the following linear equations.

a 3a + 7 = 4 b 5 − b = −5 c 4c − 4.4 = 44

d d − 4

67= 0 e 5 − 3e = −10 f

2f3

= 8

g 100 = 6g + 4.2 h h + 2

6= 5.5 i 452i − 124 = −98

j 6j − 1

17= 0 k

12 − k5

= 4 l l − 5.2

3.4= 1.5

underSTandIng

12 Write the following worded statements as a mathematical sentence and then solve for the unknown.a Seven is added to the product of x and 3, which gives the result of 4.b Four is divided by x and this result is equivalent to 2

3.

c Three is subtracted from x and this result is divided by 12 to give 25. 13 Driving lessons are usually quite expensive but a discount of $15 per lesson is given

if a family member belongs to the automobile club. If 10 lessons cost $760 (after the discount), find the cost of each lesson before the discount.

14 Anton lives in Australia and his pen pal, Utan, lives in USA. Anton’s home town of Horsham experienced one of the hottest days on record with a temperature of 46.7 °C. Utan said that his home town had experienced a day hotter than that, with the temperature reaching 113 °F. The formula for converting Celsius to Fahrenheit is F = 9

5C + 32. Was he correct?

reaSOnIng

15 Santo solved the linear equation 9 = 5 − x. His second step was to divide both sides by −1. Trudy, his mathematics buddy, said that she multiplied both sides by −1.Explain why they are both correct.

16 Find the mistake in the following working and explain what is wrong.

x5

− 1 = 2 x − 1 = 10 x = 11PrOblem SOlvIng

17 Sweet-tooth Sammy goes to the corner store and buys an equal number of 25-cent and 30-cent lollies for $16.50. How many lollies did he buy?


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18 In a cannery, cans are fi lled by two machines which together produce 16 000 cans during an 8-hour shift. If the newer machine produces 340 more cans per hour than the older machine, how many cans does each machine produce in an 8-hour shift?

19 General admission to an exhibition is $55 for an adult ticket, $27 for a child and $130 for a family of two adults and two children. a How much is saved by buying a family ticket instead of buying two

adult and two child tickets? b Is it worthwhile buying a family ticket if the family has only one child?

20 A teacher comes across a clue shown below in a cryptic mathematics cross-number. What is the value of n that the teacher is looking for?

4.3 Solving linear equations with brackets • Consider the equation 3(x + 5) = 18.

There are two good methods for solving this equation.

Method 1:

First divide both sides by 3.

3(x + 5)

3= 18

3x + 5 = 6

x = 1

Method 2:First expand the brackets.

3(x + 5) = 183x + 15 = 18

3x = 3x = 1

In this case, method 1 works well because 3 divides exactly into 18.Now try the equation 7(x + 2) = 10.

185n + 2

3n – 6

150

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112 Maths Quest 9

Method 1:First divide both sides by 7.

7(x + 2)7

= 107

x + 2 = 107

x = −47

Method 2:First expand the brackets.

7(x + 2) = 107x + 14 = 10

7x = −4

x = −47

In this case, method 2 works well because it avoids fraction addition or subtraction.Try both methods and choose the one that works best for you.WOrKed eXamPle 6

Solve each of the following linear equations.a 7(x − 5) = 28 b 6(x + 3) = 7

THInK WrITe

a 1 7 is a factor of 28, so divide both sides by 7.

a 7(x − 5) = 287(x − 5)

7= 28

7

2 Add 5 to both sides. x − 5 = 43 Write the value of x. x = 9

b 1 6 is not a factor of 7, so it will be easier to expand the brackets fi rst.

b 6(x + 3) = 76x + 18 = 7

2 Subtract 18 from both sides. 6x + 18 = 7 − 18 6x = −11

3 Divide both sides by 6. x = −116

(or −156)

Exercise 4.3 Solving linear equations with brackets IndIvIdual PaTHWaYS

⬛ PraCTISeQuestions:1a–f, 2a–h, 3a–f, 4a–f, 5, 6, 8, 10

⬛ COnSOlIdaTeQuestions:1d–i, 2d–i, 3d–i, 4d–i, 5, 7–11

⬛ maSTerQuestions:1g–l, 2g–l, 3g–l, 4g–l, 5, 7–12

FluenCY

1 WE6 Solve each of the following linear equations.a 5(x − 2) = 20 b 4(x + 5) = 8 c 6(x + 3) = 18d 5(x − 41) = 75 e 8(x + 2) = 24 f 3(x + 5) = 15


reFleCTIOnExplain why there are two possible methods for solving equations in factorised form.


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g 5(x + 4) = 15 h 3(x − 2) = −12 i 7(x − 6) = 0 j −6(x − 2) = 12 k 4(x + 2) = 4.8 l 16(x − 3) = 48

2 WE6 Solve each of the following equations.a 6(b − 1) = 1 b 2(m − 3) = 3 c 2(a + 5) = 7 d 3(m + 2) = 2 e 5(p − 2) = −7 f 6(m − 4) = −8g −10(a + 1) = 5 h −12(p − 2) = 6 i −9(a − 3) = −3j −2(m + 3) = −1 k 3(2a + 1) = 2 l 4(3m + 2) = 5

3 Solve each of the following equations.a 9(x − 7) = 82 b 2(x + 5) = 14 c 7(a − 1) = 28d 4(b − 6) = 4 e 3(y − 7) = 0 f −3(x + 1) = 7g −6(m + 1) = −30 h −4(y + 2) = −12 i −3(a − 6) = 3j −2(p + 9) = −14 k 3(2m − 7) = −3 l 2(4p + 5) = 18

4 Solve the following linear equations. Round the answers to 3 decimal places where appropriate.a 2(y + 4) = −7 b 0.3(y + 8) = 1 c 4(y + 19) = −29d 7(y − 5) = 25 e 6(y + 3.4) = 3 f 7(y − 2) = 8.7g 1.5(y + 3) = 10 h 2.4(y − 2) = 1.8 i 1.7(y + 2.2) = 7.1j −7(y + 2) = 0 k −6(y + 5) = −11 l −5(y − 2.3) = 1.6

5 MC a The best first step in solving the equation 7(x − 6) = 23 would be to:a add 6 to both sidesb subtract 7 from both sidesC divide both sides by 23d expand the brackets

b The solution to the equation 84(x − 21) = 782 is closest to:a x = 9.31 b x = 9.56C x = 30.31 d x = −11.69

underSTandIng

6 In 1974 a mother was 6 times as old as her daughter. If the mother turned 50 in the year 2000, in what year was the mother double her daughter’s age?

7 New edging is to be placed around a rectangular children’s playground. The width of the playground is x m and the length is 7 metres longer than the width.a Write down an expression for the perimeter of the playground. Write your answer in

factorised form.b If the amount of edging required is 54 m, determine the dimensions of the playground.

reaSOnIng

8 Juanita is solving the following equation: 2(x − 8) = 10. She performs the following operations to both sides of the equation in order: +8, ÷2. Explain why Juanita will not find the correct value of x using her order of inverse operations, then solve the equation.

9 As your first step to solve the equation 3(2x – 7) = 18, you are given three options: • Expand the brackets on the left-hand side.• Add 7 to both sides.• Divide both sides by 3.

Which of the options is your least preferred and why?


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114 Maths Quest 9

PrOblem SOlvIng

10 Five times the sum of 4 and a number is equal to 35. What is the number? 11 Kyle earns $55 more than Noah each week, but Callum earns three times as much as

Kyle. If Callum earns $270 a week, how much do Kyle and Noah earn each week? 12 A school wishes to hire a bus to travel to a football game. The bus will take

28 passengers, and the school will contribute $48 towards the cost of the trip. If the hiring of the bus is $300 + 10% of the cost of all the tickets, what should be the cost per person?

4.4 Solving linear equations with pronumerals on both sides • When solving equations, it is important to remember that whatever we do to one side of

an equation we must do to the other. • If the pronumeral occurs on both sides of the equation, fi rst remove it from one side, as

shown in the example below.WOrKed eXamPle 7

Solve each of the following linear equations.a 5y = 3y + 3 b 7x + 5 = 2 − 4xc 3(x + 1) = 14 − 2x d 2(x + 3) = 3(x + 7)

THInK WrITe

a 1 3y is smaller than 5y. Subtract 3y from both sides.

a 5y = 3y + 35y − 3y = 3y + 3 − 3y

2y = 3

2 Divide both sides by 2. y = 32

(or 112)

b 1 −4x is smaller than 7x. Add 4x to both sides.

b 7x + 5 = 2 − 4x7x + 5 + 4x = 2 − 4x + 4x

11x + 5 = 22 Subtract 5 from both sides. 11x + 5 − 5 = 2 − 5

11x = −3

3 Divide both sides by 11. x = −311

c 1 Expand the bracket. c 3(x + 1) = 14 − 2x3x + 3 = 14 − 2x

2 −2x is smaller than 3x. Add 2x to both sides.

3x + 3 + 2x = 14 − 2x + 2x5x + 3 = 14

3 Subtract 3 from both sides. 5x + 3 − 3 = 14 − 35x = 11

4 Divide both sides by 5. x = 115

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d 1 Expand the brackets. d 2(x + 3) = 3(x + 7)2x + 6 = 3x + 21

2 2x is smaller than 3x.Subtract 2x from both sides.

2x + 6 − 2x = 3x + 21 − 2x 6 = x + 21

3 Subtract 21 from both sides. 6 − 21 = x + 21 − 21−15 = x

4 Write the answer with the pronumeral written on the left-hand side.

x = −15

Exercise 4.4 Solving linear equations with pronumerals on both sides IndIvIdual PaTHWaYS

⬛ PraCTISeQuestions:1a–f, 2, 3a–f, 4, 5, 6a–f, 7, 8, 11

⬛ COnSOlIdaTeQuestions:1d–i, 2, 3d–i, 4, 5, 6d–i, 7–12

⬛ maSTerQuestions:1g–l, 2, 3g–l, 4, 5, 6g–l, 7–14

FluenCY

1 WE7a Solve each of the following linear equations.a 5y = 3y − 2 b 6y = −y + 7 c 10y = 5y − 15d 25 + 2y = −3y e 8y = 7y − 45 f 15y − 8 = −12yg 7y = −3y − 20 h 23y = 13y + 200 i 5y − 3 = 2yj 6 − 2y = −7y k 24 − y = 5y l 6y = 5y − 2

2 MC a To solve the equation 3x + 5 = −4 − 2x, the fi rst step is to:a add 3x to both sidesb add 5 to both sidesC add 2x to both sidesd subtract 2x from both sides

b To solve the equation 6x − 4 = 4x + 5, the fi rst step is to:a subtract 4x from both sidesb add 4x to both sidesC subtract 4 from both sidesd add 5 to both sides

3 WE7b Solve each of the following linear equations.a 2x + 3 = 8 − 3x b 4x + 11 = 1 − x c x − 3 = 6 − 2xd 4x − 5 = 2x + 3 e 3x − 2 = 2x + 7 f 7x + 1 = 4x + 10g 5x + 3 = x − 5 h 6x + 2 = 3x + 14 i 2x − 5 = x − 9j 10x − 1 = −2x + 5 k 7x + 2 = −5x + 2 l 15x + 3 = 7x − 3

reFleCTIOnDraw a diagram that could represent 2x + 4 = 3x + 1.


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116 Maths Quest 9

4 Solve each of the following linear equations.a x − 4 = 3x + 8 b 3x + 12 = 4x + 5c 2x + 9 = 7x − 1 d −2x + 7 = 4x + 19e −3x + 2 = −2x − 11 f 11 − 6x = 18 − 5xg 6 − 9x = 4 + 3x h x − 3 = 18x − 1i 5x + 13 = 15x + 3

5 MC a The solution to 5x + 2 = 2x + 23 is:a x = 3 b x = −3C x = 5 d x = 7

b The solution to 3x − 4 = 11 − 2x is:a x = 15 b x = 7C x = 3 d x = 5

6 WE7c, d Solve each of the following.a 5(x − 2) = 2x + 5 b 7(x + 1) = x − 11c 2(x − 8) = 4x d 3(x + 5) = x e 6(x − 3) = 14 − 2x f 9x − 4 = 2(3 − x)g 4(x + 3) = 3(x − 2) h 5(x − 1) = 2(x + 3)i 8(x − 4) = 5(x − 6) j 3(x + 6) = 4(2 − x)k 2(x − 12) = 3(x − 8) l 4(x + 11) = 2(x + 7)

underSTandIng

7 Aamir’s teacher gave him an algebra problem and told him to solve it. Can you help him? 3x + 7 = x2 + k = 7x + 15What is the value of k?

8 A classroom contained an equal number of boys and girls. Six girls left to play hockey, leaving twice as many boys as girls in the classroom. What was the original number of students present?

reaSOnIng

9 Express the following information as an equation, then show that n = 29 is the solution.

10 Explain what the problem is in solving the equation 4(3x – 5) = 6(2x + 3) .

PrOblem SOlvIng

11 This year Tom is 4 times as old as his daughter, while in 5 years’ time he will be only 3 times as old as his daughter. Find the ages of Tom and his daughter now.

12 If you multiply an unknown number by 6 and then add 5, the result is 7 less than the unknown number plus 1 multiplied by 3. Find the unknown number.

150 – 31n20n + 50

n – 36

–98

n – 36

n – 36


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13 You are investigating getting a business card printed for your new game store. A local printing company charges $250 for the cardboard used and an hourly rate for labour of $40.

a If h is the number of hours of labour required to print the cards, construct an equation for the cost of the cards, C.

b You have budgeted $1000 for the printing job. How many hours of labour can you afford? Give your answer to the nearest minute.

c The company estimates that it can print 1000 cards per hour of labour. How many cards will you get printed with your current budget?

d An alternative to printing is photocopying. The company charges 15 cents per side for the fi rst 10 000 cards and then 10 cents per side for the remaining cards. Which is the cheaper option for 18 750 single-sided cards and by how much?

14 A local pinball arcade offers its regular customers the following deal. For a monthly fee of $40 players get 25 $2 pinball games. Additional games cost $2 each. After a player has played 50 games in a month, all further games are $1.a If Tom has $105 to spend in a month, how many games can he play if he

takes up the special deal?b How much did Tom save by taking up the special deal.

4.5 Solving problems with linear equationsConverting worded sentences to algebraic equations • An important skill in mathematics is the ability to translate written problems into

algebraic equations in order to solve problems.

Write linear equations for each of the following statements, using x to represent the unknown. (Do not attempt to solve the equations.)a When 6 is subtracted from a certain number, the result is 15.b Three more than seven times a certain number is zero.c When 2 is divided by a certain number, the answer is 4 more than the number.

THInK WrITe

a 1 Let x be the number. a x = unknown number

2 Write x and subtract 6. This expression equals 15.

x − 6 = 15

Address: 123 The StreetMelbourneVIC 3000

Phone no: 03 1234 5678



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118 Maths Quest 9

b 1 Let x be the number. b x = unknown number

2 7 times the number is 7x. Three more than 7x equals 7x + 3. This expression equals 0.

7x + 3 = 0

c 1 Let x be the number. c x = unknown number

2 Write the term for 2 divided by a certain number.

2x

Write the expression for 4 more than the number.

x + 4

3 Write the equation. 2x

= x + 4

In a basketball game, Hao scored 5 more points than Seve. If they scored a total of 27 points between them, how many points did each of them score?

THInK WrITe

1 Defi ne a pronumeral. Let Seve’s score be x.

2 Hao scored 5 more than Seve. Hao’s score is x + 5.

3 Between them they scored a total of 27 points.

x + (x + 5) = 27

4 Solve the equation. 2x + 5 = 27 2x = 22 x = 11

5 Since x = 11, this is Seve’s score. Write Hao’s score.

Hao’s score = x + 5= 11+ 5= 16

6 Write the answer in words. Seve scored 11 points and Hao scored 16 points.



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Taxi charges are $3.60 plus $1.38 per kilometre for any trip in Melbourne. If Elena’s taxi fare was $38.10, how far did she travel?

THInK WrITe

1 The distance travelled by Elena has to be found. Defi ne the pronumeral.

Let x = distance travelled (in kilometres).

2 It costs 1.38 to travel 1 kilometre, so the cost to travel x kilometres = 1.38x. The fi xed cost is $3.60. Write an expression for the total cost.

Total cost = 3.60 + 1.38x

3 Let the total cost = 38.10. 3.60 + 1.38x = 38.10

4 Solve the equation. 1.38x = 34.50

x = 34.501.38

= 25

5 State the solution in words. Elena travelled 25 kilometres.

Exercise 4.5 Solving problems with linear equations IndIvIdual PaTHWaYS

⬛ PraCTISeQuestions:1–4, 7, 9, 11–14

⬛ COnSOlIdaTeQuestions:1–5, 7–10, 12–15

⬛ maSTerQuestions:1–16


reFleCTIOnWhy is it important to de� ne the pronumeral used in forming a linear equation to solve a problem?⬛ ⬛ ⬛ Individual pathway interactivity int-####


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120 Maths Quest 9

FluenCY

1 WE8 Write linear equations for each of the following statements, using x to represent the unknown. (Do not attempt to solve the equations.)a When 3 is added to a certain number, the answer is 5.b Subtracting 9 from a certain number gives a result of 7.c Seven times a certain number is 24.d A certain number divided by 5 gives a result of 11.e Dividing a certain number by 2 equals −9.f Three subtracted from fi ve times a certain number gives a result of −7.g When a certain number is subtracted from 14 and this result is then multiplied by 2,

the result is −3.h When 5 is added to three times a certain number, the answer is 8.i When 12 is subtracted from two times a certain number, the result is 15.j The sum of 3 times a certain number and 4 is divided by 2, which gives a result of 5.

2 MC Which equation matches the following statement?a A certain number, when divided by 2, gives a result of −12.

a x = −122

b 2x = −12

C x2

= −12 d x

12= −2

b Dividing 7 times a certain number by −4 equals 9.

a x

−4= 9 b

−4x7

= 9

C 7 + x

−4= 9 d

7x−4

= 9

c Subtracting twice a certain number from 8 gives 12. a 2x − 8 = 12 b 8 − 2x = 12C 2 − 8x = 12 d 8 − (x + 2) = 12

d When 15 is added to a quarter of a number, the answer is 10.

a 15 + 4x = 10 b 10 = x4

+ 15

C x + 15

4= 10 d 15 + 4

x= 10

underSTandIng

3 When a certain number is added to 3 and the result is multiplied by 4, the answer is the same as when the same number is added to 4 and the result is multiplied by 3. Find the number.

4 WE9 John is three times as old as his son Jack, and the sum of their ages is 48. How old is John?

5 In one afternoon’s shopping Seedevi spent half as much money as Georgia, but $6 more than Amy. If the three of them spent a total of $258, how much did Seedevi spend?

6 These rectangular blocks of land have the same area. Find the dimensions of each block, and the area.

doc-10826doc-10826doc-10826

x + 5

20

x

30


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reaSOnIng

7 A square pool is surrounded by a paved area that is 2 metres wide. If the area of the paving is 72 m2, what is the length of the pool?

8 Maria is paid $11.50 per hour, plus $7 for each jacket that she sews. If she earned $176 for one 8-hour shift, how many jackets did she sew?

9 Mai hired a car for a fee of $120 plus $30 per day. Casey’s rate was $180 plus $26 per day. If their final cost was the same, how long was the rental period?

10 WE10 The cost of producing music CDs is quoted as $1200 plus $0.95 per disk. If Maya’s recording studio has a budget of $2100, how many CDs can she have made?

11 Joseph wishes to have some flyers delivered for his grocery business. Post Quick quotes a price of $200 plus 50 cents per flyer, while Fast Box quotes $100 plus 80 cents per flyer.a If Joseph needs to order 1000 flyers, which distributor would be cheaper to use?b For what number of fliers will the cost be the same for either distributor?

2 m


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122 Maths Quest 9

PrOblem SOlvIng

12 A number is multiplied by 8 and 16 is then subtracted. The result is the same as 4 times the original number minus 8. What is the number?

13 Carmel sells three different types of healthy drinks; herbal, vegetable and citrus fi zz. One hour she sells 4 herbal, 3 vegetable and 6 citrus fi zz for $60.50. The next hour she sells 2 herbal, 4 vegetable and 3 citrus fi zz. The third hour she sells 1 herbal, 2 vegetable and 4 citrus fi zz. The total amount in cash sales for the three hours is $136.50. Carmel made $7 less in the third hour than she did in the second hour of sales.

Determine her sales in the fourth hour, if Carmel sells 2 herbal, 3 vegetable and 4 citrus fi zz.

14 A rectangular swimming pool is surrounded by a path which isenclosed by a pool fence. All measurements are in metres and are not to scale in the diagram shown.a Write an expression for the entire fenced-off area. b Write an expression for the area of the path surrounding the pool.c If the area of the path surrounding the pool is 34 m2, fi nd the dimensions of the

swimming pool.d What fraction of the fenced-off area is taken up by the pool?

4.6 Rearranging formulas • Formulas are generally written in terms of two or more pronumerals or variables. • One pronumeral is usually written on one side of the equal sign. • When rearranging formulas, use the same methods as for solving linear equations (use

inverse operations in reverse order).The difference between rearranging formulas and solving linear equations is that

rearranging formulas does not require a value for the pronumeral(s) to be found. • The subject of the formula is the pronumeral or variable that is written by itself. It is

usually written on the left-hand side of the equation.

Rearranging (transposing) formulas • A formula is simply an equation that is used for some specifi c purpose. By now you will

be familiar with many mathematical or scientifi c formulas.For example, C = 2πr relates the circumference of a circle to its radius. When the

formula is shown in this order, C is called the subject of the formula. The formula can be transposed (rearranged) to make r the subject.

C = 2πrC2π

= 2πr2π

C2π

= r

or r = C2π

Divide both sides by 2π.

Now r is the subject.

Fence

52

x + 2

x + 4

doc-6159doc-6159doc-6159


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Rearrange each formula to make x the subject.a y = kx + m b 6(y + 1) = 7(x − 2)

THInK WrITe

a 1 Subtract m from both sides. a y = kx + my − m = kx

2 Divide both sides by k. y − mk

= kxk

y − mk

= x

3 Rewrite the equation so that x is on the left-hand side.

x = y − mk

b 1 Expand the brackets. b 6(y + 1) = 7(x − 2)6y + 6 = 7x − 14

2 Add 14 to both sides. 6y + 20 = 7x

3 Divide both sides by 7. 6y + 207

= x

4 Rewrite the equation so that x is on the left-hand side. x = 6y + 20

7

WOrKed eXamPle 12

For each of the following make the variable shown in brackets the subject of the formula.a g = 6d − 3 (d)

b a = v − ut

(v)

THInK WrITe

a 1 Add 3 to both sides. a g = 6d − 3g + 3 = 6d

2 Divide both sides by 6. g + 36

= d

3 Rewrite the equation so that d is on the left-hand side. d = g + 3

6

b 1 Multiply both sides by t. b a = v − ut

at = v − u2 Add u to both sides. at + u = v3 Rewrite the equation so that v

is on the left-hand side. v = at + u




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124 Maths Quest 9

Exercise 4.6 Rearranging formulas IndIvIdual PaTHWaYS

⬛ PraCTISeQuestions:1a–f, 2a–f, 3, 6

⬛ COnSOlIdaTeQuestions:1e–h, 2e–h, 3–6, 8

⬛ maSTerQuestions:1g–l, 2g–n, 3–10

FluenCY

1 WE11 Rearrange each formula to make x the subject.a y = ax b y = ax + bc y = 2ax − b d y + 4 = 2x − 3e 6(y + 2) = 5(4 − x) f x(y − 2) = 1g x(y − 2) = y + 1 h 5x − 4y = 1i 6(x + 2) = 5(x − y) j 7(x − a) = 6x + 5ak 5(a − 2x) = 9(x + 1) l 8(9x − 2) + 3 = 7(2a −3x)

2 WE12 For each of the following, make the variable shown in brackets the subject of the formula.

a g = 4P − 3 (P) b f = 9c5

(c)

c f = 9c5

+ 32 (c) d V = IR (I)

e v = u + at (t) f d = b2 − 4ac (c)

g m = y − kh

(y) h m = y − ax − b

(y)

i m = y − ax − b

(a) j m = y − ax − b

(x)

k C = 2πr

(r) l f = ax + by (x)

m s = ut + 12at2 (a) n F = GMm

r2 (G)

underSTandIng

3 The cost to rent a car is given by the formula C = 50d + 0.2k, where d = the number of days rented and k = the number of kilometres driven. Lin has $300 to spend on car rental for her 4-day holiday. How far can she travel on this holiday?

reFleCTIOnHow does rearranging formulas differ to solving linear equations?


doc-10829doc-10829doc-10829

eles-0113eles-0113eles-0113


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4 A cyclist pumps up a bike tyre that has a slow leak. The volume of air (in cm3) after t minutes is given by the formula:

V = 24 000 − 300t

a What is the volume of air in the tyre when it is first filled?b Write an equation and solve it to work out how long it takes the tyre to go

completely flat.

reaSOnIng

5 The total surface area of a cylinder is given by the formula T = 2πr2 + 2πrh, where r = radius and h = height. A car manufacturer wants the engine’s cylinders to have a radius of 4 cm and a total surface area of 400 cm2. Show that the height of the cylinder is approximately 11.92 cm, correct to 2 decimal places. (Hint: Express h in terms of T and r.)

6 If B = 3x − 6xy, write x as the subject. Explain the process by showing all working.

PrOblem SOlvIng

7 Use algebra to show that 1v

= 1u

− 1f can also be written as u = fv

v + f.

8 Consider the formula d = "b2 − 4ac.Rearrange the formula to make a the subject.

9 Find values for a and b, such that:

4x + 1

− 3x + 2

= ax + b(x + 1)(x + 2)

10 A new game has been created by students for the school fair. To win the game you need to hit the target with 5 darts in the shaded region.

x

R

a Write an expression for the area of the shaded region.


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126 Maths Quest 9

b If R = 7.5 cm and x = 4 cm, fi nd the area of the game board, correct to 2 decimal places.

c Show that R = ÅA + x2

π by transposing the formula found in part a.

CHallenge 4.2CHallenge 4.2CHallenge 4.2CHallenge 4.2CHallenge 4.2CHallenge 4.2


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ONLINE ONLY 4.7 ReviewThe Maths Quest Review is available in a customisable format for students to demonstrate their knowledge of this topic.

The Review contains:• Fluency questions — allowing students to demonstrate the

skills they have developed to effi ciently answer questions using the most appropriate methods

• Problem Solving questions — allowing students to demonstrate their ability to make smart choices, to model and investigate problems, and to communicate solutions effectively.

A summary on the key points covered and a concept map summary of this chapter are also available as digital documents.

Review questionsDownload the Review questions document from the links found in your eBookPLUS.

www.jacplus.com.au

Link to assessON for questions to test your readiness FOr learning, your progress aS you learn and your levels OF achievement.

assessON provides sets of questions for every topic in your course, as well as giving instant feedback and worked solutions to help improve your mathematical skills.

www.assesson.com.au

int-0686int-0686int-0686

int-0700int-0700int-0700

int-3204int-3204int-3204

LanguageLanguageLanguage

algebraic equationalgebraic equationalgebraic equationalgebraic fractionalgebraic fractionalgebraic fractionalternativealternativealternativealternativealternativealternativedecomposeddecomposeddecomposeddefi nedefi nedefi neexpandexpandexpand

expressionexpressionexpressionfi xedfi xedfi xedforensic scienceforensic scienceforensic scienceformulaformulaformulainverse operationinverse operationinverse operationjustifyjustifyjustify

linear equationlinear equationlinear equationone-step equationone-step equationone-step equationsolutionsolutionsolutionsolutionsolutionsolutionsolvesolvesolvesubjectsubjectsubjecttwo-step equationtwo-step equationtwo-step equation


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128 Maths Quest 9

<InveSTIgaTIOn> FOr rICH TaSK Or <number and algebra> FOr PuZZle

Forensic science

rICH TaSK

InveSTIgaTIOn


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number and algebra

Imagine the following situation.A decomposed body was found in the bushland. A team of forensic scientists suspects that the body could be the remains of either Alice Brown or James King; they have been missing for several years. From the description provided by their Missing Persons fi le, Alice is 162 cm tall and James’ fi le indicates that he is 172 cm tall. The forensic scientists hope to identify the body based on the length of the body’s humerus.1 Complete the following tables for both males and females,

using the equations on the previous page. Calculate the body height to the nearest centimetre.

Table for males

Table for females

2 On a piece of graph paper, draw the fi rst quadrant of a Cartesian plane. Since the length of the humerus is the independent variable, place it on the x-axis. Place the dependent variable, body height, on the y-axis.

3 Plot the points from the two tables representing both male and female bodies from question 1 onto the set of axes drawn in question 2. Join the points with straight lines, using different colours to represent males and females.

4 Describe the shape of the two graphs.5 Measure the length of your humerus. Use your graph to predict your height. How accurate is the

measurement?6 The two lines of your graph will intersect if extended. At what point does this occur? Comment on this

value.

The forensic scientists measured the length of the humerus of the bone remains and found it to be 33 cm.7 Using methods covered in this activity, identify the body, justifying your decision with mathematical

evidence.


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130 Maths Quest 9

<InveSTIgaTIOn> FOr rICH TaSK Or <number and algebra> FOr PuZZle

The driest placeSolve the equations given and colour in the block containingeach answer. The letters in the remaining blocks will spell outthe puzzle’s answer.

18 – 2x = 10

3(7+ 5x ) = –9

4(15 – 3a ) = 0

–3 = 3 + 2(5 – x )

6 – 3w = –27

5x + 8 – 7x = 26

105 – 12e = 21

7 – 8f = 95

17 + 4x = 41

–1 = 5 – 2x

2(7 – 2b ) = 34

25 – 6c = 13

5 = –7 + 4f

1 – 7y = 85

8 + 3e = 2

17 + 8x = –1

5 – m3 =

7

3

M7

I5

D–4

E21

A4

S17

H2

E–20

R12

T19

I–10

N–16

Y6

C–13

H –1

I14

L–15

D9

R10

E–17

N–5

A3

T1

H–8

E16

M–3

A0

T13

C11

S–2

B–11

A18

C–6

K8

S–7

L–9

A–14

M15

A20

I–12

4

19

COde PuZZle

number and algebra


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Activities


number and algebra

4.1 Overviewvideo• The story of mathematics: The mighty Roman armies

(eles-1691)

4.2 Solving linear equationsdigital docs • SkillSHEET (doc-6150): Solving one-step equations• SkillSHEET (doc-6151): Checking solutions to equations• SkillSHEET (doc-6152): Solving equations• SkillSHEET (doc-10826): Writing equations from

worded statements• WorkSHEET 4.1 (doc-6156): Solving linear equations

4.3 Solving linear equations with bracketsdigital doc • SkillSHEET (doc-10827): Expanding brackets

4.4 Solving linear equations with pronumerals on both sidesdigital doc • SkillSHEET (doc-10828): Simplifying like terms

Interactivity• Solving equations (int-2764)

4.5 Solving problems with linear equationsdigital docs • SkillSHEET (doc-10826): Writing equations from

worded statements• WorkSHEET 4.2 (doc-6159): Solving equations with

pronumerals on both sides

4.6 rearranging formulasdigital doc • SkillSHEET (doc-10829): Transposing and

substituting into a formulaelesson• Formulas in the real world (eles-0113)

4.7 reviewInteractivities • Word search (int-0686)• Crossword (int-0700)• Sudoku (int-3204)

To access ebookPluS activities, log on to www.jacplus.com.au


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132 Maths Quest 9

number and algebra

4.2 Solving linear equations 1 a No b Yes c No d Yes e Yes f No g No h No i Yes j No k Yes l No 2 a x = 210 b x = 52 c x = 230 d x = 23 e x = 142 f x = −96 g x = 37 h x = 20 i x = 50 j x = 138 k x = 442 l x = 243 m y = −15 n y = 1.8 o y = 12.8 p y = 2 1

3

q y = 24 r y = − 118

s y = 21.5 t y = 172.5

u y = −8.32 3 a y = 5 b y = −2 c y = 0.2 d y = 1 e y = 1 f y = 2 g y = 5 1

3 h y = −1 2

3

i y = 24.5 j y = 1.2 k y = 22.25 l y = 383600

4 a x = 1 b x = −2 c x = 3 d x = −6

e x = −1 f x = 12 g x = 1 2

3 h x = −2.32

i x = 3 j x = −56 k x = 1 1

4 l x = 3 1

13 5 a x = −1 b x = 1 c x = 0 d x = 5 e x = −8.6 f x = −0.9 g x = 8.9 h x = 19 i x = −26 j x = −1 k x = −1 l x = −1

2

6 a x = 8 b x = 3 c x = 4 d x = −15 e x = 26 f x = −42 g x = 9 h x = −1 1

5

i x = 9 13 j x = −2 1

4 k x = −7 l x = 2

3

7 a z = 16 b z = 31 c z = −4 d z = 6 e z = −9 f z = −6 g z = −1.9 h z = 6.88 i z = 140 j z = 0.6 k z = −35.4 l z = 8 8 a x = 1 b x = 13 c x = −2 d x = −8 e x = −4 f x = 30 1

2 g x = −6 h x = − 7

10

i x = 10.35 j x = 0.326 k x = 22 l x = −5

9 a x = 4 b x = 37 c x = −1 1

7 d x = −6 2

3

e x = 445

f x = 8 g x = −6 h x = 7.5

i x = 5.1 j x = −6 k x = −5 1315

l x = −61 37

10 a D b D c C 11 a a = −1 b b = 10 c c = 12.1 d d = 4 e e = 5 f f = 12 g g = 15 29

30 h h = 31

i i = 13226

j j = 16 k k = −8 l l = 10.3

12 a −1 b 6 c 303 13 $91 14 No. 46.7°C ≈ 116.1°F. 15 Answers will vary. 16 The mistake is in the second line: the 1 should have been

multiplied by 5. 17 60 lollies 18 Old machine: 6640 cans; new machine: 9360 cans 19 a $34 b Yes, a saving of $7 20 17

Challenge 4.1x = −8, −2, 0, 1, 2, 3, 5, 6, 7, 8, 10, 16

4.3 Solving linear equations with brackets 1 a x = 6 b x = −3 c x = 0 d x = 56 e x = 1 f x = 0 g x = −1 h x = −2 i x = 6 j x = 0 k x = −0.8 l x = 6

2 a b = 1 16 b m = 4 1

2 c a = −1 1

2

d m = −1 13 e p = 3

5 f m = 2 2

3

g a = −1 12 h p = 1 1

2 i a = 3 1

3

j m = −2 12 k a = −1

6 l m = −1

4

3 a x = 1619 b x = 2 c a = 5 d b = 7

e y = 7 f x = −313 g m = 4 h y = 1

i a = 5 j p = −2 k m = 3 l p = 1 4 a y = −7.5 b y = −4.667 c y = −26.25 d y = 8.571 e y = −2.9 f y = 3.243 g y = 3.667 h y = 2.75 i y = 1.976 j y = −2 k y = −3.167 l y = 1.98 5 a D b C 6 1990 7 a 2(2x + 7) b Width 10 m, length 17 m 8 Answers will vary; x = 3. 9 Adding 7 to both sides is the least preferred option, as it does not

resolve the subtraction of 7 within the brackets. 10 3 11 Kyle: $90, Noah: $35 12 $10

4.4 Solving linear equations with pronumerals on both sides 1 a y = −1 b y = 1 c y = −3 d y = −5

e y = −45 f y = 827

g y = −2 h y = 20

i y = 1 j y = −115 k y = 4 l y = −2

2 a C b A 3 a x = 1 b x = −2 c x = 3 d x = 4 e x = 9 f x = 3 g x = −2 h x = 4 i x = −4 j x = 1

2 k x = 0 l x = −3

4

4 a x = −6 b x = 7 c x = 2 d x = −2 e x = 13 f x = −7 g x = 1

6 h x = − 2

17 i x = 1 5 a D b C 6 a x = 5 b x = −3 c x = −8 d x = −71

2

e x = 4 f x = 1011

g x = −18 h x = 323

i x = 23 j x = −13

7 k x = 0 l x = −15

7 −3 8 24 9 3(n − 36) − 98 = −11n + 200 10 You cannot easily divide the left-hand side by 6 or the right-hand

side by 4. 11 Daughter = 10 years, Tom = 40 years 12 The unknown number is −3. 13 a C = 40h + 250 b 18 hours, 45 minutes c 18 750 d The printing is cheaper by $1375. 14 a 65 games b $25

4.5 Solving problems with linear equations

1 a x + 3 = 5 b x − 9 = 7 c 7x = 24 d x5

= 11

e x2

= −9 f 5x − 3 = −7 g 2(14 − x) = −3

h 3x + 5 = 8 i 2x − 12 = 15 j 3x + 4

2= 5

2 a C b D c B d B 3 0

Answerstopic 4 Linear equations


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4 36 years 5 $66 6 20 × 15; 30 × 10; Area = 300 7 7 m 8 12 jackets 9 15 days 10 947 CDs 11 a Post Quick (cost = $700) b The cost is nearly the same for 333 fl yers ($366.50 and

$366.40). 12 2 13 $42.50 14 a Afenced = (5x + 20) m2

b Apath = (3x + 16) m2

c l = 8 m, w = 2 m

d 825

4.6 Rearranging formulas

1 a x =ya b x =

y − ba

c x =y + b

2a

d x =y + 7

2 e x =

8 − 6y

5 f x = 1

y − 2

g x =y + 1

y − 2 h x =

4y + 1

5 i x = −5y − 12

j x = 12a k x = 5a − 919

l x = 14a + 1393

2 a P =g + 3

4 b c =

5f

9 c c =

5(f − 32)

9

d I = VR

e t = v − ua

f c = b2 − d4a

g y = hm + k h y = m(x − b) + a

i a = y − m(x − b) j x =y − a + mb

m

k r = 2πC

l x =f − by

a m a =

2(s − ut)

t2

n G = Fr2

Mm 3 500 km 4 a 24 000 cm3 b t = 80 min = 1 h 20 min 5 Answers will vary.

6 B

3(1 − 2y) = x

7 Answers will vary.

8 a = b2 − d2

4c 9 a = 1 and b = 5 10 a A = πR2 − x2

b A = 160.71 cm2

c Answers will vary.

Challenge 4.2r = 10.608 cm

Investigation — Rich task 1 Table for males

Length of humerus l (cm) 20 25 30 35 40

Body height h (cm) 132 147 163 178 194

Table for females

Length of humerus l (cm) 20 25 30 35 40

Body height h (cm) 125 142 159 176 192

2 and 3

4 Linear 5 Answers will vary. 6 (44.6, 207.8) 7 James King

Code puzzleThe Atacama Desert in Chile


ON

LIN

E P

AG

E P

RO

OFS

Linear equations - John Wiley & Sons Maths Quest 9 number and algebra 4.2 Solving linear equations...

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Transcript of Linear equations - John Wiley & Sons Maths Quest 9 number and algebra 4.2 Solving linear equations...