NUMBER AND ALGEBRA TOC PI 4 Linear equations PROOFS...104 Jacaranda Maths Quest 9 Victorian...

TOPIC 4 Linear equations 101

c04LinearEquations.indd Page 101 22/06/17 5:34 AM

NUMBER AND ALGEBRA

TOPIC 4Linear equations

4.1 Overview Numerous videos and interactivities are embedded just where you need them, at the point of learning, in your learnON title at www.jacplus.com.au . They will help you to learn the concepts covered in this topic.

4.1.1 Why learn this? Looking for patterns in numbers, relationships and meas-urements 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.

4.1.2 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 SEQUENCE 4.1 Overview 4.2 Solving linear equations 4.3 Solving linear equations with brackets 4.4 Solving linear equations with pronumerals on both sides 4.5 Solving problems with linear equations 4.6 Rearranging formulas 4.7 Review

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102 Jacaranda Maths Quest 9 Victorian Curriculum


4.2 Solving linear equations4.2.1 What is a linear equation?

• An equation is a mathematical statement that contains an equals 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 finding 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 + 23

= 10 + 23

= 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

3 Is the equation true? LHS = RHS

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

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4.2.2 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 performing the inverse

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

operation.

Operation Inverse operation

+ −− +× ÷÷ ×

WORKED EXAMPLE 2 TI | CASIO

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

7= 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

3 Write the value of x. x = 16 23

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. = 33

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

4.2.3 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, determine 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.

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WORKED EXAMPLE 3

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 = 82y2

= 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 − 6

3 Divide both sides by −1.

−x = 2−x−1

= 2−1

4 Write the value of 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



Solve the following linear equations.

a x + 1

2= 11 b 7 − x

5= −6.3

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4.2.4 Algebraic fractions with the pronumeral in the denominator • If a pronumeral is in the denominator, there is an extra step involved in finding the solution.

Consider the following example:4x

= 32

In order to solve this equation, we first 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

THINK WRITE

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

2= 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

WORKED EXAMPLE 5

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

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Exercise 4.2 Solving linear equationsIndividual pathways

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

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

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

U U U Individual pathway interactivity: int-4489 ONLINE ONLY

To answer questions online and to receive immediate feedback and sample responses for every question, go to your learnON title at www.jacplus.com.au. Note: Question numbers may vary slightly.

Fluency1. 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

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 = −2 12

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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 = 516 d. 58 + x = 81e. x − 78 = 64 f. 209 − x = 305 g. 5x = 185 h. 60x = 1200

i. 5x = 250 j. x23

= 6 k. x17

= 26 l. x9

= 27

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. y

5= 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 = 0 d. 6y + 2 = 8e. 7 + 3y = 10 f. 8 + 2y = 12 g. 15 = 3y − 1 h. −6 = 3y − 1i. 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 = −19 d. 1 − 3x = 19e. −5 − 7x = 2 f. −8 − 2x = −9 g. 9 − 6x = −1 h. −5x − 4.2 = 7.4i. 2 = 11 − 3x j. −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 = 5 d. 5 − x = 0e. 15.3 = 6.7 − x f. 5.1 = 4.2 − x g. 9 − x = 0.1 h. 140 − x = 121i. −30 − x = −4 j. −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 − z7

= 0

e. 3 − z

2= 6 f.

−z − 5022

= −2 g. z − 4.4

2.1= −3 h.

z + 27.4

= 1.2

i. 140 − z

150= 0 j.

−z − 0.42

= −0.5 k. z − 6

9= −4.6 l.

z + 6573

= 1

8. Solve each of the following linear equations.

a. 5x + 1

3= 2 b. 2x − 5

7= 3 c.

3x + 42

= −1 d. 4x − 139

= −5

e. 4 − 3x2

= 8 f. 1 − 2x6

= −10 g. −5x − 39

= 3 h. −10x − 43

= 1

i. 4x + 2.6

5= 8.8 j. 5x − 0.7

−0.3= −3.1 k. 1 − 0.5x

4= −2.5 l. −3x − 8

14= 1

29. 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

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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 = 211. Solve each of the following linear equations.

a. 3a + 7 = 4 b. 5 − b = −5 c. 4c − 4.4 = 44 d. d − 467

= 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 − k

5= 4 l. l − 5.2

3.4= 1.5

Understanding12. 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 23.

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 = 95C + 32. Was he correct?

Reasoning15. 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 = 10x = 11

Problem solving17. Sweet-tooth Sammy goes to the corner store and buys an equal number

of 25-cent and 30-cent lollies for a total of $16.50. How many lollies did he buy?

18. In a cannery, cans are filled by two machines that 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?

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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

= 183

x + 5 = 6x = 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.

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 each question.


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

REFLECTIONHow are linear equations defined?

CHALLENGE 4.1The value of the expression 12

x − 4 is an integer. What are the possible values for x, given that x is also an

integer?

185n + 2

3n – 6

150

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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

72 Add 5 to both sides. x − 5 = 4


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

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 −1 56)

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Exercise 4.3 Solving linear equations with brackets

Individual pathways

U PRACTISEQuestions:1a–f, 2a–h, 3a–f, 4a–f, 5, 6, 8, 10

U CONSOLIDATEQuestions:1d–i, 2d–i, 3d–i, 4d–i, 5, 7–11

U MASTERQuestions:1g–l, 2g–l, 3g–l, 4g–l, 5, 7–12



Fluency1. WE6 Solve each of the following linear equations.

a. 5(x − 2) = 20 b. 4(x + 5) = 8 c. 6(x + 3) = 18 d. 5(x − 41) = 75e. 8(x + 2) = 24 f. 3(x + 5) = 15 g. 5(x + 4) = 15 h. 3(x − 2) = −12i. 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) = 2e. 5(p − 2) = −7 f. 6(m − 4) = −8 g. −10(a + 1) = 5 h. −12(p − 2) = 6i. −9(a − 3) = −3 j. −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) = 28 d. 4(b − 6) = 4e. 3(y − 7) = 0 f. −3(x + 1) = 7 g. −6(m + 1) = −30 h. −4(y + 2) = −12i. −3(a − 6) = 3 j. −2(p + 9) = −14 k. 3(2m − 7) = −3 l. 2(4p + 5) = 18

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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) = −29 d. 7(y − 5) = 25e. 6(y + 3.4) = 3 f. 7(y − 2) = 8.7 g. 1.5(y + 3) = 10 h. 2.4(y − 2) = 1.8i. 1.7(y + 2.2) = 7.1 j. −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 sides B. subtract 7 from both sides C. divide both sides by 23 D. expand the brackets b. The solution to the equation 84(x − 21) = 782 is closest to: A. x = 9.31 B. x = 9.56 C. x = 30.31 D. x = −11.69

Understanding6. 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.

Reasoning8. 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?

Problem solving10. 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. The price of each ticket is $10. If the hiring of the bus is $300 + 10% of the cost of all the tickets, what should be the cost per person?

ReflectionExplain the two possible methods for solving equations in factorised form.

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, first remove it from one side, as shown in the example below.

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diacriTech

Highlight




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 1 12)

b 1 −4x is smaller than 7x. Add 4x to both sides. b 7x + 5 = 2 − 4x7x + 5 + 4x = 2 − 4x + 4x

11x + 5 = 2

2 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

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 − 2x6 = 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

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Exercise 4.4 Solving linear equations with pronumerals on both sides

Individual pathways

U PRACTISEQuestions:1a–f, 2, 3a–f, 4, 5, 6a–f, 7, 8, 11

U CONSOLIDATEQuestions:1d–i, 2, 3d–i, 4, 5, 6d–i, 7–12

U MASTERQuestions:1g–l, 2, 3g–l, 4, 5, 6g–l, 7–14



Fluency1. WE7a Solve each of the following linear equations.

a. 5y = 3y − 2 b. 6y = −y + 7 c. 10y = 5y − 15 d. 25 + 2y = −3ye. 8y = 7y − 45 f. 15y − 8 = −12y g. 7y = −3y − 20 h. 23y = 13y + 200i. 5y − 3 = 2y j. 6 − 2y = −7y k. 24 − y = 5y l. 6y = 5y − 2

2. MC a. To solve the equation 3x + 5 = −4 − 2x, the first step is to:A. add 3x to both sides B. add 5 to both sidesC. add 2x to both sides D. subtract 2x from both sides

b. To solve the equation 6x − 4 = 4x + 5, the first step is to:A. subtract 4x from both sides B. add 4x to both sidesC. subtract 4 from both sides D. 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 − 2x d. 4x − 5 = 2x + 3e. 3x − 2 = 2x + 7 f. 7x + 1 = 4x + 10 g. 5x + 3 = x − 5 h. 6x + 2 = 3x + 14i. 2x − 5 = x − 9 j. 10x − 1 = −2x + 5 k. 7x + 2 = −5x + 2 l. 15x + 3 = 7x − 3

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

5. MC a. The solution to 5x + 2 = 2x + 23 is: A. x = 3 B. x = −3 C. x = 5 D. x = 7 b. The solution to 3x − 4 = 11 − 2x is: A. x = 15 B. x = 7 C. x = 3 D. x = 56. WE7c, d Solve each of the following.

a. 5(x − 2) = 2x + 5 b. 7(x + 1) = x − 11 c. 2(x − 8) = 4xd. 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)

Understanding7. Aamir’s teacher gave him an algebra problem and told him to solve it.

3x + 7 = x2 + k = 7x + 15Can you help him find 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?

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Reasoning9. Express the following information as an equation, then show that n = 29 is the solution.

150 – 31n20n + 50

n – 36

–98

n – 36

n – 36

10. Explain what the difficulty is when trying to solve the equation 4(3x − 5) = 6(4x + 2) without

expanding the brackets first.

Problem solving11. 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.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.

Address: 123 The StreetMelbourneVIC 3000

Phone no: 03 1234 5678

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 first 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, compared to

playing the same number of games at $2 a game?

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

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4.5 Solving problems with linear equations4.5.1 Converting 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.

WORKED EXAMPLE 8

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

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

WORKED EXAMPLE 9

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?

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WORKED EXAMPLE 10

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. Define 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 fixed 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.

THINK WRITE

1 Define 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 = 272x = 22x = 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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Exercise 4.5 Solving problems with linear equations

Individual pathways

U PRACTISEQuestions:1–4, 7, 9, 11–14

U CONSOLIDATEQuestions:1–5, 7–10, 12–15

U MASTERQuestions:1–16

U U U Individual pathway interactivity: int-4492


Fluency1. 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 five 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 = 12 C. 2 − 8x = 12 d. 8 − (x + 2) = 12d. 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 +

4x

= 10

Understanding3. 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?

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

x + 5

20

x

30

Reasoning7. 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?

2 m

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 for his car hire was $180 plus

$26 per day. If their final cost and rental period were 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?

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Problem solving12. 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 fizz. One hour she sells 4 herbal, 3 vegetable and 6 citrus fizz for $60.50. The next hour she sells 2 herbal, 4 vegetable and 3 citrus fizz. The third hour she sells 1 herbal, 2 vegetable and 4 citrus fizz. 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 fizz.

14. A rectangular swimming pool is surrounded by a path which is enclosed by a pool fence. All measurements are in metres and are not to scale in the diagram shown.a. Write an expression for the area of the entire fenced-off section.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, find the dimensions of

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

ReflectionWhy is it important to define the pronumeral used when forming a linear equation to solve a problem?

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 opera-

tions 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.

4.6.1 Rearranging (transposing) formulas • A formula is simply an equation that is used for some specific purpose. By now you will be familiar

with many mathematical or scientific 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 (rear-ranged) to make r the subject.

C = 2πr Divide both sides by 2π.C2π

= 2πr2π

C2π

= r

or r = C2π

Now r is the subject.

Fence

52

x + 2

x + 4

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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 − m

k= kx

ky − m

k= x

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

x = y − m

k

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 + 207

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 + 36

b 1 Multiply both sides by t. b

a = v − u

tat = v − u

2 Add u to both sides. at + u = v

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

v = at + u

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Exercise 4.6 Rearranging formulas

Individual pathways

U PRACTISEQuestions:1a–f, 2a–f, 3, 6

U CONSOLIDATEQuestions:1e–h, 2e–h, 3–6, 8

U MASTERQuestions:1g–l, 2g–n, 3–10

U U U Individual pathway interactivity: int-4493


Fluency1. WE11 Rearrange each formula to make x the subject.

a. y = ax b. y = ax + b c. y = 2ax − bd. y + 4 = 2x − 3 e. 6(y + 2) = 5(4 − x) f. x(y − 2) = 1g. x(y − 2) = y + 1 h. 5x − 4y = 1 i. 6(x + 2) = 5(x − y)j. 7(x − a) = 6x + 5a k. 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 − k

h (y) h. m = y − a

x − b (y) i. m = y − a

x − b (a)

j. m = y − a

x − b (x) k. C = 2π

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

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

r2 (G)

Understanding3. 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?

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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.

Reasoning5. 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)

ReflectionHow does rearranging formulas differ to solving linear equations?

CHALLENGE 4.2The volume, V, of a sphere can be calculated using the formula V = 4

3π r

3, where r is the radius of the sphere. What is the radius of a spherical ball that has the capacity to hold 5 litres of water?

4.7 Review4.7.1 Review questionsFluency

1. The linear equation represented by the sentence ‘When a certain number is multiplied by 3, the result is 5 times the certain number plus 7’ is:

a. 3x + 7 = 5x b. 5(x + 7) = 3x C. 5x + 7 = 3x d. 5x + 3x = 7

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2. The solution to the equation x3

= 5 is:

a. x = −15 b. x = 15 C. x = 1 23

d. x = 3

3. What is the solution to the equation 7 + 21 = x?a. x = 28 b. x = −28 C. x = −14 d. x = 14

4. What is the solution to the equation 5x + 3 = 37?a. x = 8 b. x = −8 C. x = 6.8 d. x = 106

5. The solution to the equation 8 − 2x = 22 is:a. x = 11 b. x = 15 C. x = −15 d. x = −7

6. The solution to the equation 4x + 3 = 7x − 33 is:a. x = −12 b. x = 12 C. x = 36

11d. x = 30

117. The solution to the equation 7(x − 15 = 28) is:

a. x = 11 b. x = 19 C. x = 20 d. x = 6.148. When rearranging y = ax + b in terms of x, we obtain:

a. x = y − a

bb. x = y − b

aC. x = b − y

ad. x = y + b

a9. Which of the following are linear equations?

a. 5x + y2 = 0 b. 2x + 3 = x − 2 c. x2

= 3

d. x2 = 1 e. 1x

+ 1 = 3x f. 8 = 5x − 2

g. 5(x + 2) = 0 h. x2 + y = −9 i. r = 7 − 5(4 − r)10. Solve each of the following linear equations.

a. 3a = 8.4 b. a + 2.3 = 1.7 c. b21

= −0.12

d. b − 1.45 = 1.65 e. b + 3.45 = 0 f. 7.53b = 5.6411. Solve each of the following linear equations.

a. 2x − 37

= 5 b. 5 − x2

= −4 c. −3x − 45

= 3

d. 6x

= 5 e. 4x

= 35

f. x + 1.7

2.3= −4.1

12. Solve each of the following linear equations.a. 5(x − 2) = 6 b. 7(x + 3) = 40 c. 4(5 − x) = 15d. 6(2x + 3) = 1 e. 4(x + 5) = 2x − 5 f. 3(x − 2) = 7(x + 4)

13. Liz has a packet of 45 Easter eggs. She saves 21 to eat tomorrow but rations the remainder so that she can eat 8 eggs each hour.a. Write a linear equation in terms of the number of hours, h, to represent this situation.b. Work out how many hours it will take to eat today’s share.

14. Solve each of the following linear equations.a. 11x = 15x − 2 b. 3x + 4 = 16 − x c. 5x + 2 = 3x + 8d. 8x − 9 = 7x − 4 e. 2x + 5 = 8x − 7 f. 3 − 4x = 6 − x

15. Translate these sentences into algebraic equations.Use x for the certain number.a. Twice a certain number is equal to 3 minus that certain number.b. When 8 is added to 3 times a certain number, the result is 19.c. Multiplying a certain number by 6 equals 4.d. Dividing 10 by a certain number is one more than dividing that number by 6.e. Multiply a certain number by 2, then add 5. Multiply this result by 7. This expression equals 0.f. Twice the distance travelled is 100 metres more than the distance travelled plus 50 metres.

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16. Samuel decides to go on a holiday. He travels a certain distance on the first day, twice that distance on the second day, three times that distance on the third day and four times that distance on the fourth day. If his total journey is 2000 km, how far did he travel on the third day?

17. For each of the following, make the variable shown in brackets the subject of the formula.a. y = 6x − 4 (x) b. y = mx + c (x) c. q = 2(P − 1) + 2r (P)

d. P = 2l + 2w (w) e. v = u + at (a) f. s = (u + v

2 )t (t)

g. v2 = u2 + 2as (a) h. 2A = h(a + b) (b)

Problem solving18. John is comparing two car rental companies, Golden Ace Rental Company and Silver Diamond Rental

Company. Golden Ace Rental Company charges a flat rate of $38 per day and $0.20 per kilometre. The Silver Diamond Rental Company charges a flat rate of $30 per day plus $0.32 per kilometre. John plans to rent a car for three days.a. Write an algebraic equation for the cost of renting a car for three days from the Golden Ace Rental

Company in terms of the number of kilometres travelled, k.b. Write an algebraic equation for the cost of renting a car for three days from the Silver Diamond

Rental Company in terms of the number of kilometres travelled, k.c. How many kilometres would John have to travel so that the cost of hiring from each company is the

same?19. Frederika has $24 000 saved for a holiday and a new stereo. Her travel expenses are $5400 and her

daily expenses are $260.a. Write down an equation for the cost of her holiday if she stays for d days.

Upon her return from holidays, Frederika wants to purchase a new stereo system that will cost her $2500.

b. How many days can she spend on her holiday if she wishes to purchase a new stereo upon her return?

20. A maker of an orange drink purchases her raw materials from two sources. The first source provides liquid with 6% orange juice, while the second source provides liquid with 3% orange juice. She wishes to make 1 litre of drink with 5% orange juice. Let x = amount of liquid (in litres) purchased from the first source.a. Write an expression for the amount of orange juice from the first supplier, given that x is the amount

of liquid.b. Write an expression for the amount of liquid from the second supplier, given that x is the amount of

liquid used from the first supplier.c. Write an expression for the amount of orange juice from the second supplier.d. Write an equation for the total amount of orange juice in the mixture of the 2 supplies, given that

1 litre of drink is mixed to contain 5% orange juice.e. How much of the first supplier’s liquid should she use?

21. Rachel, the bushwalker, goes on a 4 day journey. She travels a certain distance on the first day, half that distance on the second day, a third that distance on the third day and a fourth of that distance on the fourth day. If the total journey is 50 km, how far did she walk on the first day?

22. Svetlana, another bushwalker goes on a 5−day journey, using the same pattern as Rachel in the previous question (a certain amount, then half that amount, then one third, one fourth and one fifth). If her journey is also 50 km, how far did she travel on the first day?

23. Nile.com, the Internet bookstore, advertises its shipping cost to Australia as a flat rate of $20 for up to 10 books; while Sheds & Meager, their competitor, offers a rate of $12 plus $1.60 per book.For how many books (6, 7, 8, 9 or 10) is Nile.com’s cost a better deal?

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Language It is important to learn and be able to use correct mathematical language in order to communicate effectively. Create a summary of the topic using the key terms below. You can present your summary in writing or using a concept map, a poster or technology. algebraic equation algebraic fraction alternative decomposed defi ne expand

expression fi xed forensic science formula inverse operation justify

linear equation one-step equation solution solve subject two-step equation

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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

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Investigation 1 Rich Task Forensic science Studies have been conducted on the relationship between the height of a human and measurements of a variety of body parts. One such study relates the height of a person to the length of the upper arm bone (humerus). The relationships are different for (1) males and females and (2) for different races. Let us consider the relationships for white adult Australians.

For males, h = 3.08l + 70.45 , and for females, h = 3.36l + 57.97 , where h represents the body height in centimetres and l the length of the humerus in centimetres.

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

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

Body height h (cm)

Table for females


Body height h (cm)

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.

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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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AnswersTopic 4 Linear equationsExercise 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 13

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 13

h. y = −1 23

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 23

h. x = −2.32 i. x = 3 j. x = −56

k. x = 1 14

l. x = 3 113

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 = −12

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 15

i. x = 9 13

j. x = −2 14

k. x = −7 l. x = 23

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 12

g. x = −6 h. x = − 710

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

9. a. x = 4 b. x = 37

c. x = −1 17

d. x = −6 23

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 2930

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

Exercise 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

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2. a. b = 1 16

b. m = 4 12

c. a = −1 12

d. m = −1 13

e. p = 35

f. m = 2 23

g. a = −1 12

h. p = 1 12

i. a = 3 13

j. m = −2 12

k. a = −16

l. m = −14

3. a. x = 16 19

b. x = 2 c. a = 5 d. b = 7 e. y = 7 f. x = −3 13

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)] m 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. $20

Exercise 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 = 8

27

g. y = −2 h. y = 20 i. y = 1 j. y = −1 15

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 = 12

k. x = 0 l. x = −34

4. a. x = −6 b. x = 7 c. x = 2 d. x = −2 e. x = 13

f. x = −7 g. x = 16

h. x = − 217

i. x = 1

5. a. D b. C

6. a. x = 5 b. x = −3 c. x = −8 d. x = −7 12

e. x = 4 f. x = 1011

g. x = −18 h. x = 3 23

i. x = 23

j. x = −1 37

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

Exercise 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

4. 36 years

5. $66

6. 20 × 15; 30 × 10; Area = 300

7. 7 m

8. 12 jackets

9. 15 days

10. 947 CDs

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11. a. Post Quick (cost = $700)

b. The cost is nearly the same for 333 flyers ($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

Exercise 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 = 1y − 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 = V

R e. t = v − u

a

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 − bya

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


6. B

3(1 − 2y)= x


8. a = b2 − d2

4c 9. a = 1 and b = 5

Challenge 4.2r = 10.608 cm

4.7 Review1. C 2. B 3. C 4. C 5. D 6. B 7. B 8. B

9. b, c, f, g, i

10. a. a = 2.8 b. a = −0.6 c. b = −2.52 d. b = 3.1 e. b = −3.45 f. b = 0.749

11. a. x = 19 b. x = 13 c. x = −6 13

d. x = −1 512

e. x = 6 23

f. x = −11.13

12. a. x = 3 15

b. x = 2 57

c. x = 1 14

d. x = −1 512

e. x = −12 12

f. x = −8 12

13. a. 8h + 21 = 42 b. 3 hours

14. a. x = 12

b. x = 3 c. x = 3 d. x = 5 e. x = 2 f. x = −1

15. a. 2x = 3 − x b. 3x + 8 = 19 c. 6x = 4 d. 10x

− 1 = x6

e. 7(2x + 5) = 0 f. 2x − 100 = x + 50

16. 600 km

17. a. x =y + 4

6 b. x =

y − cm

c. P =q − 2r

2+ 1 d. w = P − 2l

2 e. a = v − u

t f. t = 2s

u + v

g. a = v2 − u2

2s h. b = 2A − ah

h 18. a. CG = 114 + 0.20k b. CS = 90 + 0.32k c. 200 km

19. a. 5400 + 260d = CH b. 61 days

20. a. 0.06x b. (1 − x) c. 0.03(1 − x)

d. 0.06x + 0.03(1 − x) = 0.05 e. 0.667 or 66.7%

21. 24 km

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22. 21 123137

≈ 21.9 km

23. 6, 7, 8, 9 or 10 books

Investigation — Rich task1. Table for males


Body height h (cm) 132 147 163 178 194

Table for females


Body height h (cm) 125 142 159 176 192

2 and 3

90

0 5 10 15 20 25Length of humerus (cm)

30 35 40 45

100110120130140150

Bod

y he

ight

(cm

)

160170180190200210

h = 3.08l + 70.45 (males)

h = 3.36l + 57.97 (females)

4. Linear


6. (44.6, 207.8)

7. James King

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