Linear equations. 4.1 Overview TOPIC 4. Why learn this? What do you know? Learning sequence. number and algebra

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1 number and algebra TOPIC 4 N LY Linear equations 4.1 Overview N AT IO 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. O Why learn this? What do you know? EV AL U 1 THInK List what you know about linear equations. Use a thinking tool such as a concept map to show your list. 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 M PL E Overview Solving linear equations Solving linear equations with brackets Solving linear equations with pronumerals on both sides Solving problems with linear equations Rearranging formulas Review ONLINE ONLY SA c04linearequations.indd 10 14/0/16 10:13 AM

2 N LY O N AT IO AL U EV E M PL SA WaTCH THIS video The story of mathematics: The mighty Roman armies Searchlight Id: eles-1691 c04linearequations.indd /0/16 10:14 AM

3 4. Solving linear equations 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 WOrKed example 1 For each of the following equations, determine whether x = 10 is a solution. a x + = 6 b x + 3 = 3x 7 c x x = 9x 10 3 THInK a 1 Substitute 10 for x in the left-hand side of the equation. WrITe a LHS = x + 3 = = 1 3 = 4 Write the right-hand side. RHS = 6 3 Is the equation true? That is, does the LHS RHS left-hand side equal the right-hand side? 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 = x + 3 = (10) + 3 = 3 Substitute 10 for x in the right-hand side. RHS = 3x 7 = 3(10) 7 = 3 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 = x x = 10 (10) = = 80 Substitute 10 for x in the right-hand side. RHS = 9x 10 = 9(10) 10 = = Maths Quest 9

4 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 performing the inverse operation. The inverse operation has the effect of undoing the original operation. WOrKed example Solve each of the following linear equations. a x 79 = 13 b x + 46 = 8 c 6x = 100 d x 7 = 19 THInK WrITe a 1 79 is subtracted from x to give 13. a x 79 = 13 Apply the inverse operation by adding 79 to both sides of the equation. TI CASIO x = Write the value of x. x = 3 b 1 46 is added to x to give 8. b x + 46 = 8 Apply the inverse operation by subtracting 46 from both sides of the equation. x = Write the value of x. x = 36 c 1 6 is multiplied by x to give 100. c 6x = 100 Perform the inverse operation by dividing both x = 100 sides of the equation by Write the value of x. x = 16 3 d 1 x is divided by 7 to give 19. d x 7 = 19 Operation Inverse operation + + Perform the inverse operation by multiplying x = 19 7 both sides of the equation by 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 confirming that it will make the equation a true statement. Topic 4 Linear equations 10

5 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. WOrKed example 3 Solve the following linear equations. a y + 4 = 1 b 6 x = 8 c x 3 4 = d 3x = 6 THInK WrITe a 1 First subtract 4 from both sides. a y + 4 = 1 y = 1 4 Divide both sides by. y = 8 y = 8 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 Subtract 6 from both sides. x = Divide both sides by 1. x = x 1 = 1 4 Write the value of x. x = c 1 Add 4 to both sides. c x 3 4 = x = + 4 x 3 = 6 Multiply both sides by 3. x 3 3 = Write the value of x. x = 18 d 1 Multiply both sides by. d 3x = 6 3x = 6 3x = 30 Divide both sides by 3. 3x 3 = Write the value of x. x = Maths Quest 9

6 WOrKed example 4 Solve the following linear equations. a x + 1 = 11 b 7 x = 6.3 THInK TI a 1 All of x + 1 has been divided by. a Multiply both sides by. CASIO WrITe x + 1 = 11 x + 1 = 11 x + 1 = 3 Subtract 1 from both sides. x = 1 b 1 All of 7 x has been divided by. b 7 x = x Multiply both sides by. = x is the same as x x = Subtract 7 from both sides. 7 x 7 = x = 38. Divide both sides by 1. x = 38. 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: 4 x = 3 In order to solve this equation, we first multiply both sides of the equations by x. 4 x x = 3 x 4 = 3x 3x or = 4 The pronumeral is now in the numerator, and the equation is easy to solve. 3x = 4 3x = 8 x = 8 3 Topic 4 Linear equations 107

7 WOrKed example Solve each of the following linear equations. a 3 a = 4 b b = reflection How are linear equations defi ned? doc-610 doc-611 doc-61 doc-1086 THInK WrITe a 1 Multiply both sides by a. a 3 a = 4 3 = 4a Multiply both sides by. 1 = 4a 3 Divide both sides by 4. a = 1 4 or a = b 1 Write the equation. b b = Multiply both sides by b. = b 3 Divide both sides of the equation by. = b Exercise 4. Solving linear equations IndIvIdual PaTHWaYS PraCTISe Questions: 1a f, a l, 3a h, 4,, 6a f, 7a f, 8a f, 9a f, 10, 11, 1, 17 COnSOlIdaTe Questions: 1d i, g r, 3d i, 4,, 6d i, 7d i, 8d i, 9d l, 10 13, Individual pathway interactivity int-4489 b = 1 master Questions: 1g l, i u, 3g l, 4,, 6g l, 7g l, 8g l, 9g l, 10 1, 14 0 FluenCY 1 WE1 For each of the following equations, determine whether x = 6 is a solution. a x + 3 = 7 b x = 7 c x = 38 d 6 (x + 1) + x = 7 e = f 3 x = 9 x 7 g x + 3x = 39 h 3(x + ) = (x 4) i x + x = 9x 6 j x = (x + 1) 14 k (x 1) = 4x + 1 l x + = x + 4 WE Solve each of the following linear equations. Check your answers by substitution. a x 43 = 167 b x 17 = 3 c x + 86 = 16 d 8 + x = 81 e x 78 = 64 f 09 x = 30 g x = 18 h 60x = 100 i x = 0 x j 3 = 6 k x 17 = 6 l x 9 = Maths Quest 9

8 m y 16 = 31 n. + y = 7.3 o y 7.3 =. p 6y = 14 q 0.y = 4.8 r 0.9y = 0.0 s y = 4.3 t y 7. = 3 u y 8 = WE3a Solve each of the following linear equations. a y 3 = 7 b y + 7 = 3 c y 1 = 0 d 6y + = 8 e 7 + 3y = 10 f 8 + y = 1 g 1 = 3y 1 h 6 = 3y 1 i 6y 7 = 140 j 4.y +.3 = 7.7 k 0.4y.7 = 6. l 600y 40 = WE3b Solve each of the following linear equations. a 3 x = 1 b 3x 1 = c 4x 7 = 19 d 1 3x = 19 e 7x = f 8 x = 9 g 9 6x = 1 h x 4. = 7.4 i = 11 3x j 3 = 6x 8 k 1 = 4 4x l 3 13x = Solve each of the following linear equations. a 7 x = 8 b 8 x = 7 c x = d x = 0 e 1.3 = 6.7 x f.1 = 4. x g 9 x = 0.1 h 140 x = 11 i 30 x = 4 j = 6 x k x + 1 = l x 1 = 0 6 WE3c, d Solve each of the following linear equations. a x = 3 b x 3 = 1 c x 8 = 1 d x 3 = e x = 8 f 4 x 6 = 11 g x 3 = 6 h x = 3 i 3x 4 = 7 j 8x 3 = 6 k x = l 3x 7 10 = 1 7 WE4 Solve each of the following linear equations. a z 1 = b z + 1 = 8 c z 4 = d 6 z = 0 e 3 z z 0 = 6 f = 7 g z 4.4 = 3 h z = 1. i 140 z = 0 10 z 0.4 j = 0. k z 6 z + 6 = 4.6 l = Solve each of the following linear equations. a x + 1 = b x = 3 c 3x + 4 = x 13 d = e 4 3x = 8 f 1 x = g x 3 = 3 h 10x 4 4x +.6 = 1 i = x 0.7 j = 3.1 k 1 0.x 3x 8 =. l = Topic 4 Linear equations 109

9 9 WE Solve each of the following linear equations. a x = 1 b 3 x = 7 c 4 x = 7 d x = 3 4 e 0.4 x = 9 f 8 x = 1 g 4 x = h 6 3 x = i x = 1 6 j 3 x = 1 k 4 x = 1 0 l x = MC a The solution to the equation 8 x = 44 is: A x = 16 B x = 16 C x = 1 D x = 38 b What is the solution to the equation x 1 = 6? A x = 14.8 B x = 14.8 C x = 10 D x = 10 c What is the solution to the equation x 1 =.3? A x = 9.6 B x = 10.6 C x = 11.6 D x = 11 Solve each of the following linear equations. a 3a + 7 = 4 b b = c 4c 4.4 = 44 d d 4 67 = 0 e 3e = 10 f f 3 = 8 g 100 = 6g + 4. h h + =. i 4i 14 = j 1 j = 0 k 1 k l. = 4 l = UNDERSTANDING 1 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. 3 c Three is subtracted from x and this result is divided by 1 to give. 13 Driving lessons are usually quite expensive but a discount of $1 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 C + 3. Was he correct? REASONING 1 Santo solved the linear equation 9 = 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. x 1 = x 1 = 10 x = 11 Problem SOlving 17 Sweet-tooth Sammy goes to the corner store and buys an equal number of -cent and 30-cent lollies for a total of $16.0. How many lollies did he buy? 110 Maths Quest 9

10 NUMBER AND ALGEBRA 18 In a cannery, cans are filled by two machines that together produce 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 $ for an adult ticket, $7 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? 0 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? CHA ALLENGE 4.1 n Solving linear equations with brackets Consider the equation 3(x + ) = 18. There are two good methods for solving this equation. Method 1: First divide both sides by 3. 3(x + ) = x + = 6 x = 1 3n Method : First expand the brackets. 3(x + ) = 18 3x + 1 = 18 3x = 3 x = 1 doc-616 In this case, method 1 works well because 3 divides exactly into 18. Now try the equation 7(x + ) = 10. Topic 4 Linear equations 111

11 reflection Explain the two possible methods for solving equations in factorised form. Method 1: First divide both sides by 7. 7(x + ) = x + = 10 7 x = 4 7 Method : First expand the brackets. 7(x + ) = 10 7x + 14 = 10 7x = 4 x = 4 7 In this case, method works well because it avoids fraction addition or subtraction. Try both methods and choose the one that works best for each question. WOrKed example 6 Solve each of the following linear equations. a 7(x ) = 8 b 6(x + 3) = 7 THInK WrITe a 1 7 is a factor of 8, so divide both sides a 7(x ) = 8 by 7. 7(x ) = Add to both sides. x = 4 3 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 first. b 6(x + 3) = 7 6x + 18 = 7 Subtract 18 from both sides. 6x + 18 = x = 11 3 Divide both sides by 6. x = 11 (or 6 1) 6 Exercise 4.3 Solving linear equations with brackets IndIvIdual PaTHWaYS PraCTISe Questions: 1a f, a h, 3a f, 4a f,, 6, 8, 10 TI CASIO COnSOlIdaTe Questions: 1d i, d i, 3d i, 4d i,, 7 11 master Questions: 1g l, g l, 3g l, 4g l,, 7 1 Individual pathway interactivity int-4490 doc-1087 FluenCY 1 WE6 Solve each of the following linear equations. a (x ) = 0 b 4(x + ) = 8 c 6(x + 3) = 18 d (x 41) = 7 e 8(x + ) = 4 f 3(x + ) = 1 11 Maths Quest 9

12 g (x + 4) = 1 h 3(x ) = 1 i 7(x 6) = 0 j 6(x ) = 1 k 4(x + ) = 4.8 l 16(x 3) = 48 WE6 Solve each of the following equations. a 6(b 1) = 1 b (m 3) = 3 c (a + ) = 7 d 3(m + ) = e (p ) = 7 f 6(m 4) = 8 g 10(a + 1) = h 1(p ) = 6 i 9(a 3) = 3 j (m + 3) = 1 k 3(a + 1) = l 4(3m + ) = 3 Solve each of the following equations. a 9(x 7) = 8 b (x + ) = 14 c 7(a 1) = 8 d 4(b 6) = 4 e 3(y 7) = 0 f 3(x + 1) = 7 g 6(m + 1) = 30 h 4(y + ) = 1 i 3(a 6) = 3 j (p + 9) = 14 k 3(m 7) = 3 l (4p + ) = 18 4 Solve the following linear equations. Round the answers to 3 decimal places where appropriate. a (y + 4) = 7 b 0.3(y + 8) = 1 c 4(y + 19) = 9 d 7(y ) = e 6(y + 3.4) = 3 f 7(y ) = 8.7 g 1.(y + 3) = 10 h.4(y ) = 1.8 i 1.7(y +.) = 7.1 j 7(y + ) = 0 k 6(y + ) = 11 l (y.3) = 1.6 MC a The best first step in solving the equation 7(x 6) = 3 would be to: A add 6 to both sides B subtract 7 from both sides C divide both sides by 3 D expand the brackets b The solution to the equation 84(x 1) = 78 is closest to: A x = 9.31 B b x = 9.6 C x = D d x = UNDERSTANDING 6 In 1974 a mother was 6 times as old as her daughter. If the mother turned 0 in the year 000, 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 4 m, determine the dimensions of the playground. REASONING 8 Juanita is solving the following equation: (x 8) = 10. She performs the following operations to both sides of the equation in order: +8,. 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(x 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? Topic 4 Linear equations 113

13 int-764 PrOblem SOlvIng 10 Five times the sum of 4 and a number is equal to 3. What is the number? 11 Kyle earns $ more than Noah each week, but Callum earns three times as much as Kyle. If Callum earns $70 a week, how much do Kyle and Noah earn each week? 1 A school wishes to hire a bus to travel to a football game. The bus will take 8 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 $ % 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, first remove it from one side, as shown in the example below.worked example 7 WOrKed example 7 Solve each of the following linear equations. a y = 3y + 3 b 7x + = 4x c 3(x + 1) = 14 x d (x + 3) = 3(x + 7) THInK CASIO a 1 3y is smaller than y. Subtract 3y from both sides. WrITe a y = 3y + 3 y 3y = 3y + 3 3y y = 3 Divide both sides by. y = 3 (or 11 ) b 1 4x is smaller than 7x. Add 4x to both sides. b 7x + = 4x 7x + + 4x = 4x + 4x 11x + = Subtract from both sides. 11x + = 11x = 3 3 Divide both sides by 11. x = 3 11 c 1 Expand the bracket. c 3(x + 1) = 14 x 3x + 3 = 14 x x is smaller than 3x. Add x to both sides. TI 3x x = 14 x + x x + 3 = 14 3 Subtract 3 from both sides. x = 14 3 x = 11 4 Divide both sides by. x = Maths Quest 9

14 d 1 Expand the brackets. d (x + 3) = 3(x + 7) x + 6 = 3x + 1 x is smaller than 3x. Subtract x from both sides. x + 6 x = 3x + 1 x 6 = x Subtract 1 from both sides. 6 1 = x = x 4 Write the answer with the pronumeral written on the left-hand side. x = 1 Exercise 4.4 Solving linear equations with pronumerals on both sides IndIvIdual PaTHWaYS PraCTISe Questions: 1a f,, 3a f, 4,, 6a f, 7, 8, 11 COnSOlIdaTe Questions: 1d i,, 3d i, 4,, 6d i, 7 1 Individual pathway interactivity int-4491 master Questions: 1g l,, 3g l, 4,, 6g l, 7 14 FluenCY 1 WE7a Solve each of the following linear equations. a y = 3y b 6y = y + 7 c 10y = y 1 d + y = 3y e 8y = 7y 4 f 1y 8 = 1y g 7y = 3y 0 h 3y = 13y + 00 i y 3 = y j 6 y = 7y k 4 y = y l 6y = y MC a To solve the equation 3x + = 4 x, the first step is to: a add 3x to both sides b add to both sides C add x to both sides d subtract x from both sides b To solve the equation 6x 4 = 4x +, the first step is to: a subtract 4x from both sides b add 4x to both sides C subtract 4 from both sides d add to both sides 3 WE7b Solve each of the following linear equations. a x + 3 = 8 3x b 4x + 11 = 1 x c x 3 = 6 x d 4x = x + 3 e 3x = x + 7 f 7x + 1 = 4x + 10 g x + 3 = x h 6x + = 3x + 14 i x = x 9 j 10x 1 = x + k 7x + = x + l 1x + 3 = 7x 3 reflection Draw a diagram that could represent x + 4 = 3x + 1. doc-1088 Topic 4 Linear equations 11

15 4 Solve each of the following linear equations. a x 4 = 3x + 8 b 3x + 1 = 4x + c x + 9 = 7x 1 d x + 7 = 4x + 19 e 3x + = x 11 f 11 6x = 18 x g 6 9x = 4 + 3x h x 3 = 18x 1 i x + 13 = 1x + 3 MC a The solution to x + = x + 3 is: A x = 3 B x = 3 C x = D x = 7 b The solution to 3x 4 = 11 x is: A x = 1 B x = 7 C x = 3 D x = 6 WE7c, d Solve each of the following. a (x ) = x + b 7(x + 1) = x 11 c (x 8) = 4x d 3(x + ) = x e 6(x 3) = 14 x f 9x 4 = (3 x) g 4(x + 3) = 3(x ) h (x 1) = (x + 3) i 8(x 4) = (x 6) j 3(x + 6) = 4( x) k (x 1) = 3(x 8) l 4(x + 11) = (x + 7) UNDERSTANDING 7 Aamir s teacher gave him an algebra problem and told him to solve it. 3x + 7 = x + k = 7x + 1 Can 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? REASONING 9 Express the following information as an equation, then show that n = 9 is the solution. n 36 n n 36 0n n 10 Explain what the difficulty is when trying to solve the equation 4(3x ) = 6(4x + ) without expanding the brackets first. Problem SOlving 11 This year Tom is 4 times as old as his daughter, while in years time he will be only 3 times as old as his daughter. Find the ages of Tom and his daughter now. 1 If you multiply an unknown number by 6 and then add, the result is 7 less than the unknown number plus 1 multiplied by 3. Find the unknown number. 116 Maths Quest 9

16 13 You are investigating getting a business card printed for your new game store. A local printing company charges $0 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 1 cents per side for the first cards and then 10 cents per side for the remaining cards. Which is the cheaper option for 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 $ pinball games. Additional games cost $ each. After a player has played 0 games in a month, all further games are $1. a If Tom has $10 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 $ a game? 4. Solving problems with linear equations 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 1. b Three more than seven times a certain number is zero. c When is divided by a certain number, the answer is 4 more than the number. THInK Address: 13 The Street Melbourne VIC 3000 Phone no: WrITe a 1 Let x be the number. a x = unknown number Write x and subtract 6. This expression equals 1. x 6 = 1 Topic 4 Linear equations 117

17 b 1 Let x be the number. b x = unknown number 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 Write the term for divided by a certain number. Write the expression for 4 more than the number. 3 Write the equation. WOrKed example 9 x x + 4 x = x + 4 In a basketball game, Hao scored more points than Seve. If they scored a total of 7 points between them, how many points did each of them score? THInK WrITe 1 Define a pronumeral. Let Seve s score be x. Hao scored more than Seve. Hao s score is x +. 3 Between them they scored a total of 7 points. x + (x + ) = 7 4 Solve the equation. x + = 7 x = x = 11 Since x = 11, this is Seve s score. Write Hao s score. Hao s score = x + = 11+ = 16 6 Write the answer in words. Seve scored 11 points and Hao scored 16 points. 118 Maths Quest 9

18 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 1 The distance travelled by Elena has to be found. Define the pronumeral. 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. WrITe Let x = distance travelled (in kilometres). Total cost = x 3 Let the total cost = x = Solve the equation. 1.38x = 34.0 x = = State the solution in words. Elena travelled kilometres. Exercise 4. Solving problems with linear equations IndIvIdual PaTHWaYS PraCTISe Questions: 1 4, 7, 9, COnSOlIdaTe Questions: 1, 7 10, 1 1 Individual pathway interactivity int-449 master Questions: 1 16 reflection Why is it important to defi ne the pronumeral used when forming a linear equation to solve a problem? Topic 4 Linear equations 119

19 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. b Subtracting 9 from a certain number gives a result of 7. c Seven times a certain number is 4. d A certain number divided by gives a result of 11. e Dividing a certain number by 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, the result is 3. h When is added to three times a certain number, the answer is 8. i When 1 is subtracted from two times a certain number, the result is 1. j The sum of 3 times a certain number and 4 is divided by, which gives a result of. MC Which equation matches the following statement? a A certain number, when divided by, gives a result of 1. A x = 1 B b x = 1 C x = 1 D d x 1 = b Dividing 7 times a certain number by 4 equals 9. x A 4 = 9 B 4x 7 = 9 C 7 + x 4 = 9 D d 7x 4 = 9 c Subtracting twice a certain number from 8 gives 1. A x 8 = 1 B b 8 x = 1 C 8x = 1 D d 8 (x + ) = 1 d When 1 is added to a quarter of a number, the answer is 10. A 1 + 4x = 10 B b 10 = x C x + 1 = 10 D d 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 x the sum of their ages is 48. How old is John? In one afternoon s shopping Seedevi spent half as x + much money as Georgia, but $6 more than Amy. If the three of them spent a total of $8, how much did 30 Seedevi spend? 0 6 These rectangular blocks of land have the same area. Find the dimensions of each block, and the area. 10 Maths Quest 9

20 REASONING 7 A square pool is surrounded by a paved area that is metres wide. If the area of the paving is 7 m, what is the length of the pool? m 8 Maria is paid $11.0 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 $10 plus $30 per day. Casey s rate for his car hire was $180 plus $6 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 $100 plus $0.9 per disk. If Maya s recording studio has a budget of $100, 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 $00 plus 0 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? Topic 4 Linear equations 11

21 doc-619 PrOblem SOlvIng 1 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 $60.0. The next hour she sells herbal, 4 vegetable and 3 citrus fizz. The third hour she sells 1 herbal, vegetable and 4 citrus fizz. The total amount in cash sales for the three hours is $ 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 herbal, 3 vegetable and 4 citrus fizz. 14 A rectangular swimming pool is surrounded by a path which is Fence enclosed by a pool fence. All measurements are in metres and x + 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. x + 4 c If the area of the path surrounding the pool is 34 m, find 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 specific purpose. By now you will be familiar with many mathematical or scientific formulas. For example, C = πr 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 = πr Divide both sides by π. C π = πr π C π = r or r = C π Now r is the subject. 1 Maths Quest 9

22 WOrKed example 11 Rearrange each formula to make x the subject. a y = kx + m b 6(y + 1) = 7(x ) THInK WrITe a 1 Subtract m from both sides. a y = kx + m y m = kx Divide both sides by k. y m = kx k k 3 Rewrite the equation so that x is on the left-hand side. y m = x k x = y m k b 1 Expand the brackets. b 6(y + 1) = 7(x ) 6y + 6 = 7x 14 Add 14 to both sides. 6y + 0 = 7x 3 Divide both sides by 7. 6y + 0 = x 7 4 Rewrite the equation so that x is 6y + 0 x = on the left-hand side. 7 WOrKed example 1 WOrKed example 1 For each of the following make the variable shown in brackets the subject of the formula. a g = 6d 3 (d) b a = v u (v) t THInK TI CASIO WrITe a 1 Add 3 to both sides. a g = 6d 3 g + 3 = 6d Divide both sides by 6. g + 3 = d 6 3 Rewrite the equation so that d d = g + 3 is on the left-hand side. 6 b 1 Multiply both sides by t. b a = v u t at = v u Add u to both sides. at + u = v 3 Rewrite the equation so that v is on the left-hand side. v = at + u Topic 4 Linear equations 13

23 reflection How does rearranging formulas differ to solving linear equations? doc-1089 eles-0113 Exercise 4.6 Rearranging formulas IndIvIdual PaTHWaYS PraCTISe Questions: 1a f, a f, 3, 6 COnSOlIdaTe Questions: 1e h, e h, 3 6, 8 Individual pathway interactivity int-4493 master Questions: 1g l, g n, 3 10 FluenCY 1 WE11 Rearrange each formula to make x the subject. a y = ax b y = ax + b c y = ax b d y + 4 = x 3 e 6(y + ) = (4 x) f x(y ) = 1 g x(y ) = y + 1 h x 4y = 1 i 6(x + ) = (x y) j 7(x a) = 6x + a k (a x) = 9(x + 1) l 8(9x ) + 3 = 7(a 3x) WE1 For each of the following, make the variable shown in brackets the subject of the formula. a g = 4P 3 (P) b f = 9c (c) c f = 9c + 3 (c) d V = IR (I) e v = u + at (t) f d = b 4ac (c) g m = y k (y) h m = y a (y) h x b i m = y a (a) j m = y a (x) x b x b k C = π r (r) l f = ax + by (x) m s = ut + 1 at (a) n F = GMm r (G) understanding 3 The cost to rent a car is given by the formula C = 0d + 0.k, 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? 14 Maths Quest 9

24 4 A cyclist pumps up a bike tyre that has a slow leak. The volume of air (in cm 3 ) after t minutes is given by the formula: V = t 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 The total surface area of a cylinder is given by the formula T = πr + πrh, 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 cm. Show that the height of the cylinder is approximately 11.9 cm, correct to 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 1 v = 1 u 1 f can also be written as u = fv v + f. 8 Consider the formula d = b 4ac. Rearrange the formula to make a the subject. 9 Find values for a and b, such that: 4 x x + = ax + b (x + 1)(x + ) CHa allenge 4. Topic 4 Linear equations 1

25 16 Maths Quest 8 for Victoria Australian Curriculum edition

26 NUMBER AND ALGEBRA ONLINE ONLY 4.7 Review The 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 efficiently 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 of the key points covered and a concept map summary of this topic are available as digital documents. int-0686 int-0700 int-304 Language algebraic equation algebraic fraction alternative decomposed defi ne expand 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. expression fi xed forensic science formula inverse operation justify Review questions Download the Review questions document from the links found in your ebookplus. linear equation one-step equation solution solve subject two-step equation The story of mathematics is an exclusive Jacaranda video series that explores the history of mathematics and how mathematics helped shape the world we live in today. The mighty Roman armies (eles-1691) details how the Romans used mathematics in the creation of their armies and the formations they went to battle in to construct one of history s most powerful empires. Topic 4 Linear equations 17

27 number and algebra <InveSTIgaTIOn> InveSTIgaTIOn FOr rich TaSK Or <number and algebra> FOr PuZZle rich TaSK SA M PL E EV AL U AT IO N O N LY Forensic science 18 Maths Quest 9 c04linearequations.indd 18 14/0/16 10:1 AM

28 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 16 cm tall and James fi le indicates that he is 17 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 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. Join the points with straight lines, using different colours to represent males and females. 4 Describe the shape of the two graphs. 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. Topic 4 Linear equations 19

29 <InveSTIgaTIOn> number and algebra FOr rich TaSK Or <number and algebra> FOr PuZZle COde PuZZle The driest place Solve the equations given and colour in the block containing each answer. The letters in the remaining blocks will spell out the puzzle s answer. 18 x = 10 3(7+ x ) = e = 19 = 7 + 4f 3 = m 4 1 7y = 8 4(1 3a ) = x = = 3 + ( x ) (7 b ) = 34 6c = w = 7 x + 8 7x = 6 1 = x e = 1 7 8f = x = 41 M 7 A 3 T 1 H 8 E 16 M 3 A 0 T 13 I 1 C 11 S B 11 A 18 C 6 K 8 S 7 L 9 A 14 M 1 A 0 I D 4 E 1 A 4 S 17 H E 0 R 1 T 19 I 10 N 16 Y 6 C 13 H 1 I 14 L 1 D 9 R 10 E 17 N 130 Maths Quest 9

30 Activities 4.1 Overview video The story of mathematics: The mighty Roman armies (eles-1691) 4. Solving linear equations Interactivity IP interactivity 4. (int-4489): Solving linear equations digital docs SkillSHEET (doc-610): Solving one-step equations SkillSHEET (doc-611): Checking solutions to equations SkillSHEET (doc-61): Solving equations SkillSHEET (doc-1086): Writing equations from worded statements WorkSHEET 4.1 (doc-616): Solving linear equations 4.3 Solving linear equations with brackets Interactivity IP interactivity 4.3 (int-4490): Solving linear equations with brackets digital doc SkillSHEET (doc-1087): Expanding brackets 4.4 Solving linear equations with pronumerals on both sides Interactivities Solving equations (int-764) IP interactivity 4.4 (int-4491): Solving linear equations with pronumerals on both sides To access ebookplus activities, log on to digital doc SkillSHEET (doc-1088): Simplifying like terms 4. Solving problems with linear equations Interactivity IP interactivity 4. (int-449): Solving problems with linear equations digital doc WorkSHEET 4. (doc-619): Solving equations with pronumerals on both sides 4.6 rearranging formulas Interactivity IP interactivity 4.6 (int-4493): Rearranging formulas digital doc SkillSHEET (doc-1089): Transposing and substituting into a formula elesson Formulas in the real world (eles-0113) 4.7 review Interactivities Word search (int-0686) Crossword (int-0700) Sudoku (int-304) digital docs Topic summary (doc-1078) Concept map (doc-1079) Topic 4 Linear equations 131

31 Answers topic 4 Linear equations 4. 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 a x = 10 b x = c x = 30 d x = 3 e x = 14 f x = 96 g x = 37 h x = 0 i x = 0 j x = 138 k x = 44 l x = 43 m y = 1 n y = 1.8 o y = 1.8 p y = 1 3 q y = 4 r y = 1 18 s y = 1. t y = 17. u y = a y = b y = c y = 0. d y = 1 e y = 1 f y = g y = 1 h y = i y = 4. j y = 1. k y =. l y = a x = 1 b x = c x = 3 d x = 6 e x = 1 f x = 1 g x = 1 3 h x =.3 i x = 3 j x = k x = 1 1 l x = a x = 1 b x = 1 c x = 0 d x = e x = 8.6 f x = 0.9 g x = 8.9 h x = 19 i x = 6 j x = 1 k x = 1 l x = 1 6 a x = 8 b x = 3 c x = 4 d x = 1 e x = 6 f x = 4 g x = 9 h x = 1 1 i x = 9 1 j x = 1 k x = 7 l x = 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 = 3.4 l z = 8 8 a x = 1 b x = 13 c x = d x = 8 e x = 4 f x = 30 1 g x = 6 h x = 7 10 i x = 10.3 j x = 0.36 k x = l x = 9 a x = 4 b x = 3 7 c x = d x = 6 3 e x = 4 4 f x = 8 g x = 6 h x = 7. i x =.1 j x = 6 k x = 13 1 l x = a D b D c C 11 a a = 1 b b = 10 c c = 1.1 d d = 4 e e = f f = 1 g g = h h = 31 i i = 13 6 j j = 1 6 k k = 8 l l = a 1 b 6 c $91 14 No C F. 1 Answers will vary. 16 The mistake is in the second line: the 1 should have been multiplied by lollies 18 Old machine: 6640 cans; new machine: 9360 cans 19 a $34 b Yes, a saving of $ Challenge 4.1 x = 8,, 0, 1,, 3,, 6, 7, 8, 10, Solving linear equations with brackets 1 a x = 6 b x = 3 c x = 0 d x = 6 e x = 1 f x = 0 g x = 1 h x = i x = 6 j x = 0 k x = 0.8 l x = 6 a b = d m = g a = 1 1 b m = 4 1 e p = 3 h p = 1 1 c a = 1 1 f m = 3 i a = j m = 1 k a = 1 l m = a x = 16 1 b x = c a = d b = 7 9 e y = 7 f x = 3 1 g m = 4 h y = 1 3 i a = j p = k m = 3 l p = 1 4 a y = 7. b y = c y = 6. d y = 8.71 e y =.9 f y = 3.43 g y = h y =.7 i y = j y = k y = l y = 1.98 a D b C a [(x + 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 Kyle: $90, Noah: $3 1 $0 4.4 Solving linear equations with pronumerals on both sides 1 a y = 1 b y = 1 c y = 3 d y = e y = 4 f y = 8 7 g y = h y = 0 i y = 1 j y = 1 1 k y = 4 l y = a C b A 3 a x = 1 b x = c x = 3 d x = 4 e x = 9 f x = 3 g x = h x = 4 i x = 4 j x = 1 k x = 0 l x = a x = 6 b x = 7 c x = d x = e x = 13 f x = 7 i x = 1 g x = 1 6 h x = 17 a D b C 6 a x = b x = 3 c x = 8 d x = 7 1 e x = 4 f x = g x = 18 h x = 3 3 i x = 3 j x = k x = 0 l x = (n 36) 98 = 11n You cannot easily divide the left-hand side by 6 or the right-hand side by Daughter = 10 years, Tom = 40 years 1 The unknown number is a C = 40h + 0 b 18 hours, 4 minutes c d The printing is cheaper by $ a 6 games b $ 4. Solving problems with linear equations 1 a x + 3 = b x 9 = 7 c 7x = 4 d x e = 9 f x 3 = 7 g (14 x) = 3 3x + 4 h 3x + = 8 i x 1 = 1 j = a C b D c B d B 3 0 x = Maths Quest 9

32 4 36 years $ ; 30 10; Area = m 8 1 jackets 9 1 days CDs 11 a Post Quick (cost = $700) b The cost is nearly the same for 333 flyers ($366.0 and $366.40) $ a A fenced = (x + 0) m b A path = (3x + 16) m c l = 8 m, w = m d Rearranging formulas 1 a x = y b x = y b c x = y + b a a a d x = y + 7 e x = 8 6y f x = 1 y g x = y + 1 h x = 4y + 1 i x = y 1 y j x = 1a k x = a 9 14a + 13 l x = a P = g + 3 b c = f (f 3) c c = d I = V e t = v u f c = b d R a 4a g y = hm + k h y = m(x b) + a i a = y m(x b) j x = y a + mb m k r = π l x = f by (s ut) m a = C a t n G = Fr Mm 3 00 km 4 a cm 3 b t = 80 min = 1 h 0 min Answers will vary. B 6 3(1 y) = x 7 Answers will vary. 8 a = b d 4c 9 a = 1 and b = Challenge 4. r = cm Investigation Rich task 1 Table for males Length of humerus l (cm) Body height h (cm) Table for females and 3 Length of humerus l (cm) Body height h (cm) Body height (cm) 10 h = 3.08l (males) h = 3.36l (females) Linear Answers will vary. 6 (44.6, 07.8) 7 James King Length of humerus (cm) Code puzzle The Atacama Desert in Chile Topic 4 Linear equations 133