n Wordbank n Chapter outline ustralian Curriculum

Transcription

1 Number and algebra 10 Equations One of the most common ways to solve comple practical problems is to use equations. By relating the various aspects of a problem using variables, we can often find the best solution, or set of solutions. Equations are used in numerous professions and trades, from accountancy to zoology. In this chapter, we will learn the techniques involved in solving equations.

2 N E W C E N T U R Y M AT H S for the A n Chapter outline One-step equations 10-0 Two-step equations 10-0 Equations with variables on both sides Equations with brackets Simple quadratic equations ¼ c Equation problems ustralian Curriculum 8 n Wordbank Proficiency strands U F R U F R U F R U U F F U F PS backtracking method A method of solving equations by undoing, or performing inverse (opposite) operations in reverse order balancing method A method of solving equations by using the same operations on both sides of the equation R R C equation A mathematical statement that two quantities are equal, involving algebraic epressions and an equals sign (¼) R C inverse operation An opposite used in solving an equation, for eample, the inverse operation of multiplying is adding solve (an equation) To find the value of an unknown variable in an equation solution The answer to an equation or problem, the correct value(s) of the variable that makes an equation true substitute To replace a variable with a number variable A quantity that can take on different values, represented by a symbol such as a letter of the alphabet

3 Chapter Equations Video tutorial Solving equations MAT08NAVT0001 n In this chapter you will: solve linear equations using algebraic techniques verify solutions by substitution solve real-life problems by using pronumerals to represent unknowns solve simple quadratic equations of the form ¼ c SkillCheck Worksheet StartUp assignment 10 MAT08NAWK Find the number represented by h each time. a h þ ¼ 10 b 4 h ¼ 8 c 1 h ¼ 4 d h 4 ¼ 9 e 9 þ h ¼ 15 f h 7 ¼ 16 g h 6 ¼ 0 h 4 4 h ¼ 8 i 10 þ h ¼ 0 j h 5 ¼ 0 k 1 h ¼ 4 l h 4 5 ¼ 4 Evaluate each epression. a 5 8 b 8 þ c ( 4) d 6 4 e þ 5 f 9 g 4 ( 9) h 1 4 ( 4) i 7 j 80 4 ( 10) k 6 15 l 8 9 If d ¼ 7, evaluate: a d þ b d 4 c d þ 1 d 18 d 4 If ¼, evaluate: a 5 þ 9 b 4 c 1 d 8 5 Simplify each epression. a n n b 8 þ c 5r r d 6t þ t 6 Epand each epression. a (m þ ) b ( ) c 4(k þ 7) d 5(d 1) e 6(a ) f 4(b þ 8) g 5(q 6) h 9(5j þ 1) 7 What is the opposite operation to: a adding? b dividing? c multiplying? d subtracting? Worksheet Equations 1 MAT08NAWK10091 Puzzle sheet Equations dominoes MAT08NAPS One-step equations An equation is a statement involving a variable (such as ), numbers and an equals (¼) sign, for eample, 5 ¼ 11. In Year 7 we learnt how to solve equations, that is, to find the value of the variable that makes the equation true. This value is called the solution to the equation. There are two algebraic methods for solving equations. The balancing method involves representing both sides of an equation as balance scales, and solving the equation by performing the same operation on both sides to keep it balanced. 90

4 NEW CENTURY MATHS for the Australian Curriculum8 The backtracking method involves using a flow chart to show what operations are performed on the variable (say, ) to create the equation, then using a reverse flow chart to undo each operation by performing the inverse (opposite) operation in reverse order. Summary To solve an equation, aim to have the variable (such as ) on one side of the equation and a number on the other side, in the form: Puzzle sheet One-step equations MAT08NAPS00046 Worksheet One-step equations MAT08NAWK0007 ¼ Check your solution by substituting it back into the equation. Operation Inverse operation þ þ 4 4 Eample 1 Solve each equation. a m þ 4 ¼ 1 b y ¼ 18 c k ¼ 5 d t 8 ¼ 6 Solution a Method 1: The balancing method m m þ 4 ¼ 1 m þ 4 4 ¼ 1 4 m Subtracting 4 from both sides. Check: 8 þ 4 ¼ 1. m ¼ 8 Method : The backtracking method + 4 m m + 4 = = m þ 4 ¼ 1 m ¼ 1 4 Undo þ 4 by subtracting 4. m ¼ 8 Check: 8 þ 4 ¼ 1. 91

5 Chapter Equations b Method 1: The balancing method y Place the 18 balls in three equal rows y y y ¼ 18 y ¼ 18 y Dividing both sides by. Check: 6 ¼ 18 y ¼ 6 Method : The backtracking method y y = = 6 18 y ¼ 18 y ¼ 18 Undo in y by dividing by. y ¼ 6 Check: 6 ¼ 18 These are all one-step equations because they require only one step or operation to solve. c Method 1: The balancing method k ¼ 5 k þ ¼ 5 þ Adding to both sides. k ¼ 8 Check: 8 ¼ 5 Method : The backtracking method k ¼ 5 k ¼ 5 þ Undo by adding. k ¼ 8 Check: 8 ¼ 5 d Method 1: The balancing method t 8 ¼ 6 t 8 ¼ 6 8 Multiplying both sides by 8. 8 t ¼ 48 Check: 48 8 ¼ 6 9

6 NEW CENTURY MATHS for the Australian Curriculum8 Method : The backtracking method t 8 ¼ 6 t ¼ 6 8 Undo 4 8 by multiplying by 8. t ¼ 48 Check: 48 8 ¼ 6 Eercise One-step equations 1 Solve each equation, showing the working. a þ 5 ¼ 1 b y þ 1 ¼ 6 c m þ 9 ¼ 1 d k þ 7 ¼ 54 e k þ 6 ¼ f s þ ¼ 1 g n þ 9 ¼ 4 h p þ 17 ¼ i m þ 8 ¼ 0 j g þ 4.5 ¼ 0 k a þ 1 4 ¼ 1 l k þ.7 ¼ 5 Solve each equation, showing the working. a t 11 ¼ 40 b w 7 ¼ 4 c a 5 ¼ d j ¼ 51 e q 1 ¼ 17 f f 4 ¼ 68 g 9 ¼ 4 h g 10 ¼ 1 i j ¼ 5 j w 9 ¼ k m 1 ¼ 7 l d 1 ¼ 1 m y 17 ¼.9 n n.1 ¼ 5.9 o b 4 5 ¼ 1 p s 1 4 ¼ 4 1 Solve each equation, showing the working. a 6 ¼ 4 b 4g ¼ 16 c 10y ¼ 90 d 5b ¼ 45 e 1d ¼ 108 f 9c ¼ 81 g p ¼ 115 h m ¼ 6 i 5 ¼ 5 j 7q ¼ 4 k c ¼ 54 l 15 ¼ 10 m 4p ¼ 7 n 8r ¼ 18 o q ¼ 10 p 9w ¼ 1 4 Solve each equation, showing the working. a s ¼ 5 b m 7 ¼ 6 c d 10 ¼ 1 d w ¼ 9 e a ¼ 4 r f ¼ 9 g r 5 ¼ h m 7 ¼ 1 i 8 ¼ 6 j t ¼ k n 4 4 ¼ 4 l ¼ Which of the following is the solution to ¼ 1? Select the correct answer A, B, C or D. A ¼ 9 B ¼ 6 C ¼ 4 D ¼ 15 6 Write an equation whose solution is: a m ¼ 4 b ¼ 0 c r ¼ 7 d d ¼ Two-step equations Two-step equations require two steps or operations to solve. See Eample 1 Worked solutions Eercise MAT08NAWS1007 Skillsheet Solving equations by balancing MAT08NASS1005 Skillsheet Solving equations by backtracking MAT08NASS1006 Skillsheet Solving equations using diagrams MAT08NASS1007 9

7 Chapter Equations Eample Homework sheet Equations 1 MAT08NAHS100 Solve each equation. a þ ¼ 17 b k 5 þ 4 ¼ 7 Solution a Method 1: The balancing method þ ¼ 17 þ ¼ 17 Step 1: Subtracting from both sides. ¼ 15 ¼ 15 Step : Dividing both sides by. Check: 5 þ ¼ 17 Worksheet Backtracking MAT08NAWK1009 ¼ 5 Method : The backtracking method þ ¼ = 5 15 = 17 ¼ 17 Step 1: Undo þ by subtracting. ¼ 15 ¼ 15 Step : Undo in by dividing by. ¼ 5 Check: 5 þ ¼ 17 b Method 1: The balancing method k 5 þ 4 ¼ 7 k þ 4 4 ¼ k 5 ¼ Step 1: Subtracting 4 from both sides. 94

8 NEW CENTURY MATHS for the Australian Curriculum8 k 5 ¼ 5 Step : Multiplying both sides by 5. 5 k ¼ 15 Check: 15 5 þ 4 ¼ þ 4 ¼ 7 Method : The backtracking method k 5 þ 4 ¼ 7 k ¼ 7 4 Step 1: Undo þ 4 by subtracting 4. 5 k 5 ¼ k ¼ 5 Step : Undo 4 5 by multiplying by 5. k ¼ 15 Check: 15 5 þ 4 ¼ þ 4 ¼ 7 Eample Solve each equation. a h ¼ 4 b þ 1 Solution a Method 1: The balancing method h ¼ 4 h ¼ 4 Step 1: Multiplying both sides by. h ¼ 1 h ¼ 1 Step : Dividing both sides by. h ¼ 6 Check: 6 ¼ 4 Method : The backtracking method h ¼ 4 h ¼ 4 Step 1: Undo 4 by multiplying by. h ¼ 1 h ¼ 1 Step : Undo in h by dividing by. h ¼ 6 Check: 6 ¼ 4 b Method 1: The balancing method þ 1 ¼ 7 þ 1 ¼ 7 Step 1: Multiplying both sides by. þ 1 ¼ 14 þ 1 1 ¼ 14 1 Step : Subtracting 1 from both sides. ¼ 1 Check: 1 þ 1 ¼ 7 ¼ 7 95

9 Chapter Equations Method : The backtracking method þ 1 ¼ 7 Step 1: Undo 4 by multiplying by. þ 1 ¼ 7 þ 1 ¼ 14 ¼ 14 1 Step : Undo þ 1 by subtracting 1. ¼ 1 Check: 1 þ 1 ¼ 7 Eercise 10-0 Two-step equations See Eample Worked solutions Eercise 10-0 MAT08NAWS1007 See Eample 1 Solve each equation, showing all steps. Remember to check your answers. a m þ 7 ¼ 19 b 5 ¼ 1 c 5k þ 1 ¼ 5 d 6w 17 ¼ 19 e 4h þ 1 ¼ 9 f 8d 5 ¼ 7 g þ 18 ¼ 4 h a 9 ¼ 6 i 14c þ ¼ 7 j 1p 10 ¼ 86 k 9y þ 11 ¼ 101 l r 15 ¼ 16 Solve each equation. a m þ 4 ¼ 1 b c 5 þ 9 ¼ 1 c d 7 þ ¼ 8 d k þ 15 ¼ 6 e h þ 7 ¼ 17 f 4 þ ¼ 4 g n 5 9 ¼ 4 h t 6 ¼ i a ¼ 6 8 j 16 ¼ 1 k v 6 15 ¼ 4 l b 4 ¼ The following is Liam s incorrect solution for 7 þ 5 ¼ 1. 7 þ 5 ¼ 1 7 ¼ 1 5 Line 1 7 ¼ 8 Line ¼ 8 7 Line ¼ Line 4 In which of the following was the error made? Select A, B, C or D. A Line 1 B Line C Line D Line 4 4 Solve each equation. a ¼ 9 b 15 9 ¼ 15 c ¼ 8 d N 5 ¼ e 6N 7 ¼ 18 f 5B ¼ 10 g ¼ h 4 5 ¼ 4 5m i ¼ 11 j ¼ 8 k 4 5 ¼ 1 l ¼ 10 5 Which of the following is the solution to k 1 ¼ 6? Select the correct answer A, B, C or D. 9 A k ¼ 7 B k ¼ 4 C k ¼ 66 D k ¼ 16 96

10 NEW CENTURY MATHS for the Australian Curriculum8 6 Solve each equation. a þ 1 ¼ b e N þ 8 ¼ 6 f þ 1 5 ¼ c N þ 5 ¼ g k 5 ¼ 1 d N 4 ¼ 5 h m þ ¼ 5 ¼ 11 7 Write 4 equations, one of each type shown in questions 1,, 4 and 5, that have the solution p ¼. Worked solutions Eercise 10-0 MAT08NAWS Equations with variables on both sides Technology worksheet Ecel: Solving linear equations MAT08NACT00017 For equations with variables on both sides, such as þ 4 ¼ þ 7, we can only use the balancing method, not the backtracking method. Summary Technology Ecel: Linear equation solver MAT08NACT0007 For equations with variables on both sides, perform operations on both sides to move: all the variables onto one side of the equation all the numbers onto the other side of the equation. Eample 4 Solve each equation. a þ 4 ¼ þ 7 b 5n ¼ n 15 c d 8 ¼ 5d 71 Video tutorial Equations with variables on both sides MAT08NAVT10018 Video tutorial Solving equations MAT08NAVT0001 Solution a þ 4 ¼ þ 7 Animated eample Formally solving equations MAT08NAAE0000 Worksheet Equations with unknowns on both sides þ 4 ¼ þ 7 þ 4 ¼ 7 Subtracting from both sides to remove it from the RHS. (RHS ¼ right-hand side ) MAT08NAWK00076 Puzzle sheet Equations with unknowns on both sides MAT08NAPS

11 Chapter Equations þ 4 4 ¼ 7 4 ¼ Subtracting 4 from both sides to remove it from the LHS. (LHS ¼ left-hand side ) Check: LHS ¼ þ 4 ¼ 1 RHS ¼ þ 7 ¼ 1 LHS ¼ RHS. b 5n ¼ n 15 5n n ¼ n 15 n n ¼ 15 n þ ¼ 15 þ n ¼ 1 n ¼ 1 n ¼ 4 Check: LHS ¼ 5 ( 4) ¼ RHS ¼ ( 4) 15 ¼ LHS ¼ RHS c d 8 ¼ 5d 71 d 8 þ 5d ¼ 5d 71 þ 5d 7d 8 ¼ 71 7d 8 þ 8 ¼ 71 þ 8 7d ¼ 6 7d 7 ¼ 6 7 d ¼ 9 Check: LHS ¼ ( 9) 8 ¼ 6 RHS ¼ 5 ( 9) 71 ¼ 6 LHS ¼ RHS. Subtracting n from both sides to remove it from the RHS. Now this is a two-step equation. Adding to both sides to remove it from the LHS. Now this is a one-step equation. Dividing both sides by. Adding 5d to both sides to remove it from the RHS. Now this is a two-step equation. Add 8 to both sides to remove it from the LHS. Now this is a one-step equation. Dividing both sides by 7. 98

12 NEW CENTURY MATHS for the Australian Curriculum8 Eercise 10-0 Equations with variables on both sides 1 Solve each equation, showing all steps. Remember to check your answers. a a þ 6 ¼ a þ 18 b k þ 4 ¼ k þ 8 c þ 6 ¼ þ 9 d 6p þ 4 ¼ p þ 19 e 6n þ 5 ¼ n þ 17 f 5q þ 6 ¼ 4q þ 1 g 5y þ 14 ¼ y þ 14 h 4a þ 7 ¼ a þ 1 i r þ 9 ¼ r þ 15 What is the solution to the equation 4 5 ¼ þ 7? Select the correct answer A, B, C or D. A ¼ 1 B ¼ C ¼ 4 D ¼ 6 Solve each equation. a 5a ¼ a þ 6 b 6 ¼ þ 6 c d 6 ¼ d 4 d 8p 15 ¼ p 10 e m þ 5 ¼ mþ 6 f 4 þ 6 ¼ 6 þ 56 g 4 þ ¼ þ 7 h r þ 18 ¼ 5r þ 11 i y 1 ¼ yþ 6 4 Here is Bree s solution for 4 ¼ þ 8. 4 ¼ þ8 þ 4 ¼ 8 Line ¼ 8 Line 5 ¼ 8 4 Line 5 ¼ 4 Line 4 See Eample 4 Worked solutions Eercise 10-0 MAT08NAWS10074 ¼ 5 4 Line 5 In which lines did Bree make mistakes? Select the correct answer A, B, C or D. A Lines 1 and B Lines and 5 C Lines 1 and 4 D Lines and 5 5 Solve each equation. a d þ 4 ¼ d b 10k ¼ 1 8k c 5 p ¼ p þ 9 d 7 þ ¼ 6 þ e 6k 11 ¼ k f m þ 0 ¼ 7m g 4t ¼ 1 4t h 8j 17 ¼ 10j i 6 q ¼ 8 q 6 Write an equation with on both sides that has the solution ¼ 7. Worked solutions Eercise 10-0 MAT08NAWS Equations with brackets Worksheet Equations MAT08NAWK1009 Summary For equations with brackets (grouping symbols), epand the epressions and then solve as usual. Video tutorial Equations with brackets MAT08NAVT10019 Worksheet Writing equations Eample 5 Solve each equation. a ( þ 5) ¼ 9 b ( y) ¼ 4y þ 4 c 5(r ) ¼ (r þ 6) MAT08NAWK10094 Puzzle sheet Equations match MAT08NAPS

13 Chapter Equations Worksheet Equations (Etension) MAT08NAWK10095 Homework sheet Equations MAT08NAHS1004 Video tutorial Solving equations MAT08NAVT0001 Solution a ( þ 5) ¼ 9 þ 15 ¼ 9 Epanding the epression to make it a twostep equation. Can you think of a way to solve this equation without epanding? þ ¼ 9 15 Subtracting 15 from both sides. ¼ 6 ¼ 6 Dividing both sides by. ¼ Check: ( þ 5) ¼ ¼ 9 b ( y) ¼ 4y þ 4 6 y ¼ 4y þ 4 Epanding the brackets. 6 y 4y ¼ 4y þ 4 4y Subtracting 4y from both sides to remove it from the RHS. 6 6y ¼ 4 Now this is a two-step equation. 6 6y 6 ¼ 4 6 Subtracting 6 from both sides to remove it from the LHS. 6y ¼ 6y 6 ¼ 6 Dividing both sides by ( 6). y ¼ 1 Check: LHS ¼ 1 ¼ 5 1 RHS ¼ 4 1 þ 4 ¼ 5 1 LHS ¼ RHS c 5(r ) ¼ (r þ 6) 5r 15 ¼ r þ 1 5r 15 r ¼ r þ 1 r r 15 ¼ 1 r 15 þ 15 ¼ 1 þ 15 r ¼ 7 r ¼ 7 r ¼ 9 Check: LHS ¼ 5 (9 ) ¼ 5 6 ¼ 0 RHS ¼ (9 þ 6) ¼ 15 ¼ 0 LHS ¼ RHS Epanding both sides. Subtract r from both sides to remove it from the RHS. Now this is a two-step equation. Add 15 to both sides to remove it from the LHS. Dividing both sides by. 400

14 NEW CENTURY MATHS for the Australian Curriculum8 Eercise Equations with brackets 1 Solve each equation, showing all steps. Remember to check your answers. a ( þ ) ¼ 8 b (m þ ) ¼ 18 c 5(k þ 1) ¼ 5 d 4(p þ 4) ¼ e 10(j þ ) ¼ 50 f 7(d þ 4) ¼ 6 g ( 4) ¼ 15 h ( 4) ¼ 1 i 5( 1) ¼ 10 j 8( 5) ¼ 4 k 4(k 5) ¼ 8 l (q 7) ¼ 6 Solve each equation. a (6 ) ¼ 6 b (4 p) ¼ 4 c (1 q) ¼ 8 d (16 h) ¼ 15 e (4 þ r) ¼ 8 f (e 16) ¼ 15 g 5(z ) ¼ 10 h 7(y þ ) ¼ 5 i 9(6 a) ¼ 0 j 5(d þ ) ¼ 0 k ( k) ¼ 1 l 4(6 c) ¼ 16 What is the solution to (1 ) ¼ 0? Select the correct answer A, B, C or D. A ¼ B ¼ 1 C ¼ 1 D ¼ 4 Solve each equation. a 6( þ 1) ¼ þ b (r þ ) ¼ r þ 5 c (p þ ) ¼ p þ d ( þ 1) ¼ e 5(p ) ¼ p f 4(e ) ¼ e þ 4 g 5(y þ ) ¼ y þ 1 h (a 5) ¼ 5 a i (w þ 1) ¼ w 15 5 Here is Robert s solution for 7(h 4) ¼ h þ 0. 7(h 4) ¼ h þ 0 7h 4 ¼ h þ 0 Line 1 7h 4 h ¼ h þ 0 h Line 4h 4 ¼ 0 Line 4h 4 þ 4 ¼ 0 þ 4 Line 4 4h ¼ 4 Line 5 4h 4 ¼ 4 4 Line 6 h ¼ 6 Line 7 In which lines did Robert make a mistake? Select the correct answer A, B, C or D. A Line 1 B Line C Line 5 D Line 7 6 Solve each equation. a 5( 1) ¼ 4( þ ) b 6( þ ) ¼ 4( þ 6) c 4(k þ ) ¼ (k ) d 7(y þ ) ¼ 4(y þ 5) e (v þ ) ¼ (v þ 5) f 4( ) ¼ ( þ 7) g (p þ 1) ¼ 5(p 1) h (s þ 1) ¼ 5(s þ ) i 5(d þ 4) ¼ (d 1) 7 Write an equation involving brackets whose solution is: a c ¼ 5 b k ¼ 1 c n ¼ d q ¼ 4 Mental skills 10 Maths without calculators Multiplying and dividing by 5, 15, 5 and 50 It is easier to multiply or divide a number by 10 than by 5. So whenever we multiply or divide a number by 5, we can double the 5 (to make 10) and then adjust the first number. 1 Study each eample. a To multiply by 5, halve the number, then multiply by ¼ ðor 9 10Þ ¼ 9 10 ¼ 90 See Eample 5 Worked solutions Eercise MAT08NAWS

15 Chapter Equations b To multiply by 50, halve the number, then multiply by ¼ ðor 1 100Þ ¼ ¼ 100 c To multiply by 5, quarter the number, then multiply by ¼ ðor Þ 4 ¼ ¼ 1100 d To multiply by 15, halve the number, then multiply by 0. e f g 8 15 ¼ ðor 4 15Þ ¼ 4 0 ¼ 10 To divide by 5, divide by 10 and double the answer. We do this because there are two 5s in every ¼ ¼ 14 ¼ 8 To divide by 50, divide by 100 and double the answer. This is because there are two 50s in every ¼ ¼ 4 ¼ 8 To divide by 5, divide by 100 and multiply the answer by 4. This is because there are four 5s in every ¼ ¼ 6 4 ¼ 4 h To divide by 15, divide by 0 and double the answer. This is because there are two 15s in every ¼ ¼ 8 ¼ 16 Now evaluate each epression. a 5 b 14 5 c 48 5 d e 5 50 f 6 5 g 8 5 h 1 5 i 1 15 j 5 k l m n o p q r s t

16 NEW CENTURY MATHS for the Australian Curriculum8 Investigation: Solving ¼ c 1 What is the inverse operation of squaring? The equation ¼ 9 has two solutions. What are the two numbers,, which when squared gives 9? What are the solutions for each equation? What is the pattern found in the answers? a ¼ 5 b ¼ 100 c ¼ 1 4 Study this eample: ¼ 49 p ¼ ffiffiffiffiffi 49 p which means ¼ ffiffiffiffiffi pffiffiffiffiffi 49 or 49 ¼7 which means ¼ 7or 7 Check: When ¼ 7, ¼ 7 ¼ 49 When ¼ 7, ¼ ( 7) ¼ 49 Use the same method to solve each equation and check your answers: a ¼ 81 b ¼ Simple quadratic equations ¼ c NSW An equation involving a pronumeral squared, such as ¼ 5 or 4 ¼ 7, is called a quadratic equation. In this section, we will solve simple quadratic equations of the type ¼ c, where c is a number. Eample 6 Solve each quadratic equation. a ¼ 6 b ¼ 11 c ¼ 40, writing the solution correct to one decimal place d ¼ 8, writing the solution as a surd Solution a ¼ 6 p ¼ ffiffiffiffiffi 6 ¼6 b ¼ 11 p ¼ ffiffiffiffiffiffiffi 11 ¼11 Finding the square root of both sides. The equation has two solutions, ¼ 6or ¼ 6. Finding the square root of both sides. The equation has solutions ¼ 11 or ¼

17 Chapter Equations c ¼ 40 p ¼ ffiffiffiffiffi 40 ¼6: : d ¼ 8 p ¼ ffiffiffiffiffi 8 The equation has solutions 6. or 6.. Leaving the answer as a surd. p ¼ ffiffiffiffiffi pffiffiffiffiffi 8 or 8 Summary The simple quadratic equation ¼ c (where c is a positive number) has two solutions, p ¼ ffiffi pffiffi p ffiffi c (which means ¼ c or ¼ c ). Eercise Simple quadratic equations ¼ c See Eample 6 1 Solve each quadratic equation. a ¼ 81 b ¼ 144 c ¼ 1 d ¼ 169 e m ¼ 5041 f u ¼ 1849 Solve each quadratic equation, writing the solution correct to two decimal places. a ¼ 1 b ¼ 54 c ¼ 88 d t ¼ 19 e h ¼ 946 f z ¼ 57 Solve each quadratic equation, writing the solution as a surd. a ¼ 41 b ¼ 0 c ¼ 48 d a ¼ 16 e p ¼ 75 f b ¼ Many simple quadratic equations have two solutions (one positive, one negative) but there is one simple quadratic equation that has only one solution. What is this equation and what is its solution? 5 The equation ¼ 100 has no solutions. Eplain why. 6 Write two more simple quadratic equations that have no solutions. 7 Copy and complete: a ¼ c has two solutions if c is. b ¼ c has no solutions if c is. c ¼ c has one solution if c is. 404

18 NEW CENTURY MATHS for the Australian Curriculum Equation problems Summary To solve word problems that require an equation to be solved: choose your pronumeral translate the words into an equation solve the equation write a sentence that answers the problem. Eample 7 a When a number is doubled and 11 is subtracted, the result is. Find the number. b Five times a number is the same as si more than three times the number. What is the number? Solution a Let the number be. Translating the words into an equation: 11 ¼ 11 ¼ Solving the equation: 11 þ 11 ¼ þ 11 ¼ 4 ¼ 4 ¼ 17 The number is 17. b Let n represent the number. 5n ¼ n þ 6 5n n ¼ n þ 6 n n ¼ 6 n ¼ 6 n ¼ The number is. Adding 11 to both sides. Dividing both sides by. Check: ¼ Subtracting from both sides. Dividing both sides by. Check: LHS ¼ 5 ¼ 15 RHS ¼ þ 6 ¼ 15 LHS ¼ RHS Worksheet Equations problems MAT08NAWK00075 Worksheet Working with formulas MAT08NAWK00077 Homework sheet Equations MAT08NAHS1004 Homework sheet Equations revision MAT08NAHS1005 Animated eample Writing equations MAT08NAAE00019 Video tutorial Solving equations MAT08NAVT0001 Worksheet Solving equations review MAT08NAWK00078 Quiz Solving equations MAT08NAQZ

19 Chapter Equations Eample 8 A repairman charges for fiing washing machines using the formula: C ¼ h þ 45 where C is the charge in dollars and h is the number of hours the job takes. Find: a the charge for a job that takes hours b the number of hours worked if the charge is $05. Solution a Substitute h ¼ into the formula. C ¼ h þ 45 ¼ þ 45 ¼ 141 The charge is $141. b Substitute C ¼ 05 into the formula and solve the equation. 05 ¼ h þ 45 h þ 45 ¼ 05 h þ ¼ h ¼ 160 h ¼ 160 h ¼ 5 The number of hours worked is 5. Swap the sides of the equation so that h is on the LHS. 406

20 N E W C E N T U R Y M AT H S for the A ustralian Curriculum 8 Eercise Equation problems 1 Solve each problem by writing an equation and then solving it. You may use diagrams to help you think about the information. a Four family tickets for a theme park cost $5. How much does each ticket cost? (Let t represent the price of a family ticket.) b Twenty cans of drink cost $ How much does each can cost? (Let c represent the cost of one can.) c A number is tripled and the result is 99. What is the number? (Let represent the unknown number.) See Eample 7 d A number has 7 added to it and the result is. Find the number. (Let n represent the unknown number.) Select the correct equation A, B, C or D for each problem. Then use the equation to solve the problem. a If is subtracted from a number, the answer is 79. What is the number? A 79 þ N ¼ B N ¼ 79 C N ¼ 79 b The sum of a number and 18 is 67. What is the number? A N þ 67 ¼ 18 B N þ 18 ¼ 67 C 18N ¼ 67 D N ¼ 79 D N 18 ¼ 67 c When a number is multiplied by, the answer is 188. What is the number? A 188N ¼ B N þ ¼ 188 C 188 N ¼ D N ¼ 188 d When a number is divided by 14, the answer is 4. What is the number? C 14N ¼ 4 D 14 þ N ¼ 4 A N 14 ¼ 4 B N ¼ 4 14 e Marika buys a car that costs $ She pays a deposit of $455. How much does she owe? A 455N ¼ C N þ 455 ¼ B N 455 ¼ D N ¼ þ 455 For each problem, write an equation and find the unknown number. a When an unknown number is divided by 5, the result is 15. b When 1 is subtracted from a number, the answer is 7. Worked solutions Eercise MAT08NAWS10076 c When a number is subtracted from 5, the answer is 5. d An unknown number is multiplied by, then 8 is subtracted. The result is 40. e An unknown number is halved, then 7 is added. The result is 1. f The sum of a number and 4 is doubled. The result is 44. g The sum of a number and 6 is divided by and the result is 7. h The product of a number and 5 is decreased by. The result is 8. 4 Translate each problem into an equation, then solve the equation to solve the problem. a Year 8 is holding a disco to raise money. Each ticket bought by students raises $5 but the costs of running the disco total $10. How many tickets must be sold to make a profit of $00? (Let n stand for the number of tickets sold.) 407

21 Chapter Equations b In eight years Kelly s age will be twice what it is now. How old is Kelly now? (Let n stand for Kelly s age now.) c Eight sheep have the same mass as three sheep and one cow. If the cow s mass is 500 kg, what is the mass of one sheep? (Let s stand for the mass of one sheep.) d If you multiply Liam s favourite number by and add 1, you get the same answer as if you multiplied the number by 5 and took away 11. What is the number? (Let represent the number.) e Mr Yen says, If you add 15 to my age and multiply by 7, the answer is 94. How old is Mr Yen? (Let a stand for his age.) f The area of a rhombus is calculated by multiplying its diagonals and dividing by. If a rhombus has an area of 88 cm and one diagonal is 11 cm, what is the length of the other diagonal? (Let d represent the length of the diagonal.) See Eample 8 Worked solutions Eercise MAT08NAWS A temperature in degrees Celsius ( C) can be converted to degrees Fahrenheit ( F) using the formula F ¼ 9C þ. 5 a Convert 5 C to F. b Convert 108 F to C. 6 The charge, $C, for hiring a hall for an event is C ¼ 150 þ N, where N stands for the number of people at the event. Find: a the charge when 5 people are at the event b the number of people at the event when the charge is $94. 7 The cost, $y, of a classified ad in the local newspaper is y ¼ 0.8w þ.5, where w is the number of words in the ad. Find: a the cost of a 1-word ad b the number of words in an ad costing $5.90 cm ( ) cm 8 The perimeter of this rectangle is 47 cm. Find: a the value of b the length of the rectangle c the width of the rectangle. 9 The profit, $P, made by a DVD store is given by P ¼ 5 900, where represents the number of DVDs sold. Find: a the profit made when 195 DVDs are sold b the number of DVDs sold if the profit is $45. m )c 17 + ( cm 10 Find the value of in this isosceles triangle. 408

22 NEW CENTURY MATHS for the Australian Curriculum8 Power plus 1 Find the value of in each of the following. a ( þ 1) þ ( 1) ¼ 1 b ( þ 4) ( 1) ¼ 9 c 4( 1) 5( ) ¼ 6 d ( þ 5) ¼ 4( þ 1) e þ 7 ¼ 6ð 1Þ ð þ 1Þ f ¼ 5ð Þ 4 g ¼ 10 6 h 4 þ 9 10 ¼ 44 Write an equation and solve it to find the unknown values in each of the following. a b c + Perimeter = 0 cm Area = 100 cm Each of the following equations has two solutions. Find them. a ¼ 5 b 7 ¼ 9 c ¼ 7 d þ 8 ¼ 0 e ( þ 4) ¼ 9 f þ ¼ 15 ( + 5) ( 40) 4 Diophantus was a famous mathematician who was the first to abbreviate his mathematical thoughts using symbols. He is known as the Greek father of algebra and when he died, one of his admirers wrote the following riddle about his life: Diophantus youth lasted of his life. He grew a beard after 1 more. After 1 7 more of his life, Diophantus married; 5 years later he had a son. The son lived eactly 1 as long as his father, and Diophantus died just 4 years after his son. All this adds to the years Diophantus lived. Write this riddle as an equation and solve it to find how long Diophantus lived. (Hint: Let years equal his life.) 409

23 Chapter 10 review Puzzle sheet Equations crossword MAT08NAPS1009 n Language of maths backtracking epand balancing formula brackets inverse operation check LHS (left-hand side) equation one-step equation pronumeral quadratic equation RHS (right-hand side) solution solve substitution two-step equation undoing unknown variable Worksheet Mind map: Equations MAT08NAWK What are the two algebraic methods for solving equations? Which method involves undoing operations? What does the word solution mean? 4 Which word in the list means opposite? 5 Why is the variable in an equation sometimes called an unknown? 6 What word means to rewrite an algebraic epression by removing the brackets? 7 What does RHS stand for: a in congruent triangles? b in solving equations? n Topic overview Which parts of this topic did you find easy? What did you already know? Give eamples of some problems that might be solved using equations. Are there any parts of this topic that you still don t understand? Talk to your teacher about them. In what sort of careers would people use equations? Copy and complete this mind map of the topic, adding detail to its branches and using pictures, symbols and colour where needed. Ask your teacher to check your work. One-and two-step equations EQUATIONS Equations with variables on both sides = Equation problems Equations with brackets Simple quadratic equations = c 410

24 Chapter 10 revision 1 Solve each of the following equations. a k þ 6 ¼ 1 b ¼ 8 c a þ ¼ 17 d a 1 ¼ 1 e ¼ 1 f 10f ¼ 10 g m 4 ¼ 7 h ¼ 8 i w þ 9 ¼ j k 5 ¼ 7 k m ¼ 16 l ¼ 1 Solve each equation. a 4p þ ¼ b m þ 17 ¼ 8 c 1 ¼ 18 d h 7 þ 5 ¼ 16 e n 8 ¼ 1 f a 5 8 ¼ 4 g y 5 ¼ 4 h þ ¼ 4 i n 5 ¼ 10 j 5 ¼ 15 k k þ 9 ¼ 5 7 Solve each equation. a þ 4 ¼ þ 6 b 5u ¼ u þ 6 c 1h 8 ¼ 8h þ 4 d v 4 ¼ 7v þ 8 e þ 9 ¼ 7 4 f 9 5t ¼ t 15 l d 1 5 ¼ 6 4 Solve each equation. a (n þ ) ¼ 18 b ( þ 1) ¼ 15 c 10( ) ¼ 10 d 5( ) ¼ þ 4 e 4(y þ 1) ¼ y þ 18 f (d ) ¼ d 0 g (r 1) ¼ (r þ 9) h 7(a þ 5) ¼ (a þ 9) i (n 4) ¼ (5 n) 5 Solve each equation: a ¼ 64 b ¼ 45, correct to one decimal place c ¼ 7, as a surd 6 a If a number is doubled and has 4 added to it, the answer is 14. What is the number? b If a number has 4 added to it and is doubled, the answer is 14. What is the number? 7 a Kay is paid $45 for each jumper she knits. If n is the number of jumpers Kay knits to earn a total of $70, which equation can be used to find the value of n? Select the correct answer A, B, C or D. A n ¼ B 45n ¼ 70 C n þ 45 ¼ 70 D 70 n ¼ 45 b Solve the equation to find the value of n. 8 Seven times a number is the same as nine more than four times the same number. Use an equation to find the number. 9 The sum (S) of the interior angles (in degrees) of a polygon is given by S ¼ 180n 60, where n is the number of sides. Find: a the sum of the interior angles when a polygon has 9 sides b the number of sides in a polygon whose angle sum is See Eercise See Eercise 10-0 See Eercise 10-0 See Eercise See Eercise See Eercise See Eercise See Eercise See Eercise