Chapter. Equations AUS_07. Y:\HAESE\AUS_07\AUS07_09\191AUS07_09.cdr Wednesday, 6 July :02:48 AM BEN

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1 Chapter 9 Equations cyan magenta yellow Y:\HAESE\AUS_07\AUS07_09\191AUS07_09.cdr Wednesday, 6 July :0:48 AM BEN Contents: A Equations B Solving simple equations C Maintaining balance D Inverse operations E Algebraic flowcharts F Solving equations G Equations with a repeated variable H Writing equations I Problem solving black AUS_07

2 19 EQUATIONS (Chapter 9) Opening problem Ray is at the Royal Show with five good friends. They decide to all go on the MegaTwister, one of the new rides at the Show. It costs $8 per ride. Ray s mum has given them $0 to share and put towards one ride for each of them. They will each have to pay some etra money as well. Suppose each of the si friends pays $ etra. Things to think about: a Eplain why the total amount paid by the si friends is (6 + 0) dollars. b Eplain why 6 +0=6 8. c Given the equation 6 +0=48, how can we find the eact value of? d How much does each of the friends pay? In this chapter we look at algebraic equations and methods used to solve them. A EQUATIONS An equation is a mathematical sentence which indicates that two epressions have the same value. An equation always contains an equal sign =. A simple equation may be a true numerical statement like 5 = 7+8. LHS equals RHS Notice that an equation has a left hand side (LHS) and a right hand side (RHS) and these are separated by the equal sign. An algebraic equation has an unknown or variable in it. For eample, 6 +0=48 contains the variable. When we solve an equation we find the value of the variable which makes the equation true. This value is called the solution of the equation. If we were to replace by a variety of numbers, most of them would make the equation false. For eample, consider the equation 6 +0=48: If =1 the LHS is = = 6 If =5 the LHS is = = 60 If = the LHS is = = 48 So, if = then the LHS = RHS, and the equation is true. We say = is the solution of the equation 6 +0=48.

3 EQUATIONS (Chapter 9) 19 Eample 1 What number can replace to make the equation true? a + =10 b =18 c 0 =4 d 8=4 a +7=10 so =7 b 6=18 so =6 c 0 5=4 so =5 d 1 8=4 so =1 EXERCISE 9A 1 State whether each of the following is an equation or an epression: a ( +) b ( +)=6 c 5 d +=6 e +4=11 f +(4 11) What number can replace to make the equation true? a 4+ =1 b +7=18 c 10 = d 9=1 e 4 = f 5=6 g 108 =9 h =7 i +=1 For each of the following, suppose is the number. Use the statement to write an equation involving. a b c d e f Three added to a number is equal to nine. Seven subtracted from a number is equal to eight. A number multiplied by si is equal to twenty four. A number when divided by three is equal to four. Two multiplied by a number is equal to ten. A number subtracted from thirteen is equal to si. B SOLVING SIMPLE EQUATIONS In this chapter we will be dealing with equations which have one unknown. Remember that in algebra: ² the sign is omitted where possible. For eample, 5 is written 5. ² the sign is usually written as a fraction. For eample, is written. In Chapter 7 we saw that given an epression involving, we can substitute a value for to evaluate the epression. For eample, consider the epression 4. When =, 4 =4 =5. In this chapter we reverse this process. We are presented with equations such as 4 =5, and our task is to work out that must be.

4 194 EQUATIONS (Chapter 9) SOLVING BY INSPECTION Some simple equations can be solved by inspection. For eample, for the equation +=8 we notice that since 6+=8, must be 6. We write: +=8 ) =6 ) is read as therefore. We use it to show that the net line of work follows from the previous line. Eample Solve by inspection: a a +6=11 b b =8 c 14 =8 d 7p =49 a a +6=11 ) a =5 fas 5+6=11g b b =8 ) b =4 fas 4 =8g c 14 =8 ) =6 fas 14 6=8g d 7p =49 ) p =7 fas 7 7=49g SOLVING BY GUESS, CHECK AND IMPROVE Another method of solving simple equations is to use guess, check and improve. This involves substituting different numbers in place of the variable until the correct solution is obtained. For eample, to solve 4 1 = we substitute different values for and summarise our trials in a table. So, =9 is the solution. EXERCISE 9B 1 Solve by inspection: much too low 5 7 getting closer 8 19 almost 9 X a 6+a =11 b b =5 c 1 = 1 c d d =18 e 11 e =4 f f 10 = g 16 = g +7 h 4h =8 i 14 i = j 5 j = 1 k 11 + k = l l l =0 m m =4 n 8 n = o = 11 One of the numbers in the brackets is the correct solution to the equation. Find the correct solution. a +5=17 f, 4, 6, 8g b =5 f8, 9, 10, 11g c 16 =6 f,, 4, 5g d 5 +=8 f6, 7, 8, 9g e 6 += 10 f0, 1,, g f 8 = f, :5, 4, 4:5g

5 EQUATIONS (Chapter 9) 195 Solve by inspection: a a = 6 b +b =1 c 5 c = 7 d d 5= 1 e e +9=5 f 5f = 10 g 1 g = h 8+h =5 i 1i =91 4 Solve by guess, check and improve: a +=17 b 5 4=41 c 15 = 18 d 4+11 =81 e 6 =8 f 6 5 = 19 Activity 1 Guess, check, and improve What to do: 1 Click on the icon to run the software. For the equation given, enter the value of which you think is the solution. The value of the left hand side of the equation will be calculated. You will be told if your guess was correct, or if it gave a value that was too large or too small. Keep guessing until you get the correct answer. 4 Compete with your class to see who is the fastest to correctly solve 10 equations. ACTIVITY C MAINTAINING BALANCE The balance of an equation is like the balance of a set of scales. Changing one side of the equation without doing the same thing to the other side will upset the balance. PERFORMING OPERATIONS ON EQUATIONS The equal sign represents the balancing point of the equation. The left hand side must balance the right hand side. For eample, the scales opposite show that 5=5 If 6 is added to both sides, the statement remains true: 5+6=5+6 ) 11 = 11 If 6 were added to one side only, then the statement would become false: 5+66= 5 ) 11 6= 5

6 196 EQUATIONS (Chapter 9) To maintain the balance, whatever is done on one side of the equal sign must also be done on the other side. Imagine a set of scales with si identical blocks on each side. The scale is balanced. If we subtract blocks from each side we get: If we add 1 block to each side we get: If we divide the number of blocks on each side by we get: If we multiply the number of blocks on each side by we get: Notice that the scales are still balanced in each case! Eample If the scales are perfectly balanced, find the relationship or connection between the objects: a b a By taking from both sides, we get: b By taking from both sides, we get: So, 1 is equal to So, 1 is equal to 1 plus 1 ) = + We take objects away until each object type appears on only one side of the scales.

7 EXERCISE 9C.1 1 The set of scales is balanced with two bananas and three strawberries on one side and 11 strawberries on the other. a Three strawberries are taken from the left side. What must be done to the right side to keep the scales balanced? b There are now two bananas on the left hand side. How many strawberries balance their weight? c How heavy is one banana in terms of strawberries? The set of scales is balanced with two golf balls and si marbles on the left and one golf ball and nine marbles on the right. a Si marbles are taken from the left side. What must be done to the right side to keep the scales balanced? b The golf ball on the right side is removed. What must be done to the left side to keep the scales balanced? c Redraw the scales if both a and b occur, ensuring balance at each step. d How heavy is one golf ball in terms of marbles? These scales are perfectly balanced. Find the relationship between the objects. a b c oo o v o o vv ooo v o c c oo c c o o o d e f EQUATIONS (Chapter 9) 197 vo oo o vv o v y y y ooo ooo ooo g ooo yo o t t t oo o o g h i e w w e w e b b Bbb B B M mm m 5 m Activity Counterfeit coins What to do: 1 You are given coins which appear identical. However, one of them is counterfeit, and has a different weight from the other two genuine coins. If you are given a set of scales, eplain how you would find which of the coins is the counterfeit coin. Now, imagine that you have 4 coins and one of them is counterfeit. How many weighings are needed to be certain of finding the bad coin? Eplain your method. Repeat with 5 and 6 coins, eplaining your answer each time.

8 198 EQUATIONS (Chapter 9) BALANCING EQUATIONS The balance of an equation will be maintained if we: ² add the same amount to both sides ² subtract the same amount from both sides ² multiply both sides by the same amount ² divide both sides by the same amount. Eample 4 istockphoto Consider the equation +5=10. What equation results when we perform the following on both sides of the equation: a add b subtract c divide by d multiply by 4? a +5=10 ) +5+=10+ ) +8=1 c +5=10 +5 ) = ) =5 b +5=10 ) +5 =10 ) +=7 d +5=10 ) 4( +5) = 4 10 ) 4( +5)=40 EXERCISE 9C. 1 Find the equation which results from adding: a to both sides of +1=5 b to both sides of +=6 c 4 to both sides of 4=7 d 6 to both sides of 6=8 e 6 to both sides of 1 = f 5 to both sides of =1 Find the equation which results from subtracting: a from both sides of +4=1 b 4 from both sides of +4=7 c from both sides of =11 d 5 from both sides of +5= e 4 from both sides of 5 +4= f from both sides of 1=5 Find the equation which results from multiplying both sides of: a = by b = by 4 c d +1= by e 1 = by f 4 Find the equation which results from dividing both sides of: =6 by a =4 by b ( +1)=8 by c +9=6 by d 4 4=8 by 4 e 5 =8 by 5 f ( ) = 18 by g =7 by 1 h (1 ) = 1 by = by

9 EQUATIONS (Chapter 9) 199 D INVERSE OPERATIONS Imagine starting with $50 in your pocket. You find $10 and then pay someone $10. You still have $50. This can be illustrated by a flowchart such as $ $60-10 $50 Observe that adding 10 and subtracting 10 have the opposite effect. One undoes the other. We say that addition and subtraction are inverse operations. Now imagine you start with $50, and your friend gives you the same amount. Your money is now doubled. If you decide to give half to your brother, you will be back to your original $50. We again illustrate the process by a flowchart: $50 * $100 / $50 Observe that multiplying by and dividing by undo each other. We say that multiplication and division are inverse operations. We can solve simple equations using inverse operations, but we must remember to keep the equation balanced by performing the same operation on both sides of the equation. For eample, consider +=7 where has been added to. +=7 ) + =7 fsubtracting is the inverse of adding g ) =4 fsimplifyingg Notice the balancing! By performing the suitable inverse operation, we are left with by itself on the LHS of the equation. This gives us the solution on the RHS. EXERCISE 9D 1 State the inverse of each of the following operations: a +4 b c d 1 e 1 f 1 g + 4 h 7 i 11 Simplify the following epressions: a 6+6 b c +4 4 d 5 5 e g 7+ 7 h f i 4 4

10 00 EQUATIONS (Chapter 9) Eample 5 Solve for using a suitable inverse operation: +5=11 +5=11 ) +5 5 =11 5 fthe inverse of +5is 5, so we take 5 from both sides.g ) =6 Find using an inverse operation: a +=9 b +6=15 c +5=0 d +7= e 5+ =1 f + =11 g +15=11 h +0:5 =0:7 i 1 + =1 Eample 6 Solve for y using a suitable inverse operation: y 6= y 6= ) y 6 +6= +6 fthe inverse of 6 is +6, so we add 6 to both sides.g ) y =4 4 Find y using an inverse operation: a y 4=5 b y 8=17 c y 7=0 d y 1 = e y 6= 1 f y = g y 8=5 h y 1: =:6 i y 1 =4 Eample 7 Solve for t using a suitable inverse operation: t = 1 t = 1 t ) = 1 ) t = 4 fthe inverse of is, so we divide both sides by.g 5 Find a using an inverse operation: a a =10 b a =1 c a =16 d 5a =0 e 4a = 16 f 6a =0 g 5a = 5 h 8a =64 i 8a = 7

11 EQUATIONS (Chapter 9) 01 Eample 8 Solve for d using a suitable inverse operation: d 7 =8 ) d 7 =8 d 7 =8 7 fthe inverse of 7 is 7, so we multiply both sides by 7.g 7 ) d =56 6 Find using an inverse operation: a = b =6 c 4 = 4 d 5 = 1 e =18 f 11 = g 6 = h =5 i =6 7 Find the unknown using a suitable inverse operation: a +a =11 b b 17 = 4 c 8c =48 d d 4=11 e e +19=1 f f = 10 g g 17 = 4 h h 7 = i 5+i = j 6j = 15 k k 1 = l 16l = 8 m 16 + m =1 n n 4 = o = 4 p p 4 =1 q q +7= 1 r r = 15 s s 5 = 1 t t 7 = 4 u 4 + u = Click on the icon to practice solving equations using a single inverse operation. INVERSE OPERATIONS

12 0 EQUATIONS (Chapter 9) E ALGEBRAIC FLOWCHARTS To solve more complicated equations it is important to understand how epressions are built up. We can study this using an algebraic flowchart. By reversing the flowchart with inverse operations, we will be able to undo the epression and find the value of the unknown. Eample 9 Complete the following flowcharts: a +5 b +5 c d a b (+5) c d Eample 10 Use a flowchart to show how 5 + is built up. Reverse it to undo the epression. Build up: Undoing: Eample 11 Use a flowchart to show how + Reverse it to undo the epression. is built up. Build up: Undoing: + +

13 EQUATIONS (Chapter 9) 0 EXERCISE 9E 1 Copy and complete the following flowcharts: a b +1 4 c d +5 Copy and complete the following flowcharts by inserting the missing operations: a?? + b?? 1 c? 7? 7 d? +4? (+4) Use a flowchart to show how to build up and then undo the following epressions: a +4 b 1 c 4 + d Use a flowchart to show how to build up and then undo the following epressions: a 1 b 1 c +5 d +5 e +8 f ( +8) g ( 6) h 6 F SOLVING EQUATIONS So far we have solved simple equations by: ² inspection ² guess, check and improve ² using one inverse operation. SOLVING BY ISOLATING THE UNKNOWN We have already seen how an epression is built up from an unknown. When we are given an equation to solve which contains a built up epression, we need to do the reverse. We use inverse operations to undo the build-up of the epression in the reverse order and thus isolate the unknown. For eample, to solve 5 +7=19 we look at how 5 +7 is built up from : 5 5 To obtain 5 +7 from, we 5, then +7. So to isolate from 5 +7, we 7 and then

14 04 EQUATIONS (Chapter 9) Eample 1 Solve for : 5 +7= =19 ) =19 7 fsubtracting 7 from both sidesg ) 5 =1 5 ) 5 = 1 5 fdividing both sides by 5g ) = 1 5 Check: LHS = = = 19 = RHS X Always check that the answer makes the original equation true. Eample 1 Solve for : 4= ) ) 4= 4 +4= +4 fadding 4 to both sidesg ) =1 =1 fmultiplying both sides by g ) = Check: LHS = 4=1 4= =RHS X EXERCISE 9F 1 Solve the following equations: a =7 b +5=8 c 5 +9=4 d 4 16 = 0 e 4= f 1=0 g 10 =18 h 6 +5= 1 i 11 = 11 j 11 =41 k 7 5=4 l 4 +15=9 Solve for : a +1= b =5 c 5 +7=1 d 4 1= 6 e = f 10 +6=

15 EQUATIONS (Chapter 9) 05 Eample 14 Solve for : 7 = = 7 µ ) 7 = 7 fmultiplying both sides by 7g 7 ) = 1 ) += 1 + fadding to both sidesg ) = 18 Check: LHS = 18 7 = 1 7 = =RHS X Solve for : a =5 b + = c 5 4 = 1 d +4 =1 e 8 = f +6 =0 g 7 +4 = h 4 = 1 Eample 15 Solve for : ( 4) = 9 ) ( 4) = 9 ( 4) = 9 fdividing both sides by g ) 4=1 ) 4 +4=1+4 fadding 4 to both sidesg ) =17 Check: LHS = (17 4) = 1 = 9 = RHS X 4 Solve for : a ( 1) = 1 b 4( +)=4 c 5( ) = 5 d ( +7)=8 e 7( 1) = 0 f 1( ) = 4 g ( ) = 9 h 5( +)= 15 i 10( +4)=60 j 6( +)=4 k 6( 1) = 18 l ( ) = 10 5 Solve for : a 7 =18 b 1=4 c ( +5)=0 d 4 =5 e = 4 f ( 6) = 0 g += 1 h +7= 5 i +1 =

16 06 EQUATIONS (Chapter 9) 6 Solve for : a 4 6=8 b +4 = c ( +7)=14 5 d + 4 = 1 e +15=0 f 9( ) = 6 g 11 =8 h 6( +1)=9 i 5 1=7 G EQUATIONS WITH A REPEATED VARIABLE In many equations, the variable appears more than once. In this course we will consider some simple cases where the variable appears more than once on the LHS. To solve these equations, we begin by simplifying the LHS and collecting like terms. Eample 16 Solve for : + =65 + =65 ) 5 =65 fcollecting like termsg ) 5 5 = 65 5 fdividing both sides by 5g ) =1 Eample 17 Solve for : +( + 10) + 50 = 180 +( + 10) + 50 = 180 ) = 180 fepanding bracketsg ) + 60 = 180 fcollecting like termsg ) = fsubtracting 60 from both sidesg ) = 10 ) = 10 fdividing both sides by g ) =60 EXERCISE 9G 1 Solve for : a + =18 b =4 c 5 + = 6 d 4 4 = 0 e 4+5 =0 f 9 5 =1 Solve for : a +( + 40) = 180 b +( 0) + 50 = 180 c +( + 0) + ( + 40) = 180

17 EQUATIONS (Chapter 9) 07 H WRITING EQUATIONS Most real life problems are given in sentences rather than in symbols. Before we can solve these problems, we must first write the problem as an equation. Eample 18 When a number multiplied by is added to 5, the result is. Write this as an algebraic equation. Suppose we represent the number with the letter n. Start with a number n multiply it by n add 5 n +5 the result is n +5= So, the algebraic equation which represents the problem is n +5=. EXERCISE 9H 1 Starting with the number n, write each of the following as an algebraic equation: a When a number is multiplied by 5, the result is 0. b When 10 is added to a number, the result is. c When a number is divided by 4, and then 6 is added, the result is 8. d When a number is subtracted from 11, and the result is divided by, the answer is. Write each of the following as an algebraic equation: a b c Paul has coins, then his father gives him 1 more. Paul now has 7 coins. A plant was initially cm high. After a week it had doubled in height, and a week after that it had grown another 10 cm. It is now 1 cm high. Carl is years old. His daughter Chloe is one third his age. His son Jarrod, who is three years older than Chloe, is 18 years old. Answer the Opening Problem on page 19. I PROBLEM SOLVING To solve worded problems, we follow these steps: Step 1: Step : Step : Step 4: Select a letter or symbol to represent the unknown quantity to be found. Write a sentence to eplain this. Write an equation using the information in the question. Solve the equation. Write the answer to the question in a sentence.

18 08 EQUATIONS (Chapter 9) Eample 19 Callum has a collection of badges. His aunt gives him 9 more badges which she found in a bo at home. Then, while Callum is on holidays, he collects enough to double his collection. He now has 1 in total. How many badges did Callum have to start with? Let be the number of badges Callum has to start with. He is given 9 by his aunt, so the number is now +9. Callum doubles his collection, so he now has ( +9). So, ( + 9) = 1 ) ( +9) = 1 fdividing both sides by g ) +9=66 ) +9 9 =66 9 fsubtracting 9 from both sidesg ) =57 Callum had 57 badges to start with. istockphoto/pagadesign EXERCISE 9I 1 90 chocolate truffles were shared equally amongst the students in a class. When Maddie gave two of her truffles to her brother, she had three left over. How many students are in Maddie s class? Nicholas went to the state car museum, where he saw cars and three motorbikes. The curator stated that there were 14 wheels in the museum in total. How many cars did Nicholas see? The average of two numbers is. If one of the numbers is 17, what is the other number? 4 David bought some boes of ice blocks. There were si ice blocks in every bo. When he opened the freezer he realised that his children had found them and had already eaten. If there are now 1 ice blocks left, how many boes did David buy? 5 In an Aussie Rules match, James scored some goals worth 6 points each, and also behinds worth 1 point each. In total he scored 7 points for his team. How many goals did James score in the match? 6 There are 14 girls in the school debating club. The members of the club are split into 6 teams of 4 students. How many boys are in the club?

19 7 A number of people ride the bus to the city. Five of them are children. The bus driver receives $50 for all the tickets. If children pay $1 per ticket, and adult tickets are $:50, how many adults are on the bus? 8 Each day a salesman is paid $80 plus one tenth of the value of the sales he makes for the day. On one day the salesman is paid $00. What value of sales did he make that day? 9 Tim works at a bank, and in his spare time he enjoys swimming. From home, he travels 0 km to the bank, then after work he goes straight to the pool. After swimming, he travels 16 km to home. Tim follows the same routine for 4 days, and travels a total of 176 km. How far is it from the bank to the pool? EQUATIONS (Chapter 9) 09 KEY WORDS USED IN THIS CHAPTER ² balance ² build up ² equation ² epression ² flowchart ² inspection ² inverse operation ² solution ² undo ² unknown 1 a Is 4 y an epression or an equation? b State the inverse of 7. c What equation results from subtracting 6 from both sides of + 6 = 8? d Solve = 1 by inspection. What number can replace to make the equation true? a 6=11 b 8 =+9 One of the numbers f6, 7, 8, 9g is the solution to the equation 11 = 10. Find the solution. 4 The following scales are perfectly balanced. Find the relationship between the objects: a b o oooo ooo oo ooo 5 State the inverse of the following operations: a 7 b 8 c +1 d 1 6 Find using an inverse operation: a +4=17 b 8= c = 15 d = 1

20 10 EQUATIONS (Chapter 9) 7 Copy and complete the following flowcharts: a?? 9 b +4 8 Use a flowchart to show how the following epressions are built up from : a ( 4) b + 9 Use a flowchart to show how to isolate in the following epressions: a 9 b ( +5) 4 10 Solve for : a 9= 5 b +4 = 1 c 9 6= d +1=10 e +7=15 f ( 8) = Charlotte saves the same amount of money each week. If she starts with $1 and after 4 weeks she has $6, how much does she save each week? 1 Solve for : a +8= 5 b 8 =45 c 5=4 1 Emma buys a packet of lollies for the movies. She eats half of the packet, and her friends share the rest. If her friends receive 6 lollies each, how many were in the packet originally? Practice test 9A Click on the link to obtain a printable version of this test. Multiple Choice PRINTABLE TEST 1 One of the numbers f, 4, 5, 6g is the solution to 4 =17. Find the solution. a Find the equation which results from adding 5 to both sides of 4 5=11. b State the inverse of multiplying by 8. Find using an inverse operation: a + =7 b 4=11 c 5 = 0 d 6 = 4 When a number is divided by four, the result is 9. What is the number? 5 Copy and complete the following flowcharts: a? 4? 4 7 b +9

21 EQUATIONS (Chapter 9) 11 6 Find the equation which results from dividing both sides of ( ) = 15 by. 7 Use a flowchart to show how the following epressions are built up from : a 4( ) b 9 8 Monica asked each of her friends to buy a charity raffle ticket for $. All but 4 of her friends bought a ticket, and she raised $18. How many friends did Monica ask? 9 Use a flowchart to show how to isolate in the following epressions: a 7 b 4( +6) 10 Solve for : a 4 +5=9 b 7= 1 c 4 5= Practice test 9C Etended response 1 A garden supplies shop has pavers on three pallets. The second pallet has 40 more pavers than the first pallet. The third pallet has 8 less pavers than the first pallet. Suppose there are pavers on the first pallet. a b c d Write an epression for the number of pavers on the: i second pallet ii third pallet. There are a total of 00 pavers. Use your answers in a to write an equation describing the situation. Simplify your equation by collecting like terms. Solve the equation, and hence find how many pavers are on each pallet. A café serves eggs, bacon, and toast for breakfast, as shown on the menu. a b c d Toast costs half as much as bacon. If bacon costs $b, write an epression for the price of toast. Using the menu and your answer to a, write an equation involving b. Solve the equation to find the cost of bacon. MENU Eggs... $4 Bacon + Toast... $6 The café adds a bacon and eggs option to the menu. How much should they charge? Eplain your answer. At the local market, pineapples are sold at various prices. a Store A sells pineapples for $ each. If I buy 8 pineapples at Store A, how much will it cost? b For the same amount of money, at Store B I can only buy 6 pineapples. i Suppose pineapples cost $p each at Store B. Write an equation to describe the situation. ii Solve the equation in i. iii Hence find out how much more Store B charges for a pineapple than Store A.

22 1 EQUATIONS (Chapter 9) 4 Philip runs a fencing company which makes the following type of fence: 1 m Each fence is made using 1 m lengths of timber. The eample above is 4 m long, and uses 17 m of timber. Philip has worked out that to make a fence l m long, he will need 4l +1 m of timber. a How many metres of timber will Philip need to make a 7 m long fence? b Philip has 61 m of timber, and wants to know how long the resulting fence will be. i Set up an equation to describe the situation. ii Solve the equation and hence state the length of the fence. 5 A school orders desks for its Year 7, 8, and 9 classes. The Year 8 class gets twice the number of desks as the Year 7 class. The Year 9 class gets less desks than the Year 8 class. a b Suppose the Year 7 class gets d desks. Write an epression for the number of desks received by: i the Year 8 class ii the Year 9 class. The school bought a total of 7 desks for the three classes. i Write an equation describing the situation. ii Simplify this equation by collecting like terms. iii Solve the equation, and hence state the number of desks given to each class. 1 m Activity 4 What to do: 1 Start with a large circle at least 0 cm across, and divide it into 1 sectors of equal size as shown. Make a series of heagons as shown in the diagram below. Continue the pattern for as long as you can. PRINTABLE WORKSHEET Si spirals can be coloured. One of these is shown. On your figure colour all si spirals in different colours. 4 Repeat the above steps by first dividing the spiral into 16 sectors. Form octagons and octagonal spirals. Polygonal spirals 0 cm