Solving equations. Pretest. Warmup


 Victor Martin
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1 Solvig equatios 8 Pretest Warmup We ca thik of a algebraic equatio as beig like a set of scales. The two sides of the equatio are equal, so the scales are balaced. If we add somethig to oe side of the scales without addig somethig to the other side, the scales will o loger be balaced. We ca solve equatios by doig the same to both sides.
2 8.1 What is a equatio? Comparig the values of expressios I mathematics, whe two expressios have the same value we use a equals symbol ( = ) to show this. For example, we kow that 4 ad 6 6 each has the value 1, so we ca write 4 = 6 6. This type of umber statemet is called a equatio because it states that the expressios o each side of the equatio are equal i value. We ca also describe such a statemet as true. Example 1 I each of the followig equatios, isert a umber o the blak lie to make the equatio true. a 1 = 17 b 4 = Workig a 1 = = 17 b 4 = 4 8 = Reasoig If the missig umber is 1, the equatio will be true. If the missig umber is 8, the equatio will be true. If two expressios are ot equal i value, the mathematicias may use the symbol to show this. The statemet 7 4 is a example of this. Sice the two expressios are ot equal i value, the statemet is false (ot true) ad the should be used istead of the = symbol 7 4. The followig symbols are ofte used to compare the value of two expressios. Symbol Meaig Example = is equal to 1 = 1 is ot equal to is greater tha is less tha Example State whether each of the followig statemets is true or false. a 1 = b 4 = 1 cotiued 70
3 Solvig equatios8 chapter Example cotiued Workig a 1 = LS = 1 = 7 RS = = = 9 LS Z RS so the statemet is false. b 4 = 1 LS = 4 = 1 RS = 1 = 10 1 = 1 LS = RS so the statemet is true. Reasoig If the statemet is true, the right side ad the left side must be equal. Here they are ot equal so the statemet is ot true. The right side ad the left side are equal so the statemet is true. 8.1 Example By chagig oly the expressio o the right side of the equatio 4 = 6 1 6, fid three other expressios which make this statemet true. Workig Reasoig Both expressios have the same value Both expressios have the same value Both expressios have the same value. exercise 8.1 Example 1 1 I each of the followig equatios, isert a umber o the blak lie to make the equatio true. a 1 7 = 1 b 1 = 9 c = 48 d 9 1 = e 1 4 = 8 f = 4 9 g 6 ( 1 ) = 6 h 1 17 i ( ) 6 18 j
4 MathsWorld 7 Australia Curriculum editio Example Example Example 4 State whether each of the followig statemets is true or false. a 4 = 4 b 4 = 7 1 c 6 = 4 d 1 1 = 7 e 16 8 = 0 f 1 = 6 g h i j k (4 1 ) l 10 4 ( 1) The equatio is true. Which oe of the followig could replace the expressio o the right side of the equatio whilst still makig it a true statemet? A 6 1 B 1 C 10 1 D 1 1 E 0 1 The statemet 4 ( 1 ) 8 8 will become false if the left side is replaced with: A 8 4 B (10 1 ) C 6 4 D E 10 4 Rai says that if a statemet such as is true, the it is also correct that Is she right? Explai your aswer. exercise 8.1 challege Example 6 Chage a sigle umber i each of the followig to tur the statemet ito a equatio. The replace the,,. or with =. a 7 1 b c d e (18 9) f g , 4 h , i j (1 8) 4 4, k (18 6) 4. ( 1 1) 4 l
5 8. Iput ad output machies The equatios cosidered so far have oly cotaied umbers, ad so you ca tell that they are true statemets. I a equatio ivolvig proumerals as well as umbers, the statemet may be true oly for certai values of the proumerals. Cosider a umber machie that takes i umbers, doubles them ad the adds. Iput Iput Output Iput 1 = Output Supposig the output umber is 11. If the proumeral represets the iput umber, we ca write 1 11 or simply, I algebra we leave out the sig is a equatio. This meas that the left side must equal the right side. There is oly oe value of that makes this equatio true. We ca see that if the iput umber was 4, the the output umber would be 4 1, which equals 11. Fidig a value of the proumeral that makes a equatio true is called solvig the equatio. A value of the proumeral that makes a equatio true is called a solutio for the equatio. The solutio to the equatio 1 11 is = 4. I example 4 the umber machie processes are tured ito algebraic expressios ad the a equatio is writte. I later sectios of this chapter we will look at ways of solvig equatios. Whe traslatig a writte statemet ito a algebraic expressio there are some key words that idicate the operatios ivolved i the expressio. Some examples are provided i the followig table. Notice that it does ot matter which proumeral we choose for the ukow umber. 7
6 MathsWorld 7 Australia Curriculum editio Additio Subtractio Statemet Expressio Statemet Expressio A umber plus two a 1 A umber mius four d 4 The sum of a umber ad five Four more tha a umber b 1 The differece betwee te ad a umber 10 e c 1 4 Six less tha a umber f 6 Multiplicatio Divisio Statemet Expressio Statemet Expressio Four times a umber 4p A umber divided by six The product of three ad a umber A umber multiplied by five q The quotiet of a umber ad eight r Oe fifth of a umber t m 6 8 Example 4 Write a equatio which represets the followig umber machies. a b Divide the iput umber by three ad the subtract oe. 8 Subtract three from the iput umber ad the divide by two. 6 Workig Reasoig a 1 8 Divide the iput umber by ad the subtract 1. Put this expressio equal to the output umber. b 6 Subtract from the iput umber ad the divide by. Put this expressio equal to the output umber. 74
7 Solvig equatios8 chapter Example Write a equatio which represets the followig umber machies. a b 8. The quotiet of the iput umber ad five. Six added to the product of the iput umber ad five. 1 1 Workig Reasoig a 1 Divide the iput umber by. Put this expressio equal to the output umber. b Multiply the iput umber by ad add 6. Put this expressio equal to the output umber. exercise 8. Example 4 1 Write a equatio which represets the followig umber machies. a b Multiply the iput umber by four. Add seve to the iput umber. 0 c d Subtract five from the iput umber. Divide the iput umber by three e f Subtract the iput umber from te. Double the iput umber ad add two
8 MathsWorld 7 Australia Curriculum editio g h Multiply the iput umber by three ad subtract oe. Halve the iput umber ad subtract seve i j Divide the iput umber by three add two. Multiply the iput umber by four ad the divide by five k Add five to the iput umber ad the multiply the result by three. Which oe of the followig umber machies is represeted by the equatio x 1 10? l Subtract two from the iput umber ad the multiply by four. 0 A Subtract oe from the iput umber, multiply the result by two ad the divide by five. 10 B Multiply the iput umber by two, divide by five ad the subtract oe from the result. 10 C Divide the iput umber by five, multiply by two ad the subtract oe from the result. 10 D Subtract oe from the iput umber, divide by five the multiply the result by two. 10 E Multiply the iput umber by two, subtract oe ad the divide the result by five
9 Solvig equatios8 chapter Which oe of the followig equatios represets the umber machie? 8. Multiply the iput umber by four, divide the result by three ad the subtract two. 8 A 4( ) 8 B 4 8 C 4 8 D 4 8 E 4a b 8 Example 4 Write a equatio that represets each of the followig umber machies. a b The sum of the iput umber ad seve. The differece betwee the iput umber ad four. 6 c d The product of the iput umber ad eight. The quotiet of the iput umber ad three e f The differece betwee five ad the iput umber. The sum of four times the iput umber ad five. 9 g h The differece betwee three times the iput umber ad eight. The product of four ad the iput umber divided by five
10 MathsWorld 7 Australia Curriculum editio i j Add three to the product of the iput umber ad seve. Six more tha the quotiet of the iput umber ad three k l Five less tha the product of the iput umber ad two. Seve more tha the product of the iput umber ad three. 7 1 exercise 8. challege 6 Which oe of the followig equatios represets the umber machie below? A B The quotiet of six times two more tha the iput umber ad five C ( 1 ) D 1 E 6( 1 ) 1 Adriee uses a twostep umber machie. The outputs for three iput values are show i the table below. Iput Output 10 8 a If the iput umber is, what is the output umber? b If the output umber is 1, what is the iput umber? c If the output umber is x, what is the output umber? 78
11 8. Solvig equatios: arithmetic strategies Fidig the value(s) of the proumeral that makes a equatio true is called solvig the equatio. A value of the proumeral that makes a equatio true is called a solutio for the equatio. This sectio cosiders some ways of solvig equatios usig arithmetic rather tha algebra. Substitutio I sectio 8. we saw that the equatio 1 11 was true if = 4. If we substitute other values for we fid that the right side (RS) is ot equal to the left side (LS). 1 Value of LS Value of RS LS RS? False False 7 11 False 9 11 False True 1 11 False False By substitutig a value for the proumeral i a equatio, we ca fid if that value makes the equatio true. Example 6 Use substitutio to determie if the umber give i brackets is a solutio to the equatio. a x 7 11 (6) b p 1 4 (9) Workig Reasoig a x 7 Work with the LS of the equatio. = 6 7 Substitute the value for x. = 18 7 Simplify. = 11 x 6 is a solutio to the equatio. LS = RS cotiued 79
12 MathsWorld 7 Australia Curriculum editio Example 6 cotiued Workig b Reasoig p 1 Work with the LS of the equatio. = 9 = 18 1 = 6 1 = 1 p 9 is ot a solutio to the equatio. Substitute the value for p. Simplify. LS, RS 4 LS Z RS Observatio The solutio to a equatio may be obvious just from lookig at the equatio. For example, we ca see that the solutio to the equatio x 1 1 is x 9. Example 7 Solve the followig equatios by observatio. a x 1 b s 4 6 c m 7 9 Workig a x 1 1 x s b s 4 c m m 16 Reasoig Write dow the equatio. Thik of a umber that ca be multiplied by to give 1. Try. Write dow the solutio to the equatio. Write dow the equatio. Thik of a umber that ca be divided by 4 to give 6. Try 4. Write dow the solutio to the equatio. Write dow the equatio. Thik of a umber from which you ca subtract 7 ad get 9. Try 16. Write dow the solutio to the equatio. Table of values We ca fid the value of the proumeral that makes a equatio true by costructig a table of values for the left side. 80
13 Solvig equatios8 chapter Example 8 Use the table of values provided to fid the value of x that makes the equatio x 1 7 = 19 true. x x Workig x = 6 Reasoig Whe x = 6, x 1 7 has the value 19. The right side of the equatio is 19. x x Guess, check ad improve Aother useful techique is guess, check ad improve try a value for the variable, ad the check whether this value makes the equatio a true statemet. You ca repeat this with other values util you get a value which does make the LS = RS. Example 9 Fid the value of x that will make the equatio x 9 a true statemet. Workig x = 10 = 0 = so the equatio is ot true for x 10 x = 6 = 1 = so the equatio is true for x 6 LS = RS Reasoig Work with the LS of the equatio. Substitute a value for x. Make a guess that x 10 might work. Simplify. LS Z RS x 10 was too big so try a smaller value. Work with the LS of the equatio Substitute a value for x. This time, make a guess that x 6 might work. Simplify. 81
14 MathsWorld 7 Australia Curriculum editio exercise 8. Example 6 Example 7 1 Use substitutio to determie if the umber give i brackets is a solutio to the equatio. y a x 1; () b ; (8) c p 1 7; (6) 4 m d x 1; (10) e x 4 7; (.) f 1 9 1; (1) 9 y g (x 4) 6; (6) h ; (6) i 4(z 1 ) 0; () j (m ) 6; (1) k x ; (4.1) l (4m 1 ) If p =, which of the followig are true statemets? a p 9 b p c p 1 d 4p e p f p 1 1 ; a 1 b I each of the followig, use the table of values provided to fid the value of the variable that makes the equatio true. a x b y 4 1 c x 1 1 = 9 x x y y x x Example 7 4 Complete each of the followig tables to fid the value of the variable that makes the equatio true. a x 1 14 b x c x = 9 x x x x x x
15 Solvig equatios8 chapter d 4p 1 1 = 1 e x 1 = 44 f p 4p x x m 1 m m 8. Example 8 Example 9 6 Solve the followig equatios by observatio. a a 1 = 6 b b 1 4 = 7 c c 1 = 1 d d 4 = 1 e e 1 8 = f f 6 = 1 g g = 1 h h = 40 i 7i = 77 j j 1 k k 4 l l 7 6 Use guess, check ad improve to fid solutios to these equatios. a x b 4x 11 c x d x exercise 8. challege Example 9 7 Use guess, check ad improve to fid solutios to these equatios. a x 1 71 b 11x 7 0 c 17x 1 19 d 1x
16 8.4 Forward trackig ad backtrackig A flow chart provides a way of visualisig a sequece of steps. For istace, before goig out at the start of the day, Sady puts o her socks the puts o her shoes ad fiishes by tyig her shoelaces. The flow chart shows the order of the steps. This is called forward trackig. Start Put o socks. Put o shoes. Tie laces. Fiish Sady puts her shoes o forward trackig! At the ed of the day, Sady reverses these steps ad uties her shoelaces takes off her shoes the takes off her socks. This reverse process is called backtrackig. Start Take off socks. Take off shoes. Utie laces. Fiish Forward trackig with umbers I a umber flow chart, there is a sequece of operatios o the iput umber. Each operatio leads us to workig out the ext umber i the flow chart. Example 10 Sady takes her shoes off backtrackig! Complete the followig flow charts ad fid the output umber. a 1 b Workig a Reasoig Perform each operatio as you move from left to right through the flow chart. 84 The output umber is 1. cotiued
17 Solvig equatios8 chapter Example 10 cotiued Workig b Reasoig Perform each operatio as you move from left to right through the flow chart. 8.4 The output umber is 4. Backtrackig with umbers Cosider the followig puzzle. Trish thiks of a umber, multiplies it by three the subtracts eight ad the result is thirtee. The puzzle ca be writte as a flow chart, show at right. To fid Trish s umber, we work our way backwards through the flow chart doig the opposite. This is backtrackig. So Trish s umber is 7. Whe we backtrack with umbers, we must udo what has bee doe to the iput umber. We do this by workig backwards i the flow chart, carryig out the opposite or iverse umber operatio; for example, if 8 has bee added, the we udo this by subtractig 8. 8 Number Operatio Iverse operatio Example 11 Use backtrackig to fid the iput umber for each of the followig flow charts. a 7 b cotiued 8
18 MathsWorld 7 Australia Curriculum editio Example 11 cotiued Workig a Reasoig Workig from right to left, the iverse of 17 is 7 so 7 is 1. The iverse of 4 is so 1 is 4. 7 The iput umber is 4. b Workig from right to left, the iverse of 4 4 is 4 so 6 4 is 4. The iverse of is 4 so 4 4 is 8. The iverse of 1 is 11 so is The iput umber is 9. Usig forward trackig to build algebraic expressios Usig flow charts we ca build up algebraic expressios. Istead of startig with a kow umber, we start with a proumeral. At each step we write the ew expressio. Supposig we start with a umber the multiply it by. This gives the expressio. Next we subtract 1, so we ow have the expressio 1. Example 1 Start 1 Write a algebraic expressio that represets each of the followig flow charts. a b Fiish x 86 Workig Reasoig a First add to. This gives 1. The divide by. This gives 1. cotiued
19 Solvig equatios8 chapter Example 1 cotiued Workig Reasoig b 4 1 First multiply x by 4. This gives 4x. x 4x 4x 1 4x 1 The add 1. This gives 4x 1 1. Fially divide by. This gives 4x We ca work out how a algebraic expressio has bee built up by thikig about the usual order of operatios with umbers. I the expressio 1 1 for example, multiplicatio would be doe before additio. So is multiplied by ad the 1 is added. Example 1 I each of these expressios, what is the first operatio that has bee carried out o? a 1 b 1 4 c ( 1 4) d Workig a is multiplied by b is divided by Reasoig is multiplied by the is added to the result. is divided by the 4 is added to the result. c 4 is added to d is subtracted from Brackets are worked out before multiplicatio. 4 is added to the the result is multiplied by. We ca thik of as ( ). is subtracted from the the result is divided by. Example 14 Costruct a flow chart for each of the followig algebraic expressios. a 4 7 b p 1 Workig Reasoig a 4 7 Let the iput umber be. Multiply by 4. This gives The subtract 7. This gives 4 7. cotiued 87
20 MathsWorld 7 Australia Curriculum editio Example 14 cotiued Costruct a flow chart for each of the followig algebraic expressios. a 4 7 b p 1 Workig Reasoig a 4 7 Let the iput umber be. Multiply by 4. This gives The subtract 7. This gives 4 7. b Let the iput umber be p. Multiply p by. This gives p. p p p p The divide by. This gives p 1. Fially add. This gives p 1. exercise 8.4 Example 10 1 Copy ad complete the followig flow charts ad fid the output umber. a b 4 7 c 10 d e 8 1. f g. h i 1 j
21 Solvig equatios8 chapter k 6 l m For parts a ad b, copy ad complete the flow chart ad use it to fid the output umber. a Kristopher starts with the umber 1, divides by the adds 9. b Toby starts with the umber 1, adds 9 ad the divides by. c Do Kristopher ad Toby obtai the same output umber? Why or why ot? Example 11 Use backtrackig to fid the iput umber for each of the followig flow charts. a 1 b 7 1 c d e 4 f g h
22 MathsWorld 7 Australia Curriculum editio i 4 j 4 7 k 11 l Example 1 4 Copy ad complete these flow charts. Write the algebraic expressio which represets each flow chart. a 4 b 7 x y c 4 d 9 p y e 7 f 1 k z g 4 1 h 7 m c i 4 1 j 4 a h Examples 1, 14 For each of the followig i state the first operatio that is carried out o. ii copy ad complete the flow chart. a 1 1 b 4 90
23 Solvig equatios8 chapter c d ( 4) 8.4 e 1 4 f g 7 19 h Thuy thiks of a umber, the adds 11, doubles the result ad fially subtracts 4. If the startig umber was. a Draw a flow chart. b Write a expressio which represets the fial umber for this puzzle. c What would the fiishig umber be if the startig umber was 7? d What would the startig umber be if the fiishig umber was? Haibal thiks of a umber, the adds 1, halves the result ad fially adds 4. If the startig umber was. a Draw a flow chart. b Write a expressio which represets the fial umber for this puzzle. c What would the fiishig umber be if the startig umber is? d What would the startig umber be if the fiishig umber is 7? exercise 8.4 challege 8 Thom thiks of a umber, the multiplies it by 4 ad the adds 6. He halves the result, the subtracts the umber he first thought of. Fially, Thom subtracts. a Draw a flow chart to represet this umber puzzle. b Write a expressio for the fial umber i terms of a startig umber. c What is the fial umber if the startig umber is i 1? ii? iii? d What do you otice about your aswers from part c? Explai why this has occurred. 91
24 8. Solvig equatios by backtrackig We saw i sectio 8.4 how flow charts are useful for forward trackig to fid the fiishig umber if we kow the startig umber. reversig the steps to fid a startig umber if we kow the fiishig umber. buildig up algebraic expressios. I this sectio we will use backtrackig to solve equatios. This meas fidig the startig umber that makes the equatio true. Example 1 For each of these umber operatios i write a equatio. ii draw a flow chart ad use backtrackig to fid the umber. a 4 is added to a umber,, ad the result is 1 b a umber, x, is multiplied by ad the result is c 7 is subtracted from a umber, a, ad the result is 11 d a umber, b, is divided by 8 ad the result is 6 Workig Reasoig a i is added to meas 1 4. Put this expressio equal to 1. ii 4 1 Udo 1 4 by subtractig = 9 b i x ii x x is multiplied by meas x. We write this as x. Put this expressio equal to. Udo by dividig by x = 7 cotiued
25 Solvig equatios8 chapter Example 1 cotiued Workig Reasoig c i a is subtracted from a umber a meas a 7. Put this expressio equal to 11. ii 7 a 11 Udo 7 by addig a = 18 d i b 8 6 b is divided by 8 meas b 4 8. We write this as b. Put this expressio equal to 6. 8 ii 8 b 6 Udo 4 8 by multiplyig by b = 48 8 Example 16 Rebus thiks of a umber, multiplies it by, ad the adds, givig a fiishig umber of 1. a Usig for Rebus umber, draw a flow chart to represet this twostep process. b Use the flow chart to write a equatio. c Use backtrackig to solve the equatio, that is, fid the startig umber. Workig a Reasoig The first step of the forward trackig process is to multiply by, ad the secod step is to add. b 1 = 1 The fiishig umber is 1. cotiued 9
26 MathsWorld 7 Australia Curriculum editio Example 16 cotiued Workig c Reasoig The first step of the backtrackig process is to subtract from 1, ad the secod step is to divide 1 by. Subtractig is the opposite (iverse) of addig. Dividig by is the opposite of multiplyig by. The startig umber is 6. = 6 Check: LS RS 1 Example 17 Costruct a flow chart to represet the left side of each of the followig equatios ad the solve each equatio usig backtrackig. a x 1 19 b x 7 c (x 1 4) 18 Workig a As a flow chart x 1 19 ca be represeted as Reasoig Start with x, multiply by, the add. x x x So backtrackig to fid the startig umber x 7 x 14 x 19 The result or output is 19. Use backtrackig to fid x. The opposite of 1 is so subtract from 19, which gives 1. The opposite of is 4 so divide 1 by, which gives 6. Write the solutio. 94 cotiued
27 Solvig equatios8 chapter Example 17 cotiued Workig x = 6 Check: LS = 6 1 = 19 RS = 19 b As a flow chart x ca be represeted as Reasoig Costruct a flow chart. Start with x, divide by ad the subtract. 8. x x x So backtrackig to fid the startig umber x 4 x 1 x 7 The result or output is 7. Use backtrackig to fid x. The opposite to is 1 so add to 7 which gives 1. The opposite to 4 is so multiply 1 by which gives 4. Write the solutio. c x 4 4 x x 4 (x 4) Start with x, add 4 the multiply the result by. The brackets are ecessary to show that the whole expressio, x 1 4, is multiplied by. Backtrack to fid the startig umber. Dividig by is the opposite of multiplyig by. Subtractig 4 is the opposite of addig 4. x = 4 Write the solutio. 9
28 MathsWorld 7 Australia Curriculum editio Example 18 Costruct a flow chart for this equatio ad use it to solve the equatio Workig Reasoig 7 7 This is a threestep process. The first thig that is doe to x is that is subtracted from it. The result is the divided by. Fially 7 is added. So backtrackig to fid the startig umber = Use backtrackig to fid. The opposite of 1 7 is 7 so subtract 7 from 10, which gives. The opposite of 4 is so multiply by, which gives 6. The opposite of is 1, so add to 6, which gives 11. exercise 8. Example 1 Example 16 1 For each of the followig oestep equatios i draw a flow chart. ii use backtrackig to fid the value of. a 1 18 b 11 c d 4 e f 4 1 g h i For each of the followig twostep equatios i draw a flow chart. ii use backtrackig to fid the value of. a 1 1 b c d 7 8 e 7( 1 ) 8 f g j 6 h 1 i k 4 l
29 Solvig equatios8 chapter Example 17 m o p 4 7 q 7 4 r For each of the followig threestep equatios i draw a flow chart. ii use backtrackig to fid the value of. a d b ( 7) 7 c ( 1 ) 19 1 e ( 1 9) 7 0 f 7 1 g 4( 1) 1 h 4( 1 1) 7 i a 1b Example 18 4 I each of the followig, complete a flow chart ad use backtrackig to fid the startig umber. a Cara thiks of a umber, divides by ad adds to the result. She gets 0 as her aswer. b Viki thiks of a umber, adds, multiplies the result by 4 ad the subtracts. She gets 4 as her aswer. c Garth thiks of a umber, subtracts 8, divides the result by ad the adds. His aswer is. Use a flow chart ad backtrackig to solve each of the followig equatios. a a (a 1 1) 1 b (b ) 0 c 1 1 d s e a f (m 1 1) 14 g 4(k ) h ( ) 1 11 i exercise 8. challege 6 For each of the followig fourstep equatios i draw a flow chart. ii use backtrackig to fid the value of. a d 4( 1 ) b 4a e ( 1 1) 1 b 6 c a 1 1b f 4a 1 1 b 97
30 8.6 Solvig equatios: balace scales Oe way to thik about the solvig process is to cosider a equatio as a balace scale where the two sides of the balace must be kept level. Cosider the equatio x 1 where x kg is the mass of a toy mouse. By writig a series of equivalet equatios we ca solve the equatio ad fid that x =. Equivalet equatios (x 1 ) Balace scales diagram (Balaced) The scale is balaced because the two sides are equal. (x 1 ) (Ubalaced) By takig kg from the left side, the scale is o loger balaced (x 1 ) (Balaced) By takig kg from the right side as well, the scale is balaced agai. (x ) (Balaced) The mass of the toy mouse is kg
31 Solvig equatios8 chapter We ca write the solutio process that we used with the scales above as follows. x x 1 x (doig the same to both sides). Example 19 Cosider the equatio x = 0. Where x kg represets the mass of a cat. a Costruct a balace scales diagram to represet the equatio x = 0, ad the use diagrams ad equivalet statemets to solve this equatio for x. b Check your aswer by substitutig the aswer i to the left side (LS) of the origial equatio to see whether it equals the right side (RS) of the equatio. Workig a Reasoig We are assumig that both cats have the same mass. 0 x = 0 Divide each side by two. This is the same as halvig each side. 10 x = 0 10 cotiued 99
32 MathsWorld 7 Australia Curriculum editio Example 19 cotiued Workig Reasoig The cat o the left side balaces the 10 kg mass o the right side. 10 x = 10 b Check metally: LS = x = 10 = 0 = RS Substitutig x = 10 ito the LS of the origial equatio shows that it is equal to the RS. If the solutio is correct the you should get the same result o the LS ad RS of the equatio. To make equivalet equatios easier to read, it is commo practice i algebra to lie up the equals sigs vertically, as i example 19. This makes it easier to idetify the left side (LS) ad right side (RS) of each equatio. The balace scales model ca be used to solve more difficult equatios; as log as we do the same thig to both sides, the balace scales will remai level. Each time you do the same to both sides you are fidig a equivalet equatio to the step before. Example 0 y kg represets the mass of a possum. Solve for y if y =. Check your aswer by substitutig your solutio ito the origial equatio. Workig y = Reasoig I the example y kg represets the mass of a possum y = y = 0 Subtractig from both sides keeps the equatio balaced. cotiued
33 Solvig equatios8 chapter Example 0 cotiued Workig y = 0 Reasoig Divide both sides by. This is the same as sharig the 0 kg equally betwee the three possums. 10 y = 10 Oe possum has a mass of 10 kg, so y = 10. Substitutig y = 10 ito the origial equatio LS = y = (10) = = RS Substitute y = 10 ito the left side of the equatio. If LS = RS, the solutio is correct. exercise 8.6 Examples 19, 0 1 The diagram below shows Ruffy the dog o a set of balace scales. Draw a equivalet scale balace diagram which shows Ruffy s mass
34 MathsWorld 7 Australia Curriculum editio The diagram below shows Mojo the cat o a set of balace scales. Draw a equivalet balace scales diagram which shows Mojo s mass Example 0 For each of the followig balace scales i write a equatio that represets the situatio show, usig x kg to represet the mass of each toy aimal. ii use equivalet equatios to fid the mass of the toy aimals. a b c d For each of the followig balace scales i write the equatio represeted. ii describe i words how to get x o its ow whilst keepig the scales balaced. iii fid the solutio to the equatio. a b x 9 0 x
35 Solvig equatios8 chapter c d 8.6 x 14 x 7 9 e f x 1 x 18 g h x 4 11 x 8 exercise 8.6 challege For each of the followig balace scales i write the equatio represeted. ii describe i words how to get x o its ow whilst keepig the scales balaced. iii fid the solutio to the equatio. a b x (x )
36 8.7 Solvig equatios: doig the same to both sides So far i this chapter we have looked at several ways of solvig equatios. I sectio 8. we used arithmetic strategies: tables of values, ispectio ad guess, check ad improve. Whe dealig with more complicated equatios, it may ot be possible to solve by ispectio. It may ot be efficiet to use guess, check ad improve. I sectio 8. we used backtrackig i flow charts. Like arithmetic strategies, backtrackig may ot be efficiet or useful whe dealig with more complicated equatios. I sectio 8.6 we used the scale balace approach of keepig the two sides balaced by doig the same to both sides. This esured that we were always makig equivalet equatios. We obviously do t wat to draw a scale balace every time we solve a equatio, but the method of doig the same to both sides is a efficiet method for solvig equatios. I this sectio we will focus o algebraic solvig of equatios by doig the same to both sides to make equivalet equatios. We will pay particular attetio to careful settig out, eve with very simple equatios that you would be able to solve metally. Example 1 Solve each of the followig equatios. I each case, check your solutio by substitutio. a m 7 = 1 b Workig a m 7 1 m m 19 Check: LS m RS b Check: LS RS Reasoig m has had 7 subtracted from it. Add 7 to both sides. Substitutig m = 19 ito the LS of the equatio gives the same value as the RS. So, m 19 is correct. has had 6 added to it. Subtract 6 from both sides. Substitutig = 8 ito the LS of the equatio gives the same value as the RS. So, = 8 is correct. 404
37 Solvig equatios8 chapter Example Solve each of the followig equatios. I each case, check your solutio by substitutio. a 7a 4 b x 8.7 Workig a 7a 4 7a a 6 Check: LS 7a RS x b x x 1 Check: LS x Reasoig a has bee multiplied by 7. Divide both sides by 7. Substitute a = 6 i the LS of the equatio. x has bee divided by. Multiply both sides by. Substitute x = 1 i the LS of the equatio. 1 RS Example Solve each of the followig equatios. I each case, check your solutio by substitutio. a 10 d 1 4 b b = 1 Workig a 10 d d d 6 d d Check: RS d 1 4 = 1 4 = = 10 = LS Reasoig We wat to get d o oe side of the equals sig by itself. I the expressio d 1 4, d is first multiplied by ad the 4 is added. We eed to udo the operatios i the reverse order. First subtract 4 from both sides. The divide both sides by. Substitute d = i the RS of the equatio. cotiued 40
38 MathsWorld 7 Australia Curriculum editio Example cotiued Workig b b 1 b b 16 b 16 Reasoig We wat to get b o oe side of the equals sig o its ow. I the expressio b, b is first multiplied by ad the is subtracted. First add to both sides. The divide both sides by. b 16 b 1 Check: LS b RS Substitute b 1 i the LS of the equatio. Example 4 Solve the equatio t Check your solutio by substitutio. Workig d t t t t t 1 Check: LS t 1 4 Reasoig We wat to get t o oe side of the equals sig o its ow. I the expressio t 1 4, t is first divided by ad the 4 is added. First subtract 4 from both sides. The multiply both sides by. Substitute t = 1 i the LS of the equatio RS 406
39 Solvig equatios8 chapter exercise Example 1 Example a Example b Example Solve the followig equatios by doig the same to both sides. a x b x 7 11 c a d 6 1 a 9 e b f b 1 0 g a 16 h d 1 14 i d 8 j m 1 8 k m.7.1 l p m x x o 7 1 t Solve the followig equatios by doig the same to both sides. a 7m 49 b 1b 60 c 8m 6 d x 6 e 9m 6 f.4h 4 g 1.k 6 h a 11 i x 6.4 j a 1 k 4b 14 l 8h 84 Solve the followig equatios by doig the same to both sides. x a 7 b b 9 7 c k 0 d g j e a 8 1 f d 4. x 9 1 h m 11 6 i m 4 1. y. k b 1. l a. 7 Matthew wats to solve the equatio 4x 1 1 for x. To do this he eeds to A add to both sides ad the multiply both sides by 4. B subtract from both sides ad the multiply both sides by 4. C add to both sides ad the divide both sides by 4. D subtract from both sides ad the divide both sides by 4. E divide both sides by 4 ad the subtract from the result. Solve each of these equatios for a by doig the same to both sides. a a b a 4 1 c a 7 d a e 6a f 11a g a h 7a 1 1 i a j 4a k 4a 1 19 l 8a 1 61 m.a a o a.6 1. p 7a q a r 10a The solutio to the equatio m is A y 9 B y 18 C y 0 D y 6 E y
40 MathsWorld 7 Australia Curriculum editio Example Solve each of these equatios for by doig the same to both sides. a b 4 7 c 4 8 d g j e f h i k l m o p q r Solve each of these equatios. a d 11 4 b d x 9 14 c y 18 x 1 9 e b 4 7 f a g m j k h k i 4x b l 1.6a 1.1. m b. a o a p 4a q h r 1 14 Haa thiks of a umber, triples it ad subtracts 4, which leaves her with. Write a equatio which describes this umber puzzle. The solve this equatio to fid what the startig umber must have bee. 408 exercise 8.7 challege 10 Solve for p. Give your aswer correct to two decimal places. a 1 = p 16 b.6p.4 = 7.8 c 7.64 = p 4 7 p d 14 =.4p 1 e = 7 f =.(p ) Solve for the ukow i the followig equatios. a v = 17 q b (a 4.) = 0. c 4 = 7 4 d 4a 10 1 e 4 a f k
41 8.8 Further equatios to solve by doig the same to both sides Equatios with brackets Whe a equatio icludes a expressio i brackets, we may be able to simplify the left side by dividig both sides of the equatio by a commo factor. Example Solve the equatio (a 1 ) 1, checkig your solutio by substitutio. Workig (a 1 ) 1 (a 1 ) 1 a 1 6 a 1 6 a Check: LS = (a 1 ) = ( 1 ) = 6 = 1 = RS Reasoig (a 1 ) has bee multiplied by. There is a commo factor of o each side of the equatio. Divide both sides by a has added to it. Subtract from both sides. Substitute a = i the LS of the equatio. If there is o commo factor o the two sides, we ormally expad the brackets usig the distributive law. For example, (a 1 ) a 1 Example 6 a 1 10 Solve the equatio (a 1 ) 11, checkig your solutio by substitutio. cotiued 409
42 MathsWorld 7 Australia Curriculum editio Example 6 cotiued Workig (a 1 ) 11 a 1 11 a a a a 4 4 a 1 Reasoig (a 1 ) has bee multiplied by. Multiply each term iside the brackets by. a has 6 added to it. Subtract 6 from both sides. a is multiplied by. Divide both sides by. Check: LS = (a 1 ) = a 1 1 b Substitute a 1 equatio. i the LS of the = 1 = 11 = RS Example 7 Solve ( a) = 0 for a. Check your aswer usig substitutio. Workig ( a) = 0 ( a) = 0 ( a) = 18 ( 1 a) = 18 a =.6 a =.6 a = 0.6 Check: LS = ( a) = ( 0.6) =.6 = 18 = 0 = RS Reasoig Subtract from both sides of the equatio. Divide both sides of the equatio by. Subtract from both sides of the equatio. Substitute a = 0.6 ito the LS of the equatio. 410
43 Solvig equatios8 chapter A algebraic fractio with a expressio such as x 1 7 i the umerator should be treated as if it were i brackets. After expadig brackets, it may be possible to simplify the left side before doig the same to both sides. 8.8 Example 8 Solve the x 1 7 Workig x 1 7 b 8 (x 1 7) 8 x x x 17 Check: LS x RS 8. Check your solutio by substitutio. Reasoig (x 1 7) has bee divided by. Udo by multiplyig both sides by. x has 7 added to it. Subtract 7 from both sides. Substitute x = 17 i the LS of the equatio. Equatios with egative umbers Durig the solvig of some equatios, the proumeral may ed up o its ow o the left side with a egative coefficiet. Multiplyig both sides of the equatio by 1 will chage the coefficiet of the proumeral to a positive umber. Example 9 Solve the equatio 1 x. Check your solutio by substitutio. Workig Method 1 1 x 1 1 x 1 x 9 x (1) 9 (1) x 9 Reasoig Subtract 1 from both sides. 1 9 The coefficiet of x is egative so multiply both sides by 1. x (1) 1x ad 9 (1) 19 cotiued 411
44 MathsWorld 7 Australia Curriculum editio Example 9 cotiued Workig Method 1 x 1 x 1 x 1 x 1 1 x 1 1 x 9 x Check: LS = 1 x = 1 9 = = RS Reasoig x has bee subtracted from 1. Add x to both sides. Subtract from both sides. Substitutig = 8 ito the LS of the equatio gives the same value as the RS. Example 0 Solve these equatios, checkig the solutio by substitutio. a a b 7b 6 Workig a a a a 7 Check: LS = a 1 11 = = 4 = RS b 7b 6 7b b 9 Check: LS = 7b = 7 (9) = 6 = RS Reasoig Subtract 11 from both sides Substitute a 7 i the LS. Divide both sides by 7. A egative umber divided by a positive umber is egative. Substitute b 9 i the LS. 41
45 Solvig equatios8 chapter Example 1 Solve the equatio x , checkig the solutio by substitutio. 8.8 Workig x x x.4 x.4 x 0.8 Check: LS = x 1.7 = = = 0. = RS Reasoig Subtract.7 from both sides Divide both sides by. A egative umber divided by a positive umber is egative. Substitute x 0.8 i the LS. Equatios with the proumeral o both sides I some equatios the proumeral occurs o both sides of the equatio. We subtract the term cotaiig the proumeral from oe side of the equatio. Example If x represets the mass i kilograms of oe toy possum i this balace scales diagram, a write a equatio to represet these balace scales. b solve the equatio to fid the value of x. c check the solutio Workig Reasoig a x 1 1 x 1 6 Mass i kilograms of LS = x 1 1 Mass i kilograms of RS = x 1 6 cotiued 41
46 MathsWorld 7 Australia Curriculum editio Example cotiued Workig b x 1 1 x 1 6 x 1 1 x x 1 6 x x x x Check: LS x RS x LS RS Reasoig Take oe possum from each side. There is ow oe possum o the left side ad oe o the right side. Subtract x from both sides so that x is removed from the right side. Subtract 1 from both sides. Substitute x = ito the LS ad evaluate. Substitute x = ito the RS ad evaluate. exercise 8.8 Example Example 6 Example 7 Example Solve the followig equatios. a (p 1 ) b 4 (c 1 1) c (m ) 16 d (x ) 0 e 1 (m 1 7) f 6 4(k 1 1) g (z ) 1 h 14 7(x ) i 9(4x 1 ) 6 Solve the followig equatios, givig the values of m i fractio form. a (m 1 4) 11 b 4(m 1 4) 0 c (m ) 17 d (m 7) e (m 1 1) 10 f (m 1 8) 6 g (m 1) h 4(m ) 7 i (m 1 ) 11 Solve the followig equatios, givig the value of a i decimal form where appropriate. a (a 1 ) 1 7 b (a 1 7) 1 c (a 1 4) 1 11 d 4(a 1 ) 7 19 e (a 1 ) 1 9 f (a ) 1 7 g (a 1 4) 1 14 h (a 1 7) 9 i 10(a 1 ) Solve the followig equatios. x 1 4 a b d g x 1 x 1 4 e 11 h x 1 11 x x c f 7. i x 6 x x
47 Solvig equatios8 chapter Example 9 Example 0a Example 0b 6 7 Solve the followig equatios. a 6 b 4 b 1 b 8 c 1 b 16 d 9 b 1 e 17 b 9 f b 10 g 6 b 11 h 41 8b 9 i 17 b 6 Solve the followig equatios. a x b x 1 14 c x d x 7 e x 8 10 f x 1 11 g x h x i x 8 1 Solve the followig equatios. a m 18 b m c m d m 1 e m 8 f m 1 g m 9 h m 6 11 i m 4 7 j m 7 k m 4 l m 8.8 Example Solve the followig equatios. a 19 7 b c d 1 e 8 16 f 1 1 g h 8 4 i 1 14 For each of these balace scales use x to represet the mass i kilograms of oe toy aimal. i Write a equatio to represet these balace scales. ii Solve the equatio to fid the value of x. iii Check the solutio. a b c d
48 MathsWorld 7 Australia Curriculum editio e f Example 10 Solve these equatios for x. a 4x 1 7 x 1 8 b 7x 11 x 1 9 c x 1 x 1 1 d 6x 11 x 1 7 e 8x 1 4x 7 f.4x x 1 8 g x 1 x 1 7 h 1x 11x 8 11 Gemma starts with a umber ad the divides by, ad subtracts from the result. If she the doubles this, she eds up with the umber 8. a Write a equatio which represets this umber puzzle. b Solve this equatio to fid what the startig umber must have bee. 1 Radha starts with a umber ad the subtracts, ad divides this aswer by the adds 9 ad the result is 11. a Write a equatio which represets this umber puzzle. b Solve this equatio to fid what the startig umber must have bee. 1 Grat thiks of a umber, adds, multiplies the result by 4, the subtracts. His fial umber is 8. What is the origial umber? exercise Fid the ukow value i each case. (p 1) a b a m 4 1 1b 1 1 c am 1b 9 challege d 4a k 6 b 1 e aa 1b f 4 (a ) = 7 g 4(x 1) 1 6 h a m 4 b 1 i (x 4)
49 8.9 Solvig problems with equatios May worded problems, icludig those with diagrams, ca be solved usig algebra. The process of traslatig the words of a problem ito algebra is referred to as formulatio. As a startig poit, it is importat to read the questio carefully ad idetify the ukow, ad the try to costruct a equatio that ca be solved. A fourstep strategy exists for settig up ad solvig worded problems with algebra. It ca be summarised as follows. 1 Traslate the words ito algebra. Decide o the ukow ad give it a proumeral, the formulate a equatio usig this proumeral. Solve the equatio. Solve by doig the same to both sides. Check the solutio. Substitute your solutio back ito the origial equatio to check that the LS RS. 4 Traslate the algebra back ito words. Express your solutio i terms of the problem wordig. Example Katy has brought home some baby chicks from school. Her sister Rachel brigs home aother four, so they have a total of 11 chicks. a Formulate a equatio that represets the umber of chicks i the home. b Solve your equatio to fid out how may chicks Katy brought home. Workig a Let c be the umber of chicks that Katy brought home. c 4 = 11 b c = 7, so Katy brought home seve chicks. Reasoig Total chicks = umber of chicks Katy has brought home 1 umber of chicks Rachel brought home So total chicks = c 1 4 If the total umber of chicks is 11, this meas that c 1 4 = 11 c 4 = 11 is a equatio that expresses the umber of chicks. By ispectio, it ca be see that c =
50 MathsWorld 7 Australia Curriculum editio Equatios ca be used to solve problems relatig to cosecutive umbers. Cosecutive umbers are whole umbers that follow oe aother. For example, 8, 9, 10 are cosecutive umbers. If we let represet a whole umber, the the ext cosecutive umber is oe more tha, that is, 1 1. The ext cosecutive umber is 1. Example 4 The sum of two cosecutive whole umbers is 11. Fid the umbers. Workig Let be the first umber, ad 1 be the ext umber. ( 1) = 11 1 = = 11 1 = 10 = 10 = Substitute = ito the origial equatio. LS = ( 1) = ( 1) = 6 = 11 = RS The two cosecutive umbers are ad 6. Reasoig Step 1: Words ito algebra Decidig o the variable is the first step, ad the writig a equatio that uses this variable is the ext step. Step : Solve the equatio Subtract 1 from both sides. Divide both sides by. Step : Check the solutio Substitutig the solutio = ito the LS. Step 4: Algebra ito words The first umber is, which is. The secod umber is 1, which is 6. Express your aswer i terms of the problem. Example Formulate a equatio which shows the perimeter of this shape i terms of the legth of each side, w cm, give that the perimeter is kow to be 1 cm. The solve it for w. w cm cotiued 418
51 Solvig equatios8 chapter Example cotiued Workig If w is the side legth, ad the shape is a square, the a equatio represetig this is w w w w = 1 4w = 1 4w 4 = 1 4 Reasoig The ukow variable w (side legth) is give. The sides are of equal legth, ad the total perimeter is 1 cm. The expressio 4w has the same value as 1. Divide both sides by w = 4 or.7 Check: If w =.7, LS = 4w = 4.7 = 1 = RS Example 6 Josef works i a supermarket o the weeked. He ears $8.0 a hour. If Josef ears $10.7 oe week but that amout icludes a $ bous for workig a extra shift, use the fourstep approach above to fid out how may hours he worked i that week. Workig Let h = the umber of hours Josef works. The amout of moey Josef will ear is 8.h 8.h = h = h = h = h 8. = h = 9. If h = 9., substitute this ito the origial equatio 8.h = 10.7 LS = 8.h = = 80.7 = 10.7 RS Reasoig Step 1: Words ito algebra Choose a proumeral for the ukow. Write a equatio. Step : Solve the equatio Subtract from both sides. Divide both sides by 8.. h is a umber so we write h = 9., ot h = 9. hours. Step : Check the solutio Substitute the solutio h = 9. ito the RS. cotiued 419
52 MathsWorld 7 Australia Curriculum editio Example 6 cotiued Workig Josef works for 9 1 hours i that week. Step Reasoig 4: Algebra ito words Express your aswer i terms of the problem. exercise 8.9 I this exercise, use the four steps for solvig algebra word problems. 40 Example Example Example Example 1 4 Pearlie has bee give a hadful of jelly beas, ad is the give seve more. a Let b represets the umber of jelly beas that Pearlie started with. Write a expressio for the umber of jelly beas that Pearlie ow has. b If she eds up with jelly beas, write a equatio which represets this situatio. c Solve this equatio to fid the value of b. d How may jelly beas did Pearlie have i her had to start with? There are 71 chocolate buttos i two piles. The secod pile has 1 more tha the first pile. a If there are c chocolate buttos i the first pile, write a expressio for the umber of chocolate buttos i the secod pile. b Write a equatio to describe this situatio. c Solve the equatio to fid the value of c. d How may chocolate buttos are i each pile? Aiela has a bag of baaa lollies that she shares with three frieds. Each perso receives k lollies ad there are three left over. a Write a expressio for the total umber of lollies i the bag. b If there are lollies i the bag, write a equatio to fid how may lollies each perso receives. c Solve the equatio to fid the value of k. d How may lollies does each perso receive? Graph paper is sold i packets cotaiig x sheets. Narelle s folder holds six packets of graph paper. She has used five sheets from the folder. a Write a expressio for the umber of sheets of graph paper i Narelle s folder, b If there are 17 sheets of graph paper i Narelle s folder, write a equatio to fid how may sheets come i a packet? c Solve the equatio to fid the value of x. d How may sheets come i a packet?
53 Solvig equatios8 chapter 6 Two childre are tryig to work out the age of their two gradparets, Sarah ad William. Here is what their gradmother told them. William is 10 years older tha I am. If you add together our two ages you get 14 years. a Write a equatio to represet this situatio usig x to represet Sarah s age i years. b How old are the gradparets? A teacher asked her class to write dow a equatio for the followig setece: If you add 6 to a ukow umber ad the divide the result by, the aswer is. The resposes of two studets are show below. Which studet do you thik is correct? For the studet who is icorrect, what is the error? 8.9 Deise + 6 = Michael + 6 = Example 4 Example 4 Example 7 The sum of two cosecutive whole umbers is 9. a Let be the smallest of the two umbers. Write a equatio to represet this. b Solve the equatio to fid the value of. c What are the two umbers? 8 The sum of three cosecutive umbers is 7. a Let be the smallest of the three umbers. Write a equatio. b Solve the equatio to fid the value of. c What are the three umbers? 9 A square garde bed has side legth x metres. a Write dow a expressio for the perimeter of the garde bed i terms of x. b If the perimeter of the garde bed is 76 metres, write a equatio to fid the value of x. c Solve the equatio to fid the value of x. 10 For each of these shapes, the perimeter is kow, but there are ukow side legths. For each shape i write a equatio usig the give iformatio. Do ot iclude cm i your equatio. ii solve the equatio to fid the value of the proumeral. iii list all the side legths. a b c x y x cm Perimeter = 64 cm Perimeter = 4.8 cm Perimeter = 6 cm 41
54 MathsWorld 7 Australia Curriculum editio Example 6 Example 6 11 If you multiply a decimal umber by 6 ad the subtract 4, the result is 4.1. a Usig to stad for the ukow umber, write a equatio. b Solve the equatio to fid the ukow umber. 1 Ali plats petuias i 1 rows. There are p petuias i each row ad 1 petuias i total. a Write this iformatio as a equatio. b How may petuias are i each row? 1 Alex works i a supermarket o the weeked. He ears $7.0 a hour. a If Alex works for hours, write a expressio for the amout of moey he ears. Do ot iclude the $ sig i your expressio. b If Alex ears $7.0 oe week, write a equatio to work out how may hours he worked. c Solve the equatio ad write dow how may hours Alex worked. 14 Madeleie works as a waitress at weekeds ad ears $16.0 per hour. Oe weeked she eared $6 which icluded $4 i tips. a Let be the umber of hours Madeleie worked. Write a equatio usig the give iformatio. Do ot iclude the $ sig i your equatio. b Solve your equatio. c How may hours did Madeleie work? 1 A urse worked a total of 48 hours i a week. She worked four ormal shifts ad 1 hours of overtime. a Let the legth of her ormal shift be hours. Write a equatio usig the give iformatio. Do ot iclude hours i your equatio. b Solve your equatio. c How log is her ormal shift? exercise 8.9 challege 16 Lucy is three years older tha Peter ad she is six years older tha Domiic. If you add together the ages of the three childre you get 0 years. Let d stad for Domiic s age. a Write a expressio for Lucy s age i terms of d. b Write a expressio for Peter s age i terms of d. c Write a equatio to represet the sum of the three ages. d Solve the equatio for d. e List the ages of the three childre. 4
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