2Linear and UNCORRECTED SAMPLE PAGES. simultaneous equations. Australian curriculum. What you will learn. Chapter 2A 2B 2C 2D 2E 2F 2G 2H 2I

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1 A B C D E F G H I J K Chpter Wht you will lern Liner n simultneous equtions Algeri epressions (Consoliting) Simplifying lgeri epressions (Consoliting) Epning lgeri epressions Solving liner equtions Equtions with rkets n pronumerls on oth sies Solving wor prolems Inequlities Using formuls Simultneous equtions: sustitution (Etening) Simultneous equtions: elimintion (Etening) Applitions of simultneous equtions (Etening) Austrlin urriulum NUMBER AND ALGEBRA Ptterns n lger Apply the istriutive lw to the epnsion of lgeri epressions, inluing inomils, n ollet like terms where pproprite

2 On ollision ourse Liner equtions with two vriles, suh s + 3y = 1, hve n infi nite numer of solutions. If we plot them on numer plne we get line. Eh point on the line represents possile solution. These equtions n moel mny ifferent situtions or systems in rel life. For instne the two vriles might e totl ost n quntity of n item, or the istne of n ojet from its strting point over time, when it moves t uniform spee in stright line. Although there is no single solution to suh istne-time eqution for one ojet, there is only one solution tht stisfi es oth equtions for two ojets moving t onstnt spee in the sme plne (provie they re not moving on prllel trks). In this sitution the two equtions re lle simultneous equtions. The one solution tht stisfi es the simultneous equtions n e foun using lger, n this solution represents the position n time t whih they meet or ollie. Solving simultneous equtions n e pplie to prolems suh s working out when n where one ship will interept nother, ut nvigtion is only one of mny res where this spet of lger hs pplitions. Online resoures Chpter pre-test Vieos of ll worke emples Intertive wigets Intertive wlkthroughs Downlole HOTsheets Aess to HOTmths Austrlin Curriulum ourses

3 78 Chpter Liner n simultneous equtions A Algeri epressions CONSOLIDATING Key ies Alger is entrl to the stuy of mthemtis n is ommonly use to solve prolems in vst rry of theoretil n prtil prolems. Alger involves the represention n mnipultion of unknown or vrying quntities in mthemtil ontet. Pronumerls (or vriles) re use to represent these unknown quntities. Let s strt: Rememering the voulry Stte the prts of the epression 5 y + (4) tht mth these wors. pronumerl (or vrile) term oeffiient onstnt term squre term Vriles representing unknown quntities re use in wie rnge of jos n ouptions. In lger, letters re use to represent numers. These letters re lle pronumerls or vriles. An epression is omintion of numers n pronumerls onnete y the four opertions +,, n. Brkets n lso e use. For emple: 5 + 4y 1 n 3( + ) y 5 A term is omintion of numers n vriles onnete with only multiplition n ivision. Terms re seprte with the opertions + n. For emple: 5 + 7y is two-term epression. Coeffiients re the numers eing multiplie y pronumerls. For emple: the 3 in 3 n 1 in re oeffiients. Constnt terms onsist of numer only. For emple: in + 4 (The sign must e inlue.) Epressions n e evlute y sustituting numer for pronumerl. For emple: if = then + 6 = + 6 = 4 Orer of opertions shoul e followe when evluting epressions: 1 Brkets Powers 3 Multiplition n ivision 4 Aition n sutrtion

4 Numer n Alger 79 Emple 1 Writing lgeri epressions for wor prolems Write n lgeri epression for the following: the numer of tikets neee for 3 oys n r girls the ost of P pies t $3 eh the numer of grms of penuts for one hil if 300 g of penuts is shre eqully mong C hilren. SOLUTION EXPLANATION 3 + r 3 tikets plus the numer of girls 3P 3 multiplie y the numer of pies 300 C Emple Converting wors to epressions Write n lgeri epression for the following: five less thn the sum of n is ivie y 4 SOLUTION 300 g ivie into C prts three more thn twie the squre of the sum of n y EXPLANATION 5 5 sutrte from + 3 Twie plus The sum of n is one first ( + ) n the result ivie y 4. ( + y) The sum of n y is one first n then the result is squre.

5 80 Chpter Liner n simultneous equtions Emple 3 Sustituting vlues into epressions Evlute these epressions if = 5, = n = 3. 7 ( ) SOLUTION 7 ( ) = 7 5 (5 3) = 35 = 35 4 = 31 = ( ) 5 3 = 4 15 = 11 Eerise A 1 Stte the numer of terms in these epressions. EXPLANATION 5 + y Mth n item in the left olumn with n item in the right olumn. A Prout Division B C D E F Sum Differene Quotient 1 3 Stte the oeffiient in these terms. 5y e f Sustitute the vlues for n. When using orer of opertions, evlute rkets efore moving to multiplition n ivision then ition n sutrtion. Evlute powers efore the other opertions. ( ) = ( ) = Sutrtion Multiplition Aition the reiprol of the squre of 3 5 UNDERSTANDING

6 Numer n Alger 81 Emple 1 Emple 4 5 Write n lgeri epression for the following. The numer of tikets require for: i 4 oys n r girls iii oys n g girls ii iv t oys n girls oys, y girls n z ults The ost of: i P pies t $6 eh ii 10 pies t $n eh iii D rinks t $ eh iv P pies t $5 n D rinks t $ The numer of grms of lollies for one hil if 500 g of lollies is shre eqully mong C hilren. Write n lgeri epression for eh of the following. The sum of n The sum of n y 5 less thn The prout of n 3 e The ifferene etween 3 n y f Three times the vlue of p g Four more thn twie h The sum of n y is ivie y 5 i 10 less thn the prout of 4 n j The squre of the sum of m n n k The sum of the squres of m n n l The squre root of the sum of n y m The sum of n its reiprol n The ue of the squre root of Emple 3 6 Evlute these epressions if = 4, = 3 n = 7. e + f + 7 Evlute these epressions if =, y = 1 n z = 1 6. y + z y + 8 A retngulr gren e is 1 m long n 5 m wie. Fin the re of the gren e. The length is inrese y m n the with is erese y y m. Fin the new length n with of the gren. Write n epression for the re of the new gren e. g 4, 5 6(½) 4, 5 6(½), 7 1 ( ) yz 8, 9 8, 9 h 3 z + 1 y 4, 5 6(½), 7 9, 10 FLUENCY PROBLEM-SOLVING A

7 8 Chpter Liner n simultneous equtions A 9 The epression for the re of trpezium is 1 ( + )h where n re the lengths of the two prllel sies n h is the istne etween the two prllel sies. Fin the re of the trpezium with = 5, = 7 n h = 3. A trpezium hs h = 4 n re 1. If n re positive integers, wht possile vlues n the vrile hve? 10 The ost of 10 ientil puzzles is $P. Write n epression for the ost of one puzzle. Write n epression for the ost of n puzzles. 11 For eh of these shpes, write n epression for: i the perimeter y p ii the re 1 Deie if the following sttements refer to the sme or ifferent epressions. If they re ifferent, write n epression for eh sttement. A Twie the sum of n y B The sum of n y A The ifferene etween hlf of n hlf of y B Hlf of the ifferene etween n y 13 For right-ngle tringle with hypotenuse n shorter sies n, Pythgors theorem sttes tht = +. Whih of these two esriptions lso esries Pythgors theorem? A The squre of the hypotenuse is equl to the squre of the sum of the two shorter sies. B The squre of the hypotenuse is equl to the sum of the squres of the two shorter sies. For the inorret esription, write n eqution to mth , 1 5 y 1, 13 PROBLEM-SOLVING REASONING

8 Numer n Alger 83 The sum of the first n positive integers 14 The rule for the sum of the first n positive integers is given y: The prout of n n one more thn n ll ivie y. Write n epression for the ove esription. Test the epression to fin these sums i (n = 4) ii (n = 10) Another wy to esrie the sme epression is: The sum of hlf of the squre of n n hlf of n. Write the epression for this esription. Chek tht your epressions in prts n re equivlent (the sme) y testing n = 4 n n = e (n + n) is lso equivlent to the ove two epressions. Write this epression in wors (5 + 1) A igrm representing the sum of the first five positive integers rrnge oring to the epression in Question ENRICHMENT A

9 84 Chpter Liner n simultneous equtions B Simplifying lgeri epressions CONSOLIDATING Key ies Just s = 4, so = 4 or 4. We sy tht the epression is simplifie to 4. Similrly, = 8 n 8 3 = 5. All these epressions hve like terms n n e simplifie to n epression with smller numer of terms. A single term suh s 5 10 n lso e simplifie using multiplition n ivision, so 5 10 = =. Let s strt: Are they equivlent? All these epressions n e seprte into two groups. Group them so tht the epressions in eh group re equivlent. y 4 10 y y + y + 1 y y ( 6 + 0y 1 ) 1 y The symols for multiplition ( ) n ivision ( ) re usully not shown in simplifie lgeri terms. For emple: 5 = 5 n 7 y = 7 y When iviing lgeri epressions ommon ftors n e nelle. For emple: 7 14 =, = 1 1 = 7y 14y = 15 n 10 = 3 5 = 3 5 Like terms hve the sme pronumerl ftors. For emple: 5 n 7 re like terms n 3 n re like terms. Sine = then n re lso like terms. The pronumerl prt of term is often written in lphetil orer. Like terms n e ollete (e n sutrte) to form single term. For emple: = 13 4 y y = y Unlike terms o not hve the sme pronumerl ftors. For emple: 5,, y n 4yz re ll unlike terms. 5

10 Numer n Alger 85 Emple 4 Multiplying lgeri terms Simplify the following. 3 SOLUTION 3 = 3 = 6 3 = 3 = 6 Emple 5 Diviing lgeri terms Simplify the following SOLUTION = (3) = = 4 Emple 6 Colleting like terms Simplify the following y olleting like terms EXPLANATION Multiply the oeffiients. Multiply the oeffiients n simplify. 1 (3) EXPLANATION Del with numerls n promumerls seprtely, nelling where possile. Write s frtion first. Cnel where possile, rell =. SOLUTION = = y y EXPLANATION Collet like terms (3 n ). The sign elongs to the term tht follows. Comine their oeffiients 3 = y y = y + 7y = 7 + 9y Collet like terms n omine their oeffiients.

11 86 Chpter Liner n simultneous equtions Emple 4 Emple 4 Emple = = 7 6 Eerise B 1 Write the missing wor or epression. Like terms hve the sme ftors. 3 + simplifies to. 3 n re terms. Simplify these frtions Collet like terms. Rememer = n = = 7 n = 6 e 4 f 5 g h Deie if the following pirs of terms re like terms. 4 n 3 n 7y 3t n 6tw e 7yz n yz g 5 n h 3 y n 7y Simplify the following. 5 m 6 e 3p 6r f 4m 4n i 4 3 j 3 5 Simplify the following. 4n 6n 7 3 g 3y 4y e h Simplify the following y nelling. e i 8 5y 0y 5 9 f j 6 10st 6t 7y y g k 3q q 5mn ( 3n) 4 ( ) 1, 3(½) 3(½) (½), 9 11(½) g 3 5p 7y 4r 3 s 4 6 u v u f i f i n 4m mn n 9nm 4 n (½) h l 5s s 3gh ( 6h) mn 3mn h 3 y 5m ( 3n) 5j ( 4) k 3mn 6n 5r s 8rs 4 11(½) UNDERSTANDING FLUENCY

12 Numer n Alger 87 Emple 5 Emple 6 Emple 6 Emple 6 7 Simplify the following y first writing in frtion form. 5 4 ( 3) 11mn 3 1 e 10 (gh) f 8 g 3y (y) h 7mn (3m) i 7pq (6p) j 4 (8) k 5 y ( 5y) l 9m n (18mn) 8 Simplify the following. 4 y 5 p 6 ( ) ( 3) () e 7 (5m) n f 5s (t) 4 g 6 4mn (3m) h 8 3y (8) i 3 1 (9) j 4 3y () k 10m 4mn (8mn) l 3pq pq p Simplify the following y olleting like terms n + 3n e 6 g 4y 3y + 8 h j k 5m m Simplify the following y olleting like terms y + + y 6t + 5 t e y y 4 f 3mn 4 + 4nm 5 g h 3st 8ts + st + 3ts f i l 1y 4y 7mn + mn mn 6y + y + 4y 4 + Simplify the following y olleting like terms. 5y 4y m n 6nm + m n 7p q p q 4p q e y 4y + 5y f 10rs + 3rs 6r s g 7 3 h i 10pq qp 3pq 6pq j 1m n mn 4m n + mn 1 A frmer hs pigs n y hikens. Write n epression for the totl numer of hes. Write n epression for the totl numer of legs. 13 A retngle s length is three times its with metres. Write n epression for: the retngle s perimeter the retngle s re. 1, FLUENCY PROBLEM-SOLVING B

13 88 Chpter Liner n simultneous equtions B 14 A right-ngle tringle hs sie lengths 5 m, 1 m n 13 m. Write n epression for: the tringle s perimeter the tringle s re. 15 The verge (men) mrk on test for 0 stuents is. Another stuent who sores 75 in the test is e to the list. Write n epression for the new verge (men). 16 Deie if the following re lwys true for ll rel numers. = = = + = + e = f 1 1 = ( 0) 17 The igrm shows the route tken y slesperson who trvels from A to D vi B n C. If the slesperson then returns iretly to A, write n epression (in simplest form) for the totl istne trvelle. If y = + 1, write n epression for the totl istne the slesperson trvels in terms of only. Simplify your epression. When y = + 1, how muh woul the istne hve een reue y (in terms of ) if the slesperson h trvelle iretly from A to D n stright k to A? Higher powers 18 For this question, note this emple: 3 4 = = Simplify these epressions with higher powers A 3y D + y e 5 3 f 16, 17 B C PROBLEM-SOLVING REASONING ENRICHMENT g h i j k l 10 10

14 Numer n Alger 89 C Epning lgeri epressions A mentl tehnique to fin the prout of 5 n 3 might e to fin 5 0 n 5 3 to give 115. This tehnique uses the istriutive lw over ition. So 5 3 = 5 (0 + 3) = Sine vriles (or pronumerls) represent numers, the sme lw pplies for lgeri epressions. Let s strt: Retngulr istriutions This igrm shows two joine retngles with the given imensions. Fin two ifferent wys to write epressions for the omine re of the two retngles. Compre your two epressions. Are they equivlent? This igrm shows retngle of length reue y length of 3. Fin two ifferent wys to write epressions for the remining re (she). Compre your two epressions. Are they equivlent? 5 Emples of the istriutive lw n often e foun in everyy tsks. The istriutive lw is use to epn n remove rkets. A term on the outsie of the rkets is multiplie y eh term insie the rkets. ( + ) = + or ( ) = ( + ) = or ( ) = + If the numer in front of the rket is negtive, the sign of eh of the terms insie the rkets will hnge when epne. For emple: ( 3) = + 6 sine = n ( 3) = Key ies

15 90 Chpter Liner n simultneous equtions Emple 7 Epning simple epressions with rkets Epn the following. 3( + 4) 5( 11) ( 5) SOLUTION EXPLANATION 3( + 4) = = 3 n 3 4 = 1 5( 11) = = 5 n 5 ( 11) = 55 ( 5) = + 10 = n ( 5) = +10 Emple 8 Epning rkets n simplifying Epn the following. 4( + 3y) (4 3) SOLUTION 4( + 3y) = y = 4 + 1y (4 3) = 4 + ( ) ( 3) = Emple 9 Simplifying y removing rkets EXPLANATION Epn the following n ollet like terms. 3( 4) 3( + y) (3 + y) SOLUTION 3( 4) = (3 1) = = 14 3 Multiply eh term insie the rkets y 4. 4 = 4 n 4 3 y = 1y. Eh term insie the rkets is multiplie y. 4 = 8, = n ( 3) = 6 3( + y) (3 + y) = 3 + 6y 3 y = y y = 5y EXPLANATION 3( 4) = 3 1. (3 1) = 1(3 1), so multiplying y negtive 1 hnges the sign of eh term insie the rkets. (3 + y) = 1 (3 + y) = 3 y. Collet like terms n simplify.

16 Numer n Alger 91 Eerise C 1 (½) Emple 7, Emple 7 Emple 8 Emple 9 1 This igrm shows two joine retngles with the given imensions. Write n epression for the re of: i the lrger retngle ( y 5) ii the smller retngle ( y 5) Use your nswers from prt to fin the omine re of oth retngles. Write n epression for the totl sie length of the sie involving. Use your nswer from prt to fin the omine re of oth retngles. e Complete this sttement: 5( + ) = + Sustitute the vlue = 5 into these epressions. i ( + 3) ii Do your nswers from prt suggest tht ( + 3) = + 6? When epne wht shoul ( + 3) equl? Sustitute the vlue = 10 into these epressions. i 3( 1) ii 3 3 Do your nswers from prt suggest tht 3( 1) = 3 3? When epne wht shoul 3( 1) equl? 3 6(½) 3 7(½) Epn the following. ( + 3) 5( + 1) ( 7) 7( 9) e 3( + ) f 7(3 ) g 4(7 ) h ( 6) Epn the following. 3( + ) ( + 11) 5( 3) 6( 6) e 4( ) f 13(5 + ) g 0(9 + ) h 300(1 ) Epn the following. ( + ) 5( ) 3(m 4) 8( + 5) e 3(4 + 5) f 4( y) g 9t(y 3) h (3 + 4) i ( 5) j (3 5) k (4 + 1) l 5y(1 3y) Epn the following n ollet like terms. 3 + ( + 4) 4 + 6( 3) + 5(3 1) 5 + (3 4) e 3 + 4( ) f 7 + ( 3) g 3( + ) h 1 5( + 4) i 5 ( 6) j 9 ( 3) k 5 (3 + ) l 4 (3 ) 5 3 7(½) UNDERSTANDING FLUENCY

17 9 Chpter Liner n simultneous equtions C Emple 9 7 Epn the following n ollet like terms. ( + 3) + 3( + ) ( 3) + ( 1) 3( + 1) + 5( 1) 4(3 + ) + 5( 3) e 3( + 1) + ( 3) f ( + ) + 3( 1) g (4 3) (3 1) h 3(4 + 3) 5(3 1) i ( + 3) 3( + 5) j ( 4) 3(3 + 5) k 3(3 1) ( ) l 4(5 ) ( 5) 8 A retngle s length is 4 more thn its with,. Fin n epne epression for its re. 9 Fin the re of these si shpes in epne form. All ngles t verties re e , f Gry gets onus $0 for every omputer he sells over n ove his quot of 10 per week. If he sells n omputers in week n n > 10, write n epression for Gry s onus in tht week (in epne form). 11 Jill pys t t 0 in the ollr for every ollr erne over $ If Jill erns $ n > , write n epression for Jill s t in epne form. 1 1, 13 1 Ientify the errors in these epressions then write out the orret epnsion. ( + 6) = + 6 ( 4) = 4 3( + 4) = ( 7) = 7 49 e 5 ( 7) = 5 14 = 9 f 4( ) 3( + ) = = 13 In Yer 7 n 8 we eplore how to use the istriutive lw to fin prouts mentlly. For emple: = 7 ( ) n 4 98 = 4 (300 ) = = = 78 = 119 Use the istriutive lw to evlute these prouts mentlly e 3 99 f g h , 13 FLUENCY PROBLEM-SOLVING REASONING

18 Numer n Alger 93 Pronumerls n tes A progressive inome t system inreses the t rte for higher inomes. Here is n emple. Inome T 0 $0 000 $0 $0 001 $ $0 + 0% of inome ove $0 000 $ $ % of inome ove $ $ % of inome ove $ Fin the vlues of n in the ove tle. Fin the t pyle for these inomes. i $ ii $7 000 iii $ Fin n epression for the t pyle for n inome of $ if: i ii < iii < iv > Chek tht you hve fully epne n simplifie your epressions for prt. A steps if require. e Use your epressions from prts n to hek your nswers to prt y hoosing prtiulr inome mount n heking ginst the tle ove. ENRICHMENT C

19 94 Chpter Liner n simultneous equtions D Key ies Solving liner equtions A mthemtil sttement ontining n equls sign, left-hn sie n right-hn sie is lle n eqution. 5 = 10, 3 = 9, + 1 = 10 n 1 = re emples of equtions. Liner 5 equtions n e written in the form + = where the power of is = 6, 3 = ( + 1) n 5 3 = + 1 re ll liner equtions. Equtions re solve y fining the vlue of the vrile 4 (or pronumerl) tht mkes the eqution true. This n e one y inspetion for very simple liner equtions (for emple, if 3 = 15 then = 5 sine 3 5 = 15). More omple liner equtions n e solve through series of steps where eh step proues n equivlent eqution. Let s strt: Why re they equivlent? The following list of equtions n e tegorise into two groups. The equtions in eh group shoul e equivlent. 5 = 0 1 = 3 = 4 1 = 3 7 = = = 1 = 1 5 Disuss how you ivie the equtions into the two groups. How n you hek to see if the equtions in eh group re equivlent? Equivlent equtions re rete y: ing numer to or sutrting numer from oth sies of the eqution multiplying or iviing oth sies of the eqution y the sme numer (not inluing 0). Solve liner eqution y reting equivlent equtions using inverse opertions (sometimes referre to s ktrking). The solution to n eqution n e heke y sustituting the solution into the originl eqution n heking tht oth sies re equl.

20 Numer n Alger 95 Emple 10 Solving simple liner equtions Solve eh of the following equtions. + 3 = = 7 5 = = e = 9 SOLUTION + 3 = 4 = 1 = 1 ( 1 Chek: + 3 = 4 ) 4 3 = 7 4 = 10 = 40 Chek: (40) 4 3 = 10 3 = 7 5 = 1 = 7 = 7 ( Chek: 5 7 ) = = = 7 3 = = 6 = 3 Chek: (3) + 5 = + 5 = 7 3 EXPLANATION Sutrt 3 from oth sies. Divie oth sies y. Chek the nswer y sustituting = 1 into the originl eqution. A 3 to oth sies. Multiply oth sies y 4. Chek the nswer y sustituting = 40 into 3. Sine this equls 7, = 40 is the solution. 4 Sutrt 5 from oth sies. Divie oth sies y. Chek the nswer. Sutrt 5 from oth sies first then multiply oth sies y 3. Divie oth sies y. Chek the nswer.

21 96 Chpter Liner n simultneous equtions Emple 10 Emple 10 e = + 4 = 18 = 14 = 7 Chek: (7) + 4 = = Eerise D Multiply oth sies y 9 first to eliminte the frtion. Solve the remining eqution y sutrting 4 from oth sies n then iviing oth sies y. Chek the nswer. 1 Write own the vlue of tht is the solution to these equtions. No written working is require. 3 = 9 4 = = 1 7 = 1 Use guess n hek (tril n error) metho to solve these equtions. No written working is require. + 1 = = 1 4 = = 5 e + 3 = 6 f = 16 g = 1 7 h 3 + = Whih of the following equtions re equivlent to 3 = 1? = = 1 1 = 1 3 = e 3 4 = 3 f = 4 g 1 (½), = 10 Solve eh of the following equtions. Chek your nswers. + 5 = = 11 3m 4 = 8 4 = 6 e n + 13 = 7 f + 5 = 7 g + 15 = 7 h 3y = 13 i 3 + = 7 j = 5 k 4 = 10 l 6 5 = 3 m 7y 3 = 8 n + 1 = 1 5n 1 4 o 4 = 1 Solve eh of the following equtions = = 5 e i = 4 5 = f y 5 4 = g 4 8(½) = = 1 4 8(½) h h 3 = 1 t + 5 = s 3 = 7 j m 4 = 3 k 1 y 5 = l 5 = 4 4 8(½) UNDERSTANDING FLUENCY

22 Numer n Alger 97 Emple 10 Emple 10 Emple 10e 6 Solve eh of the following equtions. 1 = 18 7 = = 5 3 = 13 e 5 = 9 f 4 7 = 3 g 5 8 = h 3 4 = Solve these equtions. 3 = 6 e 3 4 = 1 f 3 = 9 5n 4 = 1 5 i = 3 j = 7 g k 4 3 = = 7 3f 11 = Solve eh of the following equtions. Chek your nswers y = 4 = 5 = 3 3 e i 1 = = 9 3 f 5 3 = 3m 1 g = 4 5 j 3 6 = 5 k 4 y = h l h l 5 = = 7 3 4z 3 = = 3 + = t = 3 9 For eh of the following, write n eqution n solve it to fin the unknown vlue. Use s the unknown vlue. If 8 is e to ertin numer, the result is 34. Seven less thn ertin numer is 1. I think of numer, oule it n 4. The result is 10. I think of numer, hlve it n sutrt 4. The result is 10. e Four less thn three times numer is 0. f A numer is multiplie y 7 n the prout is ivie y 3. The finl result is 8. g Five Ester eggs re e to my initil olletion of Ester eggs. I shre them etween myself n friens n eh person gets etly four. How mny were there initilly? h My weekly py is inrese y $00 per week. Hlf of my py now goes to py the rent n $100 to uy groeries. If this leves me with $450, wht ws my originl weekly py? 9(½) FLUENCY PROBLEM-SOLVING D

23 98 Chpter Liner n simultneous equtions D 10 Desrie the error me in eh of these inorret solutions = 4 = 7 5 = = = 7 5 = 3 3 = 5 5 = 15 = = 4 = 6 = An eqution like ( + 3) = 8 n e solve without epning the rkets. The first step is to ivie oth sies y Use this pproh to solve these equtions. i 3( 1) = 1 ii 4( + ) = 4 iii 7(5 + 1) = 14 iv 5(1 ) = 10 v (3 + 1) = 3 vi 5(1 4) = 1 10, 11 By onsiering your solutions to the equtions in prt, when o you think this metho is most pproprite? Chnging the sujet 1 Mke the sujet of eh of the following equtions. e i = = + = f j + = = 1 = g k FPO = 3 = 6 = e h l = = 3 4 = 3f e 1 REASONING ENRICHMENT

24 Numer n Alger 99 E Equtions with rkets n pronumerls on oth sies More omple liner equtions my hve pronumerls on oth sies of the eqution n/or rkets. Emples re 3 = 5 1 or 4( + ) = 5. Brkets n e remove y epning n equtions with pronumerls on oth sies n e solve y olleting like terms using ition n sutrtion. Let s strt: Steps in the wrong orer The steps to solve 3( ) = ( 1) re liste here in the inorret orer. 3( ) = ( 1) = = 6 3 = 4 + Arrnge them in the orret orer working from top to ottom. By onsiering ll the steps in the orret orer, eplin wht hs hppene in eh step. Solving prolems in lger (like mny other proeures n puzzles) requires steps to e one in the right orer. Equtions with rkets n e solve y firstly epning the rkets. For emple: 3( + 1) = eomes =. If n eqution hs pronumerls on oth sies, ollet to one sie y ing or sutrting one of the terms. For emple: = 3 eomes + 4 = 3 y sutrting from oth sies. Emple 11 Solving equtions with rkets n pronumerls on oth sies Solve eh of the following equtions. (3 4) = 11 ( + 3) 4 = 8 5 = 3 4 3( + 4) = 8( + 1) SOLUTION (3 4) = = 11 6 = 19 = 19 6 or EXPLANATION Epn the rkets, (3 4) = 3 + ( 4). A 8 to oth sies then ivie oth sies y 6, leving your nswer in frtion form. Key ies

25 100 Chpter Liner n simultneous equtions Emple 11 ( + 3) 4 = = = 8 = = 1 5 = 3 4 = 4 = = 1 3( + 4) = 8( + 1) = = = = = Eerise E Epn the rkets n ollet ny like terms, i.e. 4 =. Sutrt 6 from oth sies. Divie y. Collet terms on one sie y sutrting 3 from oth sies. A to oth sies n then ivie oth sies y. Epn the rkets on eh sie. Sutrt 6 from oth sies, lterntively sutrt 8 to en up with + 1 = 8. (Sutrting 6 keeps the -oeffiient positive.) Solve the eqution n mke the sujet. 1, 1 Epn these epressions n simplify. 3( 4) + (1 ) + 3( 1) + ( 3) 5(1 ) + e 3(3 ) f 7( ) 5( ) Show the net step only for the given equtions n instrutions. ( + 3) = 5 (epn the rkets) 5 + ( 1) = 7 (epn the rkets) = 6 (sutrt from oth sies) 4 3 = + 1 (sutrt from oth sies) 3 3 5(½) 3 6(½) Solve eh of the following equtions y first epning the rkets. ( + 3) = 11 5( + 3) = 8 3(m + 4) = 31 5(y 7) = 1 e 4(p 5) = 35 f (k 5) = 9 g 4(5 ) = 18 h (1 m) = 13 i 5(3 ) = 19 j 7( + 1) = 8 k 4(3 ) = 30 l 3(3n ) = 0 m 5(3 ) = 6 n 6(1 y) = 8 o 4(3 ) = (½) UNDERSTANDING FLUENCY

26 Numer n Alger 101 Emple 11 Emple 11 Emple Epn n simplify then solve eh of the following equtions. ( + 4) + = 14 ( 3) 3 = 4 4( 1) + 1 = 0 3( + 3) 1 4 = 4 e 3( 4) + ( + 1) = 15 f ( + 1) 3( ) = 8 g 6( + 3) + = 6 h 3( + ) + 5 = 46 i 3( 3) + = 1 j (3 + 1) + 3 = 19 Solve eh of the following equtions. 5 = = 7 4 4t = 10 t 3m 8 = m e 5 3 = f = g 1 3 = h 3y + 6 = y i 5m 4 = 1 6m Solve eh of the following equtions. 5( ) = 13 3( + 1) = (y + 4) = y 6 ( + 5) = 4 e 5 4 = 6( + ) f (4m 5) = 4m + g 3( 3) = 5( + ) h 4( 3) = 3(3 + 1) i 3( ) = 5( + 4) j 3(n ) = 4(n + 5) k ( + 5) = ( + 3) l 4( + ) = 3( + 1) 7 Using for the unknown numer, write own n eqution then solve it to fin the numer. The prout of n 3 more thn numer is 7. The prout of 3 n 4 less thn numer is 4. When less thn 3 lots of numer is oule the result is 5. When 5 more thn lots of numer is triple the result is 10. e more thn 3 lots of numer is equivlent to 8 lots of the numer. f more thn 3 times the numer is equivlent to 1 less thn 5 times the numer. g 1 less thn oule numer is equivlent to 5 more thn 3 lots of the numer. 8 Sine Tr strte work her originl hourly wge hs een triple, then erese y $6. It is now to e oule so tht she gets $18 n hour. Write n eqution n solve it to fin Tr s originl hourly wge. 9 At the strt of lunh Jimmy n Jke eh rought out new g of mrles to ply with their friens. By the en of lunh they were surprise to see they still h the sme numer s eh other even though overll Jimmy h gine 5 mrles n Jke h ene up with two lots of 3 less thn his originl mount. How mny mrles were originlly in the gs? 7, 8 7, 8 8, 9 FLUENCY PROBLEM-SOLVING E

27 10 Chpter Liner n simultneous equtions E 10 Consier the eqution 3( ) = 9. Solve the eqution y firstly iviing oth sies y 3. Solve the eqution y firstly epning the rkets. Whih of the ove two methos is preferle n why? 11 Consier the eqution 3( ) = 7. Solve the eqution y firstly iviing oth sies y 3. Solve the eqution y firstly epning the rkets. Whih of the ove two methos is preferle n why? 10, 11 10, 11 1 Consier the eqution = 5 7. Solve the eqution y firstly sutrting 3 from oth sies. Solve the eqution y firstly sutrting 5 from oth sies. Whih metho ove o you prefer n why? Desrie the ifferenes. Literl solutions with ftoristion 13 Literl equtions ontin vrile (suh s ) n other vriles (or pronumerls) suh s, n. To solve suh n eqution for, ftoristion n e use s shown here. = + = ( ) = = Sutrt from oth sies Ftorise y tking out Divie oth sies y ( ) Note: Solve eh of the following for in terms of the other pronumerls y using ftoristion. = = = = 4 5 e = f ( ) = g = + h ( + ) = ( ) 10 1 Using CAS lultor E: Solving equtions This tivity is in the intertive tetook in the form of printle PDF. 13 REASONING ENRICHMENT

28 Numer n Alger 103 F Solving wor prolems Mny types of prolems n e solve y writing n solving liner equtions. Often prolems re epresse only in wors. Reing n unerstning the prolem, efining vrile n writing n eqution eome importnt steps in solving the prolem. Let s strt: Too muh television? Three friens, Rik, Kte n Sue ompre how muh television they wth in week t home. Kte wthes 3 times the mount of television of Rik n Sue wthes 4 hours less television thn Kte. In totl they wth 45 hours of television. Fin the numer of hours of television wthe y Rik. Let hours e the numer of hours of television wthe y Rik. Write epressions for the numer of hours of television wthe y Kte n y Sue. Write n eqution to represent the informtion ove. Solve the eqution. Answer the question in the originl prolem. To solve wor prolem using lger: Re the prolem n fin out wht the question is sking for. Define vrile n write sttement suh s: Let e the numer of The vrile is often wht you hve een ske to fin. Write n eqution using your efine vrile to show the reltionship etween the fts in the question. Solve the eqution. Answer the question in wors. Emple 1 Turning wor prolem into n eqution Five less thn ertin numer is 9 less thn three times the numer. Write n eqution n solve it to fin the numer. SOLUTION Let e the numer. 5 = = 9 4 = = The numer is. EXPLANATION Define the unknown s pronumerl. 5 less thn is 5 n this equls 9 less thn three times, i.e Sutrt from oth sies n solve the eqution. Write the nswer in wors. Key ies

29 104 Chpter Liner n simultneous equtions Emple 13 Solving wor prolems Emple 1 Emple 13 1 Dvi n Mith me 54 runs etween them in riket mth. If Mith me 68 more runs thn Dvi, how mny runs i eh of them mke? SOLUTION Let the numer of runs for Dvi e r. Numer of runs Mith me is r r + (r + 68) = 54 r + 68 = 54 r = 186 r = 93 Dvi me 93 runs n Mith me = 161 runs. Eerise F EXPLANATION Define the unknown vlue s pronumerl. Write ll other unknown vlues in terms of r. Write n eqution: numer of runs for Dvi + numer of runs for Mith = 54. Sutrt 68 from oth sies n then ivie oth sies y. Epress the nswer in wors. 1 1(½) For eh of the following emples, mke the unknown numer n write n eqution. Three less thn ertin numer is 9 less thn four times the numer. Seven is e to numer n the result is then multiplie y 3. The result is 9. I think of numer, tke wy 9, then multiply the result y 4. This gives n nswer of 1. A numer when oule results in numer tht is 5 more thn the numer itself. e Eight less thn ertin numer is more thn three times the numer. Leonie n Emm sore 8 gols etween them in netll mth. Leonie sore 8 more gols thn Emm. e Define vrile for the numer of gols sore y Emm. Write the numer of gols sore y Leonie in terms of the vrile in prt. Write n eqution in terms of your vrile to represent the prolem. Solve the eqution in prt to fin the unknown vlue. How mny gols i eh of them sore? 6 7, 4, 5, 7 9 UNDERSTANDING FLUENCY

30 Numer n Alger A retngle is four times s long s it is wie n its perimeter is 560 m. Define vrile for the unknown with. Write n epression for the length in terms of your vrile in prt. Write n eqution involving your vrile to represent the prolem. Drw n lel retngle to help you. Solve the eqution in prt. e Wht is the length n with of the retngle? 4 Toy rente r for totl ost of $90. If the rentl ompny hrge $40 per y, plus hiring fee of $50, for how mny ys i Toy rent the r? 5 Anrew wlke ertin istne, n then rn twie s fr s he wlke. He then ught us for the lst km. If he trvelle totl of 3 km, fin how fr Anrew wlke n rn. 6 A prize of $1000 is ivie etween Aele n Benit so tht Aele reeives $80 more thn Benit. How muh i eh person reeive? 7 Kte is three times s ol s her son. If Kte is 30 yers oler thn her son, wht re their ges? 8 A trin sttion is etween the towns Antville n Bugville. The sttion is four times s fr from Bugville s it is from Antville. If the istne from Antville to Bugville is 95 km, how fr is it from Antville to the sttion? Hint: rw igrm to help you piture the prolem. 9 Anrew, Bren n Cmmi ll work prt-time t supermrket. Cmmi erns $0 more thn Anrew n Bren erns $30 less thn twie Anrew s wge. If their totl omine wge is $400, fin how muh eh of these workers erns. 10, 11, 13 10, 1, 13 10, My ought totl of 1 fition n non-fition ooks. The fition ooks ost $1 eh n the non-fition ooks ost $5 eh. If she pi $48 ltogether, how mny of eh kin of ook i she purhse? Define the numer of non-fition ooks ought in terms of the numer of fition ooks ought. 11 If I multiply my ge in si yers time y three, the resulting ge is my mother s ge now. If my mother is urrently 48 yers ol, how ol m I? FLUENCY PROBLEM-SOLVING F 1 Twelve yers go Eri s fther ws seven times s ol s Eri ws. If Eri s fther is now 54 yers ol, how ol is Eri now?

31 106 Chpter Liner n simultneous equtions F 13 In yht re the seon leg ws hlf the length of the first leg, the thir leg ws two-thirs of the length of the seon leg, n the lst leg ws twie the length of the seon leg. If the totl istne ws 153 km, fin the length of eh leg. 14 The Ae Biyle Shop hrges flt fee of $4, plus $1 per hour, for the hire of iyle. The Best Biyle Shop hrges flt fee of $8, plus 50 ents per hour. Connie n her friens hire three iyles from Ae, n Dvi n his rother hire two iyles from Best. After how mny hours will their hire osts e the sme? 15 Cr A left Melourne for Aelie t 11:00 m n trvelle t n verge spee of 70 km per hour. Cr B left Melourne for Aelie t 1:00 pm on the sme y n trvelle t n verge spee of 90 km per hour. At wht time will Cr B th Cr A? 16 Two poks in the shpes shown elow re to e fene with wire. If the sme totl mount of wire is use for eh pok, wht re the imensions of eh pok in metres? w + 5 w w Conseutive integers n e represente lgerilly s, + 1, + et. Fin three onseutive numers tht to 84. i Write three onseutive even numers strting with. ii Fin three onseutive even numers tht to 18. i Write three onseutive o numers strting with. ii Fin three onseutive o numers tht to 51. i Write three onseutive multiples of 3 strting with. ii Fin three onseutive multiples of 3 tht to , 17 17, 18 PROBLEM-SOLVING REASONING

32 i i i i 107 F U N SA C O M R PL R E EC PA T E G D ES 18 Teo proues tey er whih sells for $4. Eh tey er osts the ompny $8 to mnufture n there is n initil strt-up ost of $700. Write rule for the totl ost, $T, of prouing tey ers. If the ost of prtiulr proution run ws $9600, how mny tey ers were mnufture in tht run? If tey ers re sol, write rule for the revenue, $R, reeive y the ompny. How mny tey ers were sol if the revenue ws $8400? e If they wnt to mke n nnul profit of $54 000, how mny tey ers o they nee to sell? REASONING Numer n Alger An rt urtor ws investigting the prie trens of two rt works tht h the sme initil vlue. The first pinting, Green poles, oule in vlue in the first yer then lost $8000 in the seon yer. In the thir yer its vlue ws three-qurters of the previous yer. ENRICHMENT Wore hllenges The seon pinting, Orhis, e $ to its vlue in the first yer then the seon yer its vlue ws only thir of the previous yer. In the thir yer its vlue improve to oule tht of the previous yer. If the vlue of the pintings ws the sme in the thir yer, write n eqution n solve it to fin the initil vlue of eh pinting. 0 Juli rove to her holiy estintion over perio of five ys. On the first y she trvels ertin istne, on the seon y she trvels hlf tht istne, on the thir y thir of tht istne, on the fourth y one-qurter of the istne n on the fifth y one-fifth of the istne. If her estintion ws 1000 km wy write n eqution n solve it to fin how fr she trvelle on the first y, to the nerest kilometre. 1 Ann King is yers ol. Her rother Henry is two-thirs of her ge n her sister Chloe is three times Henry s ge. The twins who live net oor re 5 yers oler thn Ann. If the sum of the ges of the King hilren is equl to the sum of the ges of the twins, fin the ges of ll the hilren. i i i i

33 108 Chpter Liner n simultneous equtions Progress quiz 38pt A 1 Write n lgeri epression for the following: The numer of tikets require for 5 oys, girls n y ults 15 less thn the prout of 4 n y The sum of the squres of k n p The squre of the ifferene etween m n n e The squre root of 16 more thn 38pt A Evlute these epressions if = 4, y =. 38pt B 38pt B 38pt C 38pt C 38pt D 38pt E 38pt F y 5y 3( y) Simplify the following. 7 5pq ( 8p) + y 1k 3km Simplify the following y olleting like terms. 6y + 7 4y 7 3mk + 4km + 4 8y 3 3y + y Epn the following. 4( + 7) 3( 5) Epn the following n ollet like terms. 10 3( ) 4( + y) 3( + y) 3y 30y 3 ( 5y) Solve eh of the following equtions. 3 7 = = 9 p = = 7 e 3t 5 4 = 8 f k 3 = 1 6 g 8 3 = 14 3k 6 h = 5 3 Solve eh of the following equtions. 3( 4) = 7 4(3 + 1) 3 = 31 7m 4 = 3m 1 3(y + 1) = 8(y ) 38pt F 10 For eh of the following emples, mke the unknown numer n write n eqution. Five is e to numer n the result is then multiplie y. The result is 14. Nine less thn ertin numer is 6 more thn four times the numer. If I multiply my ge in five yers time y four, the resulting ge is my grnfther s ge now. If my grnfther is urrently 88 yers ol, set up n solve n eqution to etermine how ol I m now.

34 Numer n Alger 109 G Inequlities An inequlity (or ineqution) is mthemtil sttement whih uses <,, > or sign. Some emples of inequlities inlue: < 6, 5 1, n > 4. Inequlities n represent n infinite set of numers. For emple, the inequlity < 6 mens tht < 3 n this is the infinite set of ll rel numers less thn 3. Let s strt: Infinite solutions Greg, Kevin n Gret think tht they ll hve orret solution to the eqution Greg sys = 4 is solution. Kevin sys = 10 is solution. Gret sys = 100 is solution. Use sustitution to show tht they re ll orret. Cn you fin the smllest whole numer whih is solution to the inequlity? Cn you fin the smllest numer (inluing frtions) whih stisfies the inequlity? Wht metho les you to your nswer? Inequlities n e illustrte using numer line euse line represents n infinite numer of points. Use n open irle when showing > (greter thn) or < (less thn). For emple: > < Use lose irle when showing (greter thn or equl to) or (less thn or equl to). For emple: A set of numers my hve oth n upper n lower oun. For emple: < Key ies

35 110 Chpter Liner n simultneous equtions Key ies Liner inequlities n e solve in similr wy to liner equtions. All the numers tht stisfy n equlity re lle solution set. If, we multiply or ivie oth sies of n inequlity y negtive numer, the inequlity sign is reverse. For emple: 5 < 8 ut 5 > 8 so if > 1 then < 1. If we swp the sies of n inequlity, then the inequlity sign is reverse. For emple: 3 < 7 ut 7 > 3 so if > then <. Emple 14 Representing inequlities on numer line Show eh of the following emples on numer line: is less thn 6 ( < 6) is greter thn or equl to ( ) is greter thn ut less thn or equl to 3 ( < 3). SOLUTION < < Emple 15 Solving inequlities EXPLANATION An open irle is use to inite tht 6 is not inlue. A lose irle is use to inite tht is inlue. An open irle is use to inite tht is not inlue n lose irle is use to inite tht 3 is inlue. Fin the solution set for eh of the following inequlities. 3 < 7 5 > SOLUTION 3 < 7 < 10 EXPLANATION A 3 to oth sies

36 Numer n Alger 111 Emple 14 Emple 15 5 > 3 > < Eerise G Sutrt 5 from oth sies. Divie oth sies y, n reverse the inequlity sign. A 3 to oth sies. Multiply oth sies y 4; the inequlity sign oes not hnge. Gther pronumerls on one sie y sutrting from oth sies. Sutrt 3 from oth sies n then ivie oth sies y 4. Ple the vrile on the left n reverse the inequlity. 1 Insert the symol < or > to mke eh sttement true , (½) (½) Show eh of the following inequlities on numer line. > 1 4 < e 1 f < 8 g 1 < < 1 h i 0 3 j 5 k 3 < 4 l 6 < Fin the solution set for eh of the following inequlities. + 5 < 8 > 3 y 8 > 1 + m < 7 e 5 15 f 4t > 0 3 6(½) 3 7(½) g 3 4 h y i 3m 7 < 11 j k 7 5 < l 7 > 9 3 7(½) UNDERSTANDING FLUENCY

37 11 Chpter Liner n simultneous equtions G Emple 15 Emple 15 Emple Fin the solution set for eh of the following inequlities. 4 3 > 8 4n e 5 11 f 7 3 g 3 > 9 h 4t + 10 i 6m 14 < 15 Fin the solution set for eh of the following inequlities < 4 7 e 3 4 > 6 f Solve eh of the following inequlities. 4( + ) < 1 3( + 5) > 9 5(3 ) 5 (3 ) > 1 e 5(y + ) < 6 f 7(1 ) < 11 7 Fin the solution set for eh of the following inequlities t + > t 1 3 < 4 e 1 3m 7 4m f y y 7 5 > Weny is yers ol n Jy is 6 yers younger. The sum of their ges is less thn 30. Write n inequlity involving n solve it. Wht n you sy out Weny s ge? 9 The perimeter of prtiulr retngle nees to e less thn 50 m. If the length of the retngle is 1 m n the with is w m, write n inequlity involving w n solve it. Wht with oes the retngle nee to e? 10 How mny integers stisfy oth of the given equtions? n > 10 n > n 3 3 > < n 6 3 < 7 11 The with of retngulr re is 10 m n its height is ( 4) m. If the re is less thn 80 m, wht re the possile integer vlues for? 1 Two r rentl ompnies hve the following pyment plns: Crz: $90 per week n 15 per kilometre Rent: $110 per week n 10 per kilometre Wht is the mimum whole numer of kilometres tht n e trvelle in one week with Crz if it is to ost less thn it woul with Rent? P < 50 m 1 m 10 1 w m FLUENCY PROBLEM-SOLVING

38 Numer n Alger , 14 14, Consier the inequlity >. i List 5 vlues of etween 1 n whih mke the inequlity true. ii Wht must e true out ll the vlues of if the inequlity is true? Consier the inequlity < 5. i List 5 vlues of whih mke the inequlity true. ii Wht must e true out ll the vlues of if the inequlity is true. Complete these sttements. i If > then ii If < then 14 Consier the eqution 9 > 3. Solve the eqution y firstly ing to oth sies then solve for. Solve the eqution y firstly sutrting 9 from oth sies. Wht i you hve to rememer to o in prt to ensure tht the nswer is the sme s in prt? 15 Comine ll your knowlege from this hpter so fr to solve these inequlities. ( + 1) > e 3 < 1 7( 3) 1 > 4 Literl inequlities f 4( 1) (3 ) 4 16 Given,, n re positive numers (suh tht 1 < < ), solve eh of the following for. > g ( + ) < e + < > j + k + > 1 h f i l ( ) > 16 REASONING ENRICHMENT G

39 114 Chpter Liner n simultneous equtions H Key ies Using formuls A formul (or rule) is n eqution tht reltes two or more vriles. You n fin the vlue of one of the vriles if you re given the vlue of ll others. Some ommon formuls ontin squres, squre roots, ues n ue roots. The following re some emples of formuls. A = p r is the formul for fining the re, A, of irle given its rius, r. F = 9 C + 3 is the formul for onverting egrees Celsius, C, to egrees Fhrenheit, F. 5 = vt is the formul for fining the istne,, given the veloity, v, n time, t. A, F n re si to e the sujets of the formuls given ove. Let s strt: Common formuls As lss group, try to list t lest 10 formuls tht you know. Write own the formuls n esrie wht eh vrile represents. Whih vrile is the sujet of eh formul? The sujet of formul is vrile tht usully sits on its own on the left-hn sie. For emple, the C in C = p r is the sujet of the formul. A vrile in formul n e evlute y sustituting numers for ll other vriles. A formul n e trnspose (rerrnge) to mke nother vrile the sujet. C = p r n e trnspose to give r = C p To trnspose formul use similr steps s you woul for solving n eqution, sine vriles represent numers. Note tht = if 0 n + = + provie or is not zero. Emple 16 Sustituting vlues into formuls Sustitute the given vlues into the formul to evlute the sujet. S =, when = 3 n r = r E = 1 mv, when m = 4 n v = 5 SOLUTION S = 1 r S = = = 5 EXPLANATION Sustitute = 3 n r = 0.4 n evlute.

40 Numer n Alger 115 E = 1 mv E = = = 50 Emple 17 Fining the unknown vlue in formul Sustitute m = 4 n v = 5 n evlute. Note: squre the vlue of v efore multiplying y the vlue of m. The re of trpezium is given y A = 1 ( + )h. Sustitute A = 1, = 5 n h = 4 then fin the vlue of. SOLUTION A = 1 ( + )h 1 = 1 (5 + ) 4 1 = (5 + ) 6 = 5 + = 1 Emple 18 Trnsposing formuls EXPLANATION Write the formul n sustitute the given vlues of A, n h. Then solve for. 1 4 = n ivie oth sies y sine is ftor of 1. Alterntively, you n epn the rkets. Trnspose eh of the following to mke the sujet. = ( + ) = + ( > 0) SOLUTION = ( + ) = + = = ( or = ) EXPLANATION Divie oth sies y. Sutrt from oth sies. Mke the sujet on the left sie. An lterntive nswer hs ommon enomintor, whih will lso e the nswer formt if you epn the rkets first.

41 116 Chpter Liner n simultneous equtions Emple 16 Emple 17 = + ( > 0) = + = = = Eerise H 1 Stte the letter whih is the sujet of these formuls. 3 A = 1 h Squre oth sies to remove the squre root. Sutrt from oth sies. Mke the sujet. Tke the squre root of oth sies, = s is positive. D = 4 M = + A = p r 1, (½) (½) Sustitute the given vlues into eh of the following formuls to evlute the sujet. Roun to two eiml ples where pproprite. A = h, when = 3 n h = 7 F = m, when m = 4 n = 6 m = +, when = 14 n = 6 4 t = v, when = 18 n v = 3 e A = p r, when p = 3.14 n r = 1 f V = 4 3 p r3, when p = 3.14 n r = g = +, when = 1 n = h Q = gh, when g = 9.8 n h = 11.4 i I = MR, when M = 1. n R = 6.4 j = ut + 1 t when u = 0, t = 4 n = 10 Sustitute the given vlues into eh of the following formuls then solve the equtions to etermine the vlue of the unknown pronumerl eh time. Roun to two eiml ples where pproprite. m = F, when m = 1 n = 3 A = lw, when A = 30 n l = 6 A = 1 ( + )h, when A = 64, = 1 n h = 4 C = p r, when C = 6 n p = 3.14 e S = p r, when S = 7 n p = 3.14 f v = u + s, when v =, u = 6 n = 1 g m =, when m = 8 n = 4 y 3 4(½) 3 4(½) 3 4(½) UNDERSTANDING FLUENCY

42 Numer n Alger 117 Emple 18 4 Trnspose eh of the following formuls to mke the pronumerl shown in rkets the sujet. ( ) A = p rh r I = Prt (r) 100 ( ) p = m( + n) n = + () e V = p r ( ) h (r > 0) r f P = v (v > 0) (v) R g S = p rh + p r ( ) h h A = (p + q) ( ) p l i T = p (g) j A + B = 4C (A) g 5 The formul s = t t hours. gives the spee s km/h of r whih hs trvelle istne of km in Fin the spee of r whih hs trvelle 400 km in 4.5 hours. Roun to two eiml ples. i Trnspose the formul s = to mke the sujet. t ii Fin the istne overe if r trvels t 75 km/h for 3.8 hours. 6 The formul F = 9 C + 3 onverts egrees Celsius, C, to egrees Fhrenheit, F. 5 Fin wht eh of the following tempertures is in egrees Fhrenheit. i 100 C ii 38 C Trnspose the formul to mke C the sujet. Clulte wht eh of the following tempertures is in egrees Celsius. Roun to one eiml ple where neessry. i 14 F ii 98 F 5, The veloity, v m/s, of n ojet is esrie y the rule v = u + t, where u is the initil veloity in m/s, is the elertion in m/s n t is the time in seons. Fin the veloity fter 3 seons if the initil veloity is 5 m/s n the elertion is 10 m/s. Fin the time tken for oy to reh veloity of 0 m/s if its elertion is 4 m/s n its initil veloity is 1 m/s. 6 8 FLUENCY PROBLEM-SOLVING H

43 118 Chpter Liner n simultneous equtions H 8 The volume of wter (V litres) in tnk is given y V = t where t is the time in seons fter tp is turne on. Over time, oes the wter volume inrese or erese oring to the formul? Fin the volume fter minutes. Fin the time it tkes for the volume to reh 1500 litres. Roun to the nerest minute. How long, to the nerest minute, oes it tke to ompletely empty the tnk? 9 Write formul for the following situtions. Mke the first liste vrile the sujet. e f g $D given ents m given e metres The isounte prie $D whih is 30% off the mrke prie $M. The vlue of n investment $V whih is 15% more thn the initil mount $P. The ost $C of hiring r t $50 upfront plus $18 per hour for t hours. The istne km remining in 4 km mrthon fter t hours if the running spee is 14 km/h. The ost $C of ottle of soft rink if ottles ost $. 10 Write formul for the vlue of in these igrms. Perimeter P e Note: + = 9 9, 10 f Are = A 9(½), 10 PROBLEM-SOLVING REASONING

44 Numer n Alger 119 Bsketll formuls 11 The formul T = 3 + y + f n e use to lulte the totl numer of points me in sketll gme where: = numer of three-point gols y = numer of two-point gols f = numer of free throws me T = totl numer of points Fin the totl numer of points for gme where 1 three-point gols, 15 two-point gols n 7 free throws were me. Fin the numer of three-point gols me if the totl numer of points ws 36 with 5 two-point gols me n 5 free throws me. ( The formul V = p + 3r ) 3s t + f + m o n e use to lulte the g vlue, V, of sketll plyer where: p = points erne r = numer of reouns = numer of ssists s = numer of stels = numer of loks t = numer of turnovers f = numer of personl fouls m = numer of misse shots o = numer of offensive reouns g = numer of gmes plye Clulte the vlue of plyer with 350 points erne, reouns, 14 ssists, 5 stels, 3 loks, 8 turnovers, 14 personl fouls, 4 misse shots, 3 offensive reouns n 10 gmes. 11 ENRICHMENT H

45 10 Chpter Liner n simultneous equtions I Simultneous equtions: sustitution EXTENDING Key ies A liner eqution with one unknown hs one unique solution. For emple, = is the only vlue of tht mkes the eqution + 3 = 7 true. The liner eqution + 3y = 1 hs two unknowns n it hs n infinite numer of solutions. Eh solution is pir of n y vlues tht mkes the eqution true, for emple = 0 n y = 4 or = 3 n y = or = 4 1 n y = 1. However, if we re tol tht + 3y = 1 n lso A shre trer emining omputer moels of finnil t, tht y = 1, we n fin single solution tht whih n involve fining vlues tht stisfy two equtions stisfies oth equtions. Equtions like this re simultneously. lle simultneous liner equtions, euse we n fin pir of n y vlues tht stisfy oth equtions t the sme time (simultneously). Let s strt: Multiple solutions There is more thn one pir of numers n y whih stisfy the eqution y = 5. Write own t lest 5 pirs (, y) whih mke the eqution true. A seon eqution is y = 8. Do ny of your pirs tht mke the first eqution true, lso mke the seon eqution true? If not, n you fin the speil pir of numers tht stisfies oth equtions simultneously? An lgeri metho lle sustitution n e use to solve simultneous equtions. It is use when t lest one of the equtions hs single vrile s the sujet. For emple, y is the sujet in the eqution y = To solve simultneous equtions using sustitution: 1 Sustitute one eqution into the other, using rkets. Solve for the remining vrile. 3 Sustitute to fin the vlue of the seon vrile. Emple 19 Solving simultneous equtions using sustitution + 3y = 8 n y = ( + 1) = = = 8 5 = 5 = 1 y = (1) + 1 = Solve eh of the following pirs of simultneous equtions y using sustitution. + y = 10 y = 4 4 y = 6 y = y = 19 y = 8