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1 Online resources Auto-mrked chpter pre-test Video demonstrtions of ll worked emples Interctive widgets Interctive wlkthroughs Downlodble HOTsheets Access to ll HOTmths Austrlin Curriculum courses Access to the HOTmths gmes librry Epressions, equtions nd inequlities Wht you will lern A Algebric epressions REVISION B Simplifying lgebric epressions REVISION C Epnding lgebric epressions D Liner equtions with pronumerls on one side E Liner equtions with brckets nd pronumerls on both sides F Using liner equtions to solve problems G Liner inequlities H Using formuls I Liner simultneous equtions: substitution J Liner simultneous equtions: elimintion K Using liner simultneous equtions to solve problems L Qudrtic equtions of the form = c (Equtions involving lgebric frctions re locted in section D nd lso in Chpter 8 (section 8K).)

2 NSW syllbus Arcde gmes STRAND: NUMBER AND ALGEBRA SUBSTRANDS: ALGEBRAIC TECHNIQUES (S) EQUATIONS (S, 5.) Outcomes A student uses lgebric techniques to solve simple liner nd qudrtic equtions. (MA 10NA) A student solves liner nd simple qudrtic equtions, liner inequlities nd liner simultneous equtions, using nlyticl nd grphicl techniques. (MA5. 8NA) NESA, 01 Nerly every occuption uses the concept of solving equtions in their dily job. This cn be s esy s chnge equls money given subtrct money owed, to more complicted equtions relted to the physics involved in stellite lunch. The coders of computer gmes require n understnding not just of bsic lgebr, liner, qudrtics nd, t times, simultneous equtions, but of the coding progrms s well. The more complicted the gme the more detiled the equtions nd coding needs to be. Pc-Mn is fmous rcde gme from the 1980s vilble now s computer gme nd pp. It even hs its own crtoon spin-off. The equtions used in the coding of such gme cn be s simple s defining the het of Pc-Mn (het is nicknme for the property tht mkes the Ghosts wnt to chse Pc-Mn): het = positive vlue, or het = negtive vlue. Other more complicted equtions re lso incorported into the code. Equtions of the type het = k {het (up) + het (down) + } re employed throughout the code for the gme to ctivte the fetures tht mke it unique.

3 8 Chpter Epressions, equtions nd inequlities Pre-test 1 Write lgebric epressions for: more thn b the product of nd b c lots of y less d the sum of nd divided by Evlute the following if = nd b =. 5 b b + c 9 b d (b + 1) Simplify by collecting like terms. + 5 b 7y + y y c ( ) d + 1y + 5y e f y y + y y Simplify the following. b 7 ( y) c 5 Describe nd clculte in two different wys how the re of rectngle ABCD cn be found. C 6 Epnd the following. ( + ) b ( 5) c ( y) d (b 1) 8b d 9mn 6n 7 Which of the following equtions is = solution to? + = 9 b + = 5 c + 1 = d 5 = 1 8 Find the vlue of tht mkes the following true. 1 + = 1 b = 7 c + 1 = 9 d = 5 9 Stte if ech of the following re true or flse. 5 > b 6 < c d 1 < 5 e 7 > f < 1 10 Find the vlue of y in ech of the following for the given vlue of. y =, = 5 b y = + 7, = c + y = 7, = 1 d y =, = Which inequlity describes this number line? A > B C < D E < 1 Which inequlity describes this number line? A C 1 E 1 A B D B > D < 1

4 A Algebric epressions 8 A Algebric epressions REVISION Algebr is centrl to the study of mthemtics nd is commonly used to solve problems in vst rry of theoreticl nd prcticl problems. Algebr involves the represention nd mnipultion of unknown or vrying quntities in mthemticl contet. Pronumerls re used to represent these unknown quntities. Let s strt: Remembering the vocbulry Stte the prts of the epression 5 y + () tht mtch these words. pronumerl term coefficient constnt term squred term Pronumerls representing unknown quntities re used in wide rnge of jobs nd occuptions. In lgebr, letters re used to represent numbers. These letters re clled pronumerls or vribles. An epression is combintion of numbers nd pronumerls connected by the four opertions +,, nd. Brckets cn lso be used. For emple: 5 + y 1 nd ( + ) y 5 A term is combintion of numbers nd pronumerls connected with only multipliction nd division. Terms re seprted with the opertions + nd. For emple: 5 + 7y is two-term epression Coefficients re the numbers being multiplied by pronumerls. 1 For emple: the in nd in re coefficients Constnt terms consist of number only. For emple: in + (The sign must be included.) Epressions cn be evluted by substituting number for pronumerl. For emple: if = then + 6 = + 6 = Order of opertions should be followed when evluting epressions: 1 Opertions in brckets, then Powers, then Multipliction nd division (from left to right), then Addition nd subtrction (from left to right) Stge 5.# Key ides

5 8 Chpter Epressions, equtions nd inequlities Emple 1 Writing lgebric epressions for worded sttements Write n lgebric epression for the: number of tickets needed for boys nd r girls b cost of P pies t $ ech c number of grms of penuts for one child if 00 g of penuts is shred eqully mong C children SOLUTION Emple Converting words to epressions Write n lgebric epression for: five less thn b three more thn twice c the sum of nd b is divided by d the squre of the sum of nd y SOLUTION EXPLANATION 5 5 subtrcted from b + Twice plus c + b The sum of nd b is done first ( + b) nd the result is divided by. d ( + y) The sum of nd y is done first nd then the result is squred. Emple Substituting vlues into epressions Evlute these epressions if = 5, b = nd c =. 7 ( c) b b c SOLUTION 7 ( c) = 7 5 (5 ) = 5 = 5 = 1 b b c = ( ) 5 = 15 = 11 EXPLANATION + r tickets plus the number of girls b P multiplied by the number of pies c 00 C 00 g divided into C prts EXPLANATION Substitute the vlues for nd c. When using order of opertions, evlute brckets before moving to multipliction nd division then ddition nd subtrction. Evlute powers before the other opertions. ( ) = ( ) =.

6 A Algebric epressions 85 Emple 1 Emple Eercise A REVISION UNDERSTANDING AND FLUENCY 1 Consider the epression + 5d + w + 9. Complete the following. The epression contins terms b The coefficient of d is c The coefficient of w is d The constnt term is Mtch n item in the left column with n item in the right column. A Product Division B Sum b Subtrction C Difference c Multipliction D Quotient d Addition E e The reciprocl of F 1 Stte the coefficient in these terms. 5y b c f 1, 5 6(½),, 5 6(½), 7, 5 6(½), 7 The squre of d 5 Write n lgebric epression for the following. The number of tickets required for: i boys nd r girls ii t boys nd girls iii b boys nd g girls iv boys, y girls nd z dults b The cost of: i P pies t $6 ech ii 10 pies t $n ech iii D drinks t $ ech iv P pies t $5 nd D drinks t $ c The number of grms of lollies for one child if 500 g of lollies is shred eqully mong C children. 5 Write n lgebric epression for ech of the following. The sum of nd b The sum of b nd y c 5 less thn d The product of nd e The difference between nd y f Three times the vlue of p g Four more thn twice h The sum of nd y is divided by 5 i 10 less thn the product of nd j The squre of the sum of m nd n k The sum of the squres of m nd n l The squre root of the sum of nd y m The sum of nd its reciprocl n The cube of the squre root of

7 86 Chpter Epressions, equtions nd inequlities Emple 6 Evlute these epressions if =, b = nd c = 7. b c b bc c c d b c + b b e f + c 1 g ( b) c h bc 7 Evlute these epressions if =, y = 1 nd z = 1 6. y + z b y + c yz d PROBLEM-SOLVING AND REASONING z + 1 y 8 A rectngulr grden bed is 1 m long nd 5 m wide. Find the re of the grden bed. b The length is incresed by cm nd the width is decresed by y cm. Find the new length nd width of the grden. c Write n epression for the re of the new grden bed. 9 The epression for the re of trpezium is 1 h( + b) where nd b re the lengths of the two prllel sides nd h is the distnce between the two prllel sides. Find the re of the trpezium with = 5, b = 7 nd h =. b A trpezium hs h = nd re 1. If nd b re whole numbers, wht possible vlues cn the pronumerl hve? 10 The cost of 10 identicl puzzles is $P. Write n epression for the cost of one puzzle. b Write n epression for the cost of n puzzles. 11 For ech of these shpes, write n epression for the: i perimeter ii re b y p 8, 9, 11 8, 9, 11, 1 9, 10, 1, 1 1 Decide if the following sttements refer to the sme or different epressions. If they re different, write n epression for ech sttement. A Twice the sum of nd y B The sum of nd y b A The difference between hlf of nd hlf of y B Hlf of the difference between nd y c 5 y

8 A Algebric epressions 87 1 For right-ngled tringle with hypotenuse c nd shorter sides nd b, Pythgors theorem sttes tht c = + b. Which of these two descriptions lso describes Pythgors theorem? A The squre of the hypotenuse is equl to the squre of the sum of the two shorter sides. B The squre of the hypotenuse is equl to the sum of the squres of the two shorter sides. b For the incorrect description, write n eqution to mtch. ENRICHMENT The sum of the first n positive integers 1 The rule for the sum of the first n positive integers is given by: The product of n nd one more thn n ll divided by. c b Write n epression for the bove description. b Test the epression to find these sums. i (n = ) ii (n = 10) c Another wy to describe the sme epression is: The sum of hlf of the squre of n nd hlf of n. Write the epression for this description. d Check tht your epressions in prts nd c re equivlent (the sme) by testing n = nd n = e (n + n) is lso equivlent to the bove two epressions. Write this epression in words (5 + 1) This digrm represents the sum of the first five positive integers rrnged ccording to the epression in question 1.

9 88 Chpter Epressions, equtions nd inequlities Key ides B Simplifying lgebric epressions Just s =, so = or. We sy tht the epression is simplified to. Similrly, + 5 = 8 nd 8 = 5. All these epressions hve like terms nd cn be simplified to n epression with smller number of terms. A single term such s 5 10 cn lso be simplified using multipliction nd division, so: 5 10 = = Let s strt: Are they equivlent? REVISION Stge 5.# 5. All these epressions cn be seprted into two groups. Group them so tht the epressions in ech group re equivlent. y 10 y 8 y + y + 1 y y y 1 y ( ) The symbols for multipliction ( ) nd division ( ) re usully not shown in simplified lgebric terms. For emple: 5 b = 5b nd 7 y = 7 When dividing lgebric epressions common fctors cn be cncelled. 7 For emple: 1 = b = 1 b = b 7y 1y = 1 y 15 b 10 = 5 b = b 5 Like terms hve the sme pronumerl fctors. For emple: 5 nd 7 re like terms nd b nd b re like terms. Since b = b, then b nd b re lso like terms The letters in term re usully written in lphbeticl order. Like terms cn be collected (dded nd subtrcted) to form single term. For emple: 5b + 8b = 1b y y = y Unlike terms do not hve the sme pronumerl fctors. y For emple: 5,, y nd re ll unlike terms. 5

10 B Simplifying lgebric epressions 89 Emple Multiplying lgebric terms Simplify the following. b b b SOLUTION EXPLANATION b = b = 6b Multiply the coefficients. b b = b = 6 b Emple 5 Dividing lgebric terms Simplify the following. 6b 18b SOLUTION 1 6b 1 18b 1 = b 1 b (b) = 1 1 b 1 b = Emple 6 Collecting like terms Simplify the following by collecting like terms. + b + y + + 7y c 8b 9b b + b SOLUTION + = + = + b + y + + 7y = + + y + 7y = 7 + 9y c 8b 9b b + b = 8b b 9b + b = 7b 6b Multiply the coefficients nd simplify. b 1 b (b) EXPLANATION Del with numerls nd promumerls seprtely, cncelling where possible. Write s frction first. Cncel where possible, recll =. EXPLANATION Collect like terms ( nd ). The sign belongs to the term tht follows. Combine their coefficients = 1. Collect like terms nd combine their coefficients. Collect like terms. Remember b = b nd b = 1b. 8 1 = 7 nd 9 + = 6

11 90 Chpter Epressions, equtions nd inequlities Emple Emple b Emple 5 Emple 5b Emple 6 Eercise B REVISION UNDERSTANDING AND FLUENCY 1 True or Flse? 1 cn be written s b 0b is equl to 0 c b = b = b d nd re like terms Simplify these products. b b 6 c 8 d b e 5 n m f 9 Are these like terms or unlike terms? b nd b b nd 7y c 5 nd m d t nd 6tw e 7yz nd yz f mn nd 9nm g 5 nd h y nd 7y i b nd b Simplify the following. 5 m b 6b c 5p d y e p 6r f m n g 7y h 5m ( n) i c d j b 5 k r s l 5j ( ) k 5 Simplify the following. n 6n b q q c 5s s d 7 b e 5mn ( n) f gh ( 6h) g y y h b ( b) i mn mn 6 Simplify the following by cncelling. 8b b 6 10st u f g v 6t u c h b 6 5r s 8rs 1, 7(½), 9 11(½) 11(½) 5 11(½) d i mn 6n 5b 9b e j 5y 0y 7y 7 Simplify the following by first writing in frction form. 5 b ( ) c 11mn d 1b e 10 (gh) f 8 g y (y) h 7mn (m) i 7pq (6p) j b (8b) k 5 y ( 5y) l 9m n (18mn) 8 Simplify the following. y b 5 p c 6 ( ) b d ( ) (b) e 7 (5m) n f 5s (t) g 6 mn (m) h 8 y (8) i b 1bc (9bc) j y () k 10m mn (8mn) l pq pq p 9 Simplify the following by collecting like terms. 6 + b 6 c 6 d + e f g + + h + + i + + j + 7 k 7 l + 7 m n 6b b b o 7mn + mn mn p y y + 8 q r 6y + y + y s 5b + + 7b t 5m m u + y

12 B Simplifying lgebric epressions 91 Emple 6b Emple 6c 10 Simplify the following by collecting like terms. + b + + 5b b + y + + y c 6t + 5 t + 1 d e y y f mn + nm 5 g b + + b h st 8ts + st + ts 11 Simplify the following by collecting like terms. 5y y b b + b c 8m n 6nm + m n d 7p q p q p q e y y + 5y f 10rs + rs 6r s g 7 h b b + b + b i 10pq qp pq 6pq j 1m n mn m n + mn PROBLEM-SOLVING AND REASONING 1 A frmer hs pigs nd y chickens. Write n epression for the totl number of heds. b Write n epression for the totl number of legs. 1 A rectngle s length is three times its bredth metres. Write n epression for the rectngle s: perimeter b re 1 A right-ngled tringle hs side lengths 5 cm, 1 cm nd 1 cm. Write n epression for the tringle s: perimeter b re 15 The verge (men) mrk on test for 0 students is. Another student who scores 75 in the test is dded to the list. Write n epression for the new verge (men). 16 Decide if the following re lwys true for ll rel numbers. b = b b b = b c + b = b + d b = b e b = b f 1 1 = 17 The digrm shows the route tken by slesperson who trvels from A to D vi B nd C. If the slesperson then returns directly to A, write n epression (in simplest form) for the totl distnce trvelled. b If y = + 1, write n epression for the totl distnce the slesperson trvels in terms of only. Simplify your epression. c When y = + 1, how much would the distnce hve been reduced by (in terms of ) if the slesperson hd trvelled directly from A to D nd stright bck to A? ENRICHMENT Higher powers 1, 1, , For this question, note this emple: = 1 1 = 1 Simplify these epressions with higher powers. b b b c 6 d b 9b 1b 16 b e 5 f 85 g h b b 15 i b 8 j b 5 k 5 b l 5 b 7 Uncorrected 5b rd smple pges Cmbridge 7 University b Press Plmer, 6 et bl Ph b A y D + y 18 B C

13 9 Chpter Epressions, equtions nd inequlities Key ides C Epnding lgebric epressions A mentl technique to find the product of 5 nd might be to find 5 0 nd dd 5 to give 115. This technique uses the distributive lw over ddition. So 5 = 5 (0 + ) = Since pronumerls represent numbers, the sme lw pplies for lgebric epressions. Let s strt: Rectngulr distributions This digrm shows two joined rectngles with the given dimensions. Find two different wys to write epressions for the combined re of the two rectngles. Compre your two epressions. Are they equivlent? This digrm shows rectngle of length reduced by length of. Find two different wys to write epressions for the remining re (shded). Compre your two epressions. Are they equivlent? - 5 Emples of the distributive lw cn often be found in everydy tsks. The distributive lw is used to epnd nd remove brckets. A term on the outside of the brckets is multiplied by ech term inside the brckets. (b + c) = b + c or (b c) = b c (b + c) = b c or (b c) = b + c If the number in front of the brcket is negtive, the sign of ech of the terms inside the brckets will chnge when epnded. For emple: ( ) = + 6, since = nd ( ) = 6. Emple 7 Epnding simple epressions with brckets Epnd the following. ( + ) b 5( 11) c ( 5) SOLUTION Stge 5.# EXPLANATION ( + ) = + 1 = nd = 1 b 5( 11) = = 5 nd 5 ( 11) = 55 c ( 5) = + 10 = nd ( 5) =

14 C Epnding lgebric epressions 9 Emple 8 Epnding brckets nd simplifying Epnd the following. ( + y) b ( ) SOLUTION ( + y) = + y = + 1y b ( ) = + ( ) ( ) = Emple 9 Simplifying by removing brckets Eercise C UNDERSTANDING AND FLUENCY EXPLANATION Multiply ech term inside the brckets by. = nd y = 1y. Ech term inside the brckets is multiplied by. = 8 nd = nd ( ) = 6 Epnd the following nd collect like terms. ( ) b ( + y) ( + y) SOLUTION ( ) = ( 1) = + 1 = 1 b ( + y) ( + y) = + 6y y = + 6y y = 5y EXPLANATION ( ) = 1. ( 1) = 1( 1) so multiplying by negtive 1 chnges the sign of ech term inside the brckets. ( + y) = 1( + y) = y. Collect like terms nd simplify. 1, 6(½) 7(½) 7(½) 1 This digrm shows two joined rectngles with the given dimensions. Write n epression for the re of the: i lrger rectngle ( by 5 ) ii smller rectngle ( by 5 ) b Use your nswers from prt to find the combined re of both rectngles. c Write n epression for the totl side length of the side involving. d Use your nswer from prt c to find the combined re of both rectngles. e Complete this sttement: 5( + ) = + Substitute the vlue = 5 into these epressions. i ( + ) ii + 6 b Do your nswers from prt suggest tht ( + ) = + 6? When epnded wht should ( + ) equl? c Substitute the vlue = 10 into these epressions. i ( 1) ii d Do your nswers from prt c suggest tht ( 1) =? When epnded wht should ( 1) equl? 5

15 9 Chpter Epressions, equtions nd inequlities Emple 7, b Emple 7c Emple 8 Emple 9 Emple 9b Epnd the following. ( + ) b 5( + 1) c ( 7) d 7( 9) e ( + ) f 7( ) g (7 ) h ( 6) i 6( + 1) j ( w) k 9(w + h) l (d w 1) Epnd the following. ( + ) b ( + 11) c 5( ) d 6( 6) e ( ) f 1(5 + ) g 0(9 + ) h 00(1 ) i 5(w ) j (w + 8) k ( 7) l (w + 9) 5 Epnd the following. ( + b) b 5( ) c (m ) d 8( + 5) e ( + 5) f ( y) g 9t(y ) h ( + ) i d(d 5) j b(b 5) k ( + 1) l 5y(1 y) 6 Epnd the following nd collect like terms. + ( + ) b + 6( ) c + 5( 1) d 5 + ( ) e + ( ) f 7 + ( ) g ( + ) h 1 5( + ) i 5 ( 6) j 9 ( ) k 5 ( + ) l ( ) 7 Epnd the following nd collect like terms. ( + ) + ( + ) b ( ) + ( 1) c ( + 1) + 5( 1) d ( + ) + 5( ) e ( + 1) + ( ) f ( + ) + ( 1) g ( ) ( 1) h ( + ) 5( 1) i ( + ) ( + 5) j ( ) ( + 5) k ( 1) ( ) l (5 ) ( 5) PROBLEM-SOLVING AND REASONING 8 A rectngle s length is more thn its width,. Find n epnded epression for its re. 9 Find the re of these bsic shpes in epnded form. b c + 1 d e f , 9, , 1, 1(½) 9 1

16 C Epnding lgebric epressions Gry gets bonus $0 for every computer he sells over nd bove his quot of 10 per week. If he sells n computers in week nd n > 10, write n epression for Gry s bonus in tht week (in epnded form). 11 Jill pys t t 0c in the dollr for every dollr erned over $ If Jill erns $ nd > 10000, write n epression for Jill s t in epnded form. 1 Identify the errors in these epressions, then write out the correct epnsion. ( + 6) = + 6 b ( ) = c ( + ) = + 1 d 7( 7) = 7 9 e 5 ( 7) = 5 1 = 9 f ( ) ( + ) = = 1 In Yers 7 nd 8 we eplored how to use the distributive lw to find products mentlly. For emple: 7 10 = 7 (100 + ) = = 78 nd 98 = (00 ) = 00 = 119 Use the distributive lw to evlute these products mentlly. 6 5 c 5 91 e 99 b 9 10 d 6 f 7 95 ENRICHMENT Pronumerls nd tes g h A progressive income t system increses the t rte for higher incomes. Here is n emple. Income T 0 $0 000 $0 $0 001 $ $0 + 0% of income bove $0 000 $ $ % of income bove $ $ b + 50% of income bove $ Find the vlues of nd b in the bove tble. b Find the t pyble for these incomes. i $5 000 ii $7 000 iii $ c Find n epression for the t pyble for n income of $ if: i ii 0000 < iii < iv > d Check tht you hve fully epnded nd simplified your epressions for prt c. Add steps if required. e Use your epressions from prts c nd d to check your nswers to prt b by choosing prticulr income mount nd checking ginst the tble bove. 1

17 96 Chpter Epressions, equtions nd inequlities Key ides D Liner equtions with pronumerls on one side A mthemticl sttement contining n equls sign, left-hnd side nd right-hnd side is 1 clled n eqution. For emple, 5 = 10, = 9, + 1 = 10 nd = re equtions. 5 Liner equtions cn be written in the form + b = c where the power of is 1. For emple, 1 = 6, = ( + 1) nd 5 = + 1 re ll liner equtions. Equtions re solved by finding the vlue of the pronumerl tht mkes the eqution true. This cn be done by inspection for very simple liner equtions (for emple, if = 15, then = 5 since 5 = 15). More comple liner equtions cn be solved through series of steps where ech step produces n equivlent eqution. Let s strt: Why re they equivlent? The following list of equtions cn be ctegorised into two groups. The equtions in ech group should be equivlent. 5 = 0 1 = = 1 = 7 = 7 5 = = 1 = 1 5 Discuss how you divided the equtions into the two groups. How cn you check to see if the equtions in ech group re equivlent? Equivlent equtions re creted by: dding number to or subtrcting number from both sides of the eqution multiplying or dividing both sides of the eqution by the sme number (not including 0 ) swpping the left-hnd side (LHS) with the right-hnd side (RHS). Solve liner eqution by creting simpler equivlent equtions. The solution to n eqution cn be checked by substituting the solution into the originl eqution nd checking tht both sides re equl. Equtions involving lgebric frctions re introduced in this section nd etended in Chpter 8. Emple 10 Solving simple liner equtions Solve ech of the following equtions. + = b = 7 c 5 = 1 d + 5 = 7 e + = f 10 = Stge 5.#

18 D Liner equtions with pronumerls on one side 97 SOLUTION EXPLANATION + = = 1 b = 1 1 Check: LHS = ( ) + = = RHS = 7 = 10 = 0 Check: LHS = (0) = 10 = 7 = RHS c 5 = 1 = 7 = 7 or 1 7 Check: LHS = 5 = = 1 = RHS ( ) d e f + 5 = 7 = = 6 = Check: LHS = () + 5 = + 5 = 7 = RHS + = 9 + = 18 = 1 = 7 Check: LHS = (7) + = = = RHS 10 = = 10 = 5 = 15 = 7 1 Subtrct from both sides. Divide both sides by. Check the nswer by substituting = 1 into the originl eqution. Add to both sides. Multiply both sides by. Check the solution by substituting = 0 into. Since this equls 7, = 0 is the solution. Subtrct 5 from both sides. Divide both sides by. Check the solution. Subtrct 5 from both sides first then multiply both sides by. Divide both sides by. Check the solution. Multiply both sides by 9 first to eliminte the frction. Solve the remining eqution by subtrcting from both sides nd then dividing both sides by. Check the solution. Swp LHS nd RHS. Subtrct 5 from both sides. Multiply both sides by. Divide both sides by.

19 98 Chpter Epressions, equtions nd inequlities Emple 10 Emple 10b Emple 10c Emple 10d, f Eercise D UNDERSTANDING AND FLUENCY 1 (½),, 8(½), 8(½) 1 Solve these equtions by inspection. + = 6 b = 6 c 6 = d = 6 e = 6 f 6 = g 6 = h 6 = Solve the following equtions. + 1 = 7 b 11 1 = 1 c = d + 1 = 5 e = 6 f 1 = 16 g 1 = h 1 = Which of the following equtions re equivlent to = 1? + 1 = 1 b 1 = 1 c 1 = 1 d = 1 e = f = g = 10 h = 1 5 Solve ech of the following equtions. Check your nswers. + 5 = 9 b = 11 c 8 = m d = 6 e n + 1 = 7 f + 5 = 7 g b + 15 = 7 h y = 1 i 7 = + j b + 7 = 5 k = 10 l 6 5 = m 7y = 8 n + 1 = 1 o 1 = 5n 1 5 Solve ech of the following equtions. + = 5 b + = 5 c b + 5 = 9 d t + 5 = e + = f y 5 = g 7 = 1 h s = 7 i 5 = j m = k 1 y 5 = l 5 = 6 Solve ech of the following equtions. 1 = 18 b 9 = 7 c 15 5 = 5 d 1 = e 5 = 9 f = 7 g 5 8 = h 10 = 7 Solve these equtions. b = 6 b = 9 c = 9 d 5 = e = 1 f 5n = 1 g 5 1 = 7 h = 7 i + = j 5 + d f = 7 k 11 = l z = 5 m + = 0 n + 1 = 1 o = p p = p (½)

20 D Liner equtions with pronumerls on one side 99 Emple 10e 8 Solve ech of the following equtions. Check your nswers = b = 5 c + y 6 + b = d = 1 5 m 1 + e = f = g = h = 5 7 i = 9 j 5 = b 6 y k = l = 9 5t 6 + w w w m = 0 n = o = 0 p = 1 PROBLEM-SOLVING AND REASONING 9 For ech of the following, write n eqution nd solve it to find the unknown vlue. Use s the unknown vlue. If 8 is dded to certin number, the result is. b Seven less thn certin number is 1. c I think of number, double it nd dd. The result is 10. d I think of number, hlve it nd subtrct. The result is 10. e Four less thn three times number is 0. f A number is multiplied by 7 nd the product is divided by. The finl result is 8. g Five Ester eggs re dded to my initil collection of Ester eggs. I shre them between myself nd two friends nd ech person gets ectly four. How mny were there initilly? h My weekly py is incresed by $00 per week. Hlf of my py now goes to py the rent nd $100 to buy groceries. If this leves me with $50, wht is my originl weekly py? 10 Describe the error mde in ech of these incorrect solutions. 1 = b 5 + = 7 c 5 = 1 d 1 = = 7 = = 5 = 5 = 6 = 10 5 = 15 = 11 An eqution like ( + ) = 8 cn be solved without epnding the brckets. The first step is to divide both sides by. Use this pproch to solve these equtions. i ( 1) = 1 ii ( + ) = iii 7(5 + 1) = 1 iv 5(1 ) = 10 v ( + 1) = vi 5(1 ) = 1 b By considering your solutions to the equtions in prt, when do you think this method is most pproprite? ENRICHMENT Chnging the subject 1 Mke the subject of ech of the following equtions. 9, 10 9, 10 9(½), 10, 11 1 b = c b + b = c c c b = d d b = c d b e c = d f b = 1 b c g = d h b c c d = + b b d 6c d c i = d j = d k = e l = f Uncorrected crd smple pges Cmbridge c University Press Plmer, b et l e Ph

21 100 Chpter Epressions, equtions nd inequlities Key ides E Liner equtions with brckets nd pronumerls on both sides More comple liner equtions my hve pronumerls on both sides of the eqution nd/or brckets. Emples re = 5 1 nd ( + ) = 5. Brckets cn be removed by epnding, nd equtions with pronumerls on both sides cn be solved by collecting like terms using ddition nd subtrction. Let s strt: Steps in the wrong order The steps to solve ( ) = ( 1) re listed here in the incorrect order. ( ) = ( 1) = 6 + = 6 = + Arrnge them in the correct order working from top to bottom. By considering ll the steps in the correct order, eplin wht hs hppened in ech step. Equtions with brckets cn be solved by first epnding the brckets. For emple: ( + 1) = becomes + =. If n eqution hs pronumerls on both sides, collect to one side by dding or subtrcting one of the terms. For emple: + = becomes + = by subtrcting from both sides. Emple 11 Solving equtions with brckets nd pronumerls on both sides Solve ech of the following equtions. ( ) = 11 b ( + ) = 8 c 5 = d ( + ) = 8( + 1) SOLUTION ( ) = = 11 6 = 19 = 19 6 or 1 6 EXPLANATION Epnd the brckets, ( ) = + ( ). Add 8 to both sides, then divide both sides by 6, leving your nswer in frction form. Stge 5.#

22 E Liner equtions with brckets nd pronumerls on both sides 101 Emple 11 SOLUTION b ( + ) = = = 8 = = 1 c 5 = = = = 1 d ( + ) = 8( + 1) = = + 8 = = = Eercise E UNDERSTANDING AND FLUENCY 1 Epnd these epressions nd simplify. ( ) + b (1 ) + c ( 1) + ( ) d 5(1 ) + e ( ) f 7( ) 5( ) EXPLANATION Epnd the brckets nd collect ny like terms, i.e. =. Subtrct 6 from both sides. Divide by. Collect terms on one side by subtrcting from both sides. Add to both sides nd then divide both sides by. Show the net step only for the given equtions nd instructions. ( + ) = 5 (epnd the brckets) b 5 + ( 1) = 7 (epnd the brckets) c + 1 = 6 (subtrct from both sides) d = + 1 (subtrct from both sides) Epnd the brckets on ech side. Subtrct 6 from both sides. Alterntively subtrct 8 to end up with + 1 = 8. (Subtrcting 6 keeps the -coefficient positive.) Solve the eqution nd mke the subject. Solve ech of the following equtions by first epnding the brckets. ( + ) = 11 b 5( + ) = 8 c (m + ) = 1 d 5(y 7) = 1 e (p 5) = 5 f (k 5) = 9 g (5 b) = 18 h (1 m) = 1 i 5( ) = 19 j 7( + 1) = 8 k ( ) = 0 l (n ) = 0 m 5( ) = 6 n 6(1 y) = 8 o ( ) = 1 1,, 5(½), 6(½) 6(½)

23 10 Chpter Epressions, equtions nd inequlities Emple 11b Emple 11c Emple 11d Epnd nd simplify, then solve ech of the following equtions. ( + ) + = 7 b ( ) = c ( 1) + 1 = 0 d ( + ) 1 = e ( ) + ( + 1) = 15 f ( + 1) ( ) = 8 g 6( + ) + = 6 h ( + ) + 5 = 6 i ( ) + = 1 j ( + 1) + = 19 5 Solve ech of the following equtions. 5b = b + 1 b 8 = 7 c t = 10 t d m 8 = m e 5 = + 5 f 9 + = g 1 = h y + 6 = y i 5m = 1 6m 6 Solve ech of the following equtions. 5( ) = 1 b ( + 1) = + 10 c (y + ) = y 6 d ( + 5) = e 5b = 6(b + ) f (m 5) = m + g ( ) = 5( + ) h ( ) = ( + 1) i ( ) = 5( + ) j (n ) = (n + 5) k ( + 5) = ( + ) l ( + ) = ( + 1) PROBLEM-SOLVING AND REASONING 7 Using for the unknown number, write down n eqution nd then solve it to find the number. The product of nd more thn number is 7. b The product of nd less thn number is. c When less thn lots of number is doubled the result is 5. d When 5 more thn lots of number is tripled the result is 10. e more thn lots of number is equivlent to 8 lots of the number. f more thn times the number is equivlent to 1 less thn 5 times the number. g 1 less thn doubled number is equivlent to 5 more thn lots of the number. 8 Since Tr strted work her originl hourly wge hs been tripled, then decresed by $6. It is now to be doubled so tht she gets $18 n hour. Wht ws her originl hourly wge? 9 At the strt of lunch Jimmy nd Jke ech brought out new bg of mrbles to ply with their friends. By the end of lunch they were surprised to see they still hd the sme number s ech other even though overll Jimmy hd gined 5 mrbles nd Jke hd ended up with the double of less thn his originl mount. How mny mrbles were originlly in the bgs? 7, 8, 10, 11 7, 8, 10, 11 8, 9, 10 1

24 E Liner equtions with brckets nd pronumerls on both sides Consider the eqution ( ) = 9. Solve the eqution by first dividing both sides by. b Solve the eqution by first epnding the brckets. c Which of the bove two methods is preferble nd why? 11 Consider the eqution ( ) = 7. Solve the eqution by first dividing both sides by. b Solve the eqution by first epnding the brckets. c Which of the bove two methods is preferble nd why? 1 Consider the eqution + 1 = 5 7. Solve the eqution by first subtrcting from both sides. b Solve the eqution by first subtrcting 5 from both sides. c Which method bove do you prefer nd why? Describe the differences. ENRICHMENT Literl solutions with fctoristion 1 1 Literl equtions contin pronumerl (such s ) nd other pronumerls such s, b nd c. To solve such n eqution for, fctoristion cn be used s shown here. = b + c b = c Subtrct b from both sides ( b) = c Fctorise by tking out = c Divide both sides by ( b) b Solve ech of the following for in terms of the other pronumerls by using fctoristion. = b + d b + 1 = b + c 5 = b + c d + 1 = b 5 e bc = b c f ( b) = b g b c = d + bd h ( + b) = b( c)

25 10 Chpter Epressions, equtions nd inequlities Key ides F Using liner equtions to solve problems Mny types of problems cn be solved by writing nd solving liner equtions. Often problems re epressed only in words. Reding nd understnding the problem, defining pronumerl nd writing n eqution become importnt steps in solving the problem. Let s strt: Too much television? Three friends, Rick, Kte nd Sue, compre how much television they wtch in week t home. Kte wtches times the mount of television of Rick, nd Sue wtches hours less television thn Kte. In totl they wtch 5 hours of television. Find the number of hours of television wtched by Rick. Let hours be the number of hours of television wtched by Rick. Write epressions for the number of hours of television wtched by Kte nd by Sue. Write n eqution to represent the informtion bove. Solve the eqution. Answer the question in the originl problem. To solve word problem using lgebr: Red the problem nd find out wht the question is sking for. Define pronumerl nd write sttement such s: Let be the number of The pronumerl is often wht you hve been sked to find. Write n eqution using your defined pronumerl. Solve the eqution. Answer the question in words. Check tht the solution mkes sense. Emple 1 Turning word problem into n eqution Five less thn certin number is 9 less thn three times the number. Write n eqution nd solve it to find the number. SOLUTION Let be the number. 5 = 9 5 = 9 = = EXPLANATION Define the unknown s pronumerl. 5 less thn is 5 nd this equls 9 less thn three times, i.e. 9. Subtrct from both sides nd solve the eqution. The number is. Write the nswer in words. Stge 5.#

26 F Using liner equtions to solve problems 105 Emple 1 Solving word problems Emple 1 Emple 1 Dvid nd Mitch mde 5 runs between them in cricket mtch. If Mitch mde 68 more runs thn Dvid, how mny runs did ech of them mke? SOLUTION Let the number of runs for Dvid be r. EXPLANATION Define the unknown vlue s pronumerl. Number of runs Mitch mde is r Write ll other unknown vlues in terms of r. r + (r + 68) = 5 r + 68 = 5 r = 186 r = 9 Dvid mde 9 runs nd Mitch mde = 161 runs. Eercise F UNDERSTANDING AND FLUENCY Write n eqution: number of runs for Dvid + number of runs for Mitch = 5 Subtrct 68 from both sides nd then divide both sides by. Epress the nswer in words. 1 For ech of the following emples, mke the unknown number nd write n eqution. Three less thn certin number is 9 less thn four times the number. b Seven is dded to number nd the result is then multiplied by. The result is 9. c I think of number, tke wy 9, then multiply the result by. This gives n nswer of 1. d A number when doubled results in number tht is 5 more thn the number itself. e Eight less thn certin number is more thn three times the number. Leonie nd Emm scored 8 gols between them in netbll mtch. Leonie scored 8 more gols thn Emm. Define pronumerl for the number of gols scored by Emm. b Write the number of gols scored by Leonie in terms of the pronumerl in prt. c Write n eqution in terms of your pronumerl to represent the problem. d Solve the eqution in prt c to find the unknown vlue. e How mny gols did ech of them score? A rectngle is four times s long s it is wide nd its perimeter is 560 cm. Define pronumerl for the unknown width. b Write n epression for the length in terms of your pronumerl in prt. c Write n eqution involving your pronumerl to represent the problem. d Solve the eqution in prt c. e Wht is the length nd width of the rectngle? 1 6 1(½), 7,, 5, 7 9 Toby rented cr for totl cost of $90. If the rentl compny chrged $0 per dy, plus hiring fee of $50, for how mny dys did Toby rent the cr? 5 Andrew wlked certin distnce, nd then rn twice s fr s he wlked. He then cught bus for the lst km. If he trvelled totl of km, find how fr Andrew wlked nd rn. 6 A prize of $1000 is divided between Adele nd Benit so tht Adele receives $80 more thn Benit. How much did ech person receive?

27 106 Chpter Epressions, equtions nd inequlities 7 Kte is three times s old s her son. If Kte is 0 yers older thn her son, wht re their ges? 8 A trin sttion is between the towns Antville nd Bugville. The sttion is four times s fr from Bugville s it is from Antville. If the distnce from Antville to Bugville is 95 km, how fr is it from Antville to the sttion? 9 Andrew, Brend nd Cmmi ll work prt time t supermrket. Cmmi erns $0 more thn Andrew nd Brend erns $0 less thn twice Andrew s wge. If their totl combined wge is $00, find how much ech of these workers erns. PROBLEM-SOLVING AND REASONING 10 Mcy bought totl of 1 fiction nd non-fiction books. The fiction books cost $1 ech nd the non-fiction books cost $5 ech. If she pid $8 ltogether, how mny of ech kind of book did she purchse? 11 If I multiply my ge in si yers time by three, the resulting ge is my mother s ge now. If my mother is currently 8 yers old, how old m I? 1 Twelve yers go Eric s fther ws seven times s old s Eric ws. If Eric s fther is now 5 yers old, how old is Eric now? 1 In ycht rce the second leg ws hlf the length of the first leg, the third leg ws two thirds of the length of the second leg, nd the lst leg ws twice the length of the second leg. If the totl distnce ws 15 km, find the length of ech leg. 1 The Ace Bicycle Shop chrges flt fee of $, plus $1 per hour, for the hire of bicycle. The Best Bicycle Shop chrges flt fee of $8, plus 50 cents per hour. Connie nd her friends hire three bicycles from Ace, nd Dvid nd his brother hire two bicycles from Best. After how mny hours will their hire costs be the sme? 10, 11, 1, 16 10, 1, 1, 16, 17 10, 1 15, 17, Cr A left Melbourne for Adelide t 11.m. nd trvelled t n verge speed of 70 km per hour. Cr B left Melbourne for Adelide t 1 p.m. on the sme dy nd trvelled t n verge speed of 90 km per hour. At wht time will cr B ctch up to cr A? 16 Two pddocks in the shpes shown below re to be fenced with wire. If the sme totl mount of wire is used for ech pddock, wht re the side lengths of ech pddock in metres? w + 5 w w + 0

28 F Using liner equtions to solve problems Consecutive integers cn be represented lgebriclly s, + 1, + etc. Find three consecutive numbers tht dd to 8. b i Write three consecutive even numbers strting with. ii Find three consecutive even numbers tht dd to 18. c i Write three consecutive odd numbers strting with. ii Find three consecutive odd numbers tht dd to 51. d i Write three consecutive multiples of strting with. ii Find three consecutive multiples of tht dd to Tedco produces teddy ber tht sells for $. Ech teddy ber costs the compny $8 to mnufcture nd there is n initil strt-up cost of $700. Write rule for the totl cost, $T, of producing teddy bers. b If the cost of prticulr production run ws $9600, how mny teddy bers were mnufctured in tht run? c If teddy bers re sold, write rule for the revenue, $R, received by the compny. d How mny teddy bers were sold if the revenue ws $800? e If they wnt to mke n nnul profit of $5 000, how mny teddy bers do they need to sell? ENRICHMENT Worded chllenges 19 An rt curtor ws investigting the price trends of two rt works tht hd the sme initil vlue. The first pinting, Green poles, doubled in vlue in the first yer nd then lost $8000 in the second yer. In the third yer its vlue ws three qurters of the previous yer. The second pinting, Orchids, dded $ to its vlue in the first yer then the second yer its vlue ws only third of the previous yer. In the third yer its vlue improved to double tht of the previous yer. If the vlue of the pintings ws the sme in the third yer, write n eqution nd solve it to find the initil vlue of ech pinting. 0 Juli drives to her holidy destintion over period of five dys. On the first dy she trvels certin distnce, on the second dy she trvels hlf tht distnce, on the third dy third of tht distnce, on the fourth dy one qurter of the distnce nd on the fifth dy one fifth of the distnce. If her destintion is 1000 km wy, write n eqution nd solve it to find how fr she trvels on the first dy to the nerest kilometre Ann King is yers old. Her brother Henry is two thirds of her ge nd her sister Chloe is three times Henry s ge. The twins who live net door re 5 yers older thn Ann. If the sum of the ges of the King children is equl to the sum of the ges of the twins, find the ges of ll the children.

29 108 Chpter Epressions, equtions nd inequlities Key ides G Liner inequlities An inequlity (or ineqution) is mthemticl sttement tht uses <,, > or sign. Some emples of inequlities include: < 6, 5 1, nd + 1 >. Inequlities cn represent n infinite set of numbers. For emple, the inequlity < 6 mens tht < nd this is the infinite set of ll rel numbers less thn. Let s strt: Infinite solutions Greg, Kevin nd Gret think tht they ll hve correct solution for: Greg sys = is solution. Kevin sys = 10 is solution. Gret sys = 100 is solution. Use substitution to show tht they re ll correct. Cn you find the smllest whole number tht is solution to the inequlity? Cn you find the smllest number (including frctions) tht stisfies the inequlity? Wht method leds you to your nswer? Inequlities cn be illustrted using number line becuse line represents n infinite number of points. Use n open circle when showing > (greter thn) or < (less thn). For emple: > 1 < Use closed circle when showing (greter thn or equl to) or (less thn or equl to). For emple: A set my hve both n upper nd lower bound. For emple: < Liner inequlities cn be solved in similr wy to liner equtions. If, we multiply or divide both sides of n inequlity by negtive number, the inequlity sign is reversed. For emple: 5 < 8 but 5 > 8 so if > 1 then < 1. If we swp the sides of n inequlity, then the inequlity sign is reversed. For emple: < 7 but 7 > so if > then <. Stge 5.#

30 G Liner inequlities 109 Emple 1 Representing inequlities on number line Show ech of the following emples on number line. is less thn 6 ( < 6). b is greter thn or equl to ( ). c is greter thn but less thn or equl to ( < ). SOLUTION < 6 b c EXPLANATION An open circle is used to indicte tht 6 is not included. A closed circle is used to indicte tht is included. < An open circle is used to indicte tht is not included nd closed circle is used to indicte tht is included. Emple 15 Solving inequlities Find the solution set for ech of the following inequlities. d < 7 b 5 > c 11 d e SOLUTION < 7 < 10 b 5 > > < 1 d c 11 d 8 d d e EXPLANATION Add to both sides Subtrct 5 from both sides. Divide both sides by, nd reverse the inequlity sign. Add to both sides. Multiply both sides by ; the inequlity sign does not chnge. Gther pronumerls on one side by subtrcting from both sides. Subtrct from both sides nd then divide both sides by. Plce the pronumerl on the left nd reverse the inequlity. Swp LHS nd RHS nd reverse the inequlity sign. Subtrct 9 from both sides nd divide both sides by.

31 110 Chpter Epressions, equtions nd inequlities Emple 1 Emple 15 Emple 15b Emple 15c Emple 15d Eercise G UNDERSTANDING AND FLUENCY 1 Mtch the grph with the sttement. 0 1 b c d , 6(½) 7(½) 7(½) A 1 B 1 < C 1 < D > 1 Show ech of the following inequlities on number line. > b 1 c d < e 1 f < 8 g 1 < < 1 h i 0 j 5 k < l 6 < Find the solution set for ech of the following inequlities. + 5 < 8 b b > c y 8 > d 1 + m < 7 e 5 15 f t > 0 g h y i m 7 < 11 j k > 7 5 l 9 < 7 Find the solution set for ech of the following inequlities. > 8 b n 6 c 5 1 d 7 e 5 11 f 7 g > 9 h t + 10 i 6m 1 < 15 5 Find the solution set for ech of the following inequlities. 5 b 9 c d < e > 6 f Solve ech of the following inequlities. ( + ) < 1 b ( + 5) > 9 c 5 5( ) d ( ) > 1 e 5(y + ) < 6 f 11 > 7(1 ) 7 Find the solution set for ech of the following inequlities b 6t + > t 1 c 7y + 7 y d < e 1 m 7 m f 7 5b > b

32 G Liner inequlities 111 PROBLEM-SOLVING AND REASONING 8 10, , 1, , 1, 15 8 Wendy is yers old nd Jy is 6 yers younger. The sum of their ges is less thn 0. Write n inequlity involving nd solve it. Wht cn you sy bout Wendy s ge? 9 The perimeter of prticulr rectngle needs to be less thn 50 cm. If the length of the rectngle is 1 cm nd the bredth is b cm, write n inequlity P < 50 cm involving b nd solve it. Wht bredth does the rectngle need to be? 10 How mny integers stisfy both of the given equtions? 1 cm nd 5 5 b 7 > 10 nd > c nd > d < nd 6 < 7 11 The bredth of rectngle is 10 m nd the height is ( ) m. If the re is less thn 80 m, wht re the possible whole number vlues for? 1 Two cr rentl compnies hve the following pyment plns: Crz: $90 per week nd 15c per kilometre Rent: $110 per week nd 10c per kilometre Wht is the mimum whole number of kilometres tht cn be trvelled in one week with Crz if it is to cost less thn it would with Rent? 1 Consider the inequlity >. i List five vlues of between 1 nd which mke the inequlity true. ii Wht must be true bout ll the vlues of if the inequlity is true? b Consider the inequlity < 5. i List five vlues of which mke the inequlity true. ii Wht must be true bout ll the vlues of if the inequlity is true? c Complete these sttements. i If >, then. ii If <, then. 1 Consider the eqution 9 >. Solve the eqution by first dding to both sides then solve for. b Solve the eqution by first subtrcting 9 from both sides. c Wht did you hve to remember to do in prt b to ensure tht the nswer is the sme s in prt? 15 Combine ll your knowledge from this chpter so fr to solve these inequlities. ( + 1) > + 5 b + 6 c < 1 ( 1) 7( ) d + e 1 > f ( ) ENRICHMENT Literl inequlities Given, b, c nd d re positive numbers (such tht 1 < < b ), solve ech of the following for. b > c b b c b c d b c + b b b e < d f d g ( + b) < c h > d c c c i (b ) > c j + b c k + b > b 1 l b c b b cm

33 11 Chpter Epressions, equtions nd inequlities Key ides H Using formuls A formul (or rule) is n eqution tht reltes two or more pronumerls. You cn find the vlue of one of the pronumerls if you re given the vlue of ll other unknowns. Some common formuls contin squres, squre roots, cubes nd cube roots. The following re some emples of formuls. A = πr is the formul for finding the re, A, of circle given its rdius, r. F = 9 C + is the formul for converting degrees Celsius, C, to degrees Fhrenheit, F. 5 d = vt is the formul for finding the distnce, d, given the velocity, v, nd time, t. A, F nd d re sid to be the subjects of the formuls given bove. Let s strt: Common formuls As clss group, try to list t lest 10 formuls tht you know. Write down the formuls nd describe wht ech pronumerl represents. Which pronumerl is the subject of ech formul? The subject of formul is pronumerl tht usully sits on its own on the left-hnd side. For emple: the C in C = πr is the subject of the formul. A pronumerl in formul cn be evluted by substituting numbers for ll other pronumerls. A formul cn be trnsposed (rerrnged) to mke nother pronumerl the subject. C = πr cn be trnsposed to give r = C π. Emple 16 Substituting vlues into formuls Substitute the given vlues into the formul to evlute the subject. S = 1 r, when = nd r = 0. b E = 1 mv, when m = nd v = 5 SOLUTION S = 1 r S = 1 0. = 0.6 = 5 EXPLANATION b E = 1 mv E = 1 5 Substitute = nd r = 0. nd evlute. Substitute m = nd v = 5 nd evlute. Stge 5.# = 1 5 = 50

34 H Using formuls 11 Emple 17 Finding the unknown vlue in formul The re of trpezium is given by A = 1 h( + b). Substitute A = 1, = 5 nd h =, then find the vlue of b. SOLUTION A = 1 h( + b) 1 = 1 (5 + b) 1 = (5 + b) 6 = 5 + b b = 1 Emple 18 Trnsposing formuls Trnspose ech of the following to mke b the subject. c = ( + b) b c = + b (b > 0) SOLUTION c = ( + b) c = + b c = b b = c c or b = ( ) b c = + b c = + b c = b b = c b = c Eercise H UNDERSTANDING AND FLUENCY EXPLANATION Write the formul nd substitute the given vlues of A, nd h. Then solve for b. EXPLANATION 1 Stte the letter tht is the subject of these formuls. b A = 1 bh D = b c c M = + b d A = πr Tret pronumerls s you would numbers. Divide both sides by. Subtrct from both sides. Mke b the subject on the left-hnd side. Squre both sides to remove the squre root. Subtrct from both sides. Mke b the subject. Tke the squre root of both sides, b = c if b is positive. 1, (½) (½) (½)

35 11 Chpter Epressions, equtions nd inequlities Emple 16 Emple 17 Emple 18 S ubstitute the given vlues into ech of the following formuls to evlute the subject. Round to deciml plces where pproprite. A = bh, when b = nd h = 7 b F = m, when m = nd = 6 c m = + b, when = 1 nd b = 6 d t = d, when d = 18 nd v = v e A = πr, when π =.1 nd r = 1 f V = πr, when π =.1 nd r = g c = + b, when = 1 nd b = h Q = gh, when g = 9.8 nd h = 11. i I = MR, when M = 1. nd R = 6. j = ut + 1 t, when u = 0, t = nd = 10 Substitute the given vlues into ech of t h e following formuls then solve the equtions to determine the vlue of the unknown pronumerl ech time. Round to deciml plces where pproprite. m = F, when m = 1 nd = b A = lb, when A = 0 nd l = 6 c A = 1 h( + b), when A = 6, b = 1 nd h = d C = πr, when C = 6 nd π =.1 e S = πr, when S = 7 nd π =.1 f v = u + s, when v =, u = 6 nd = 1 g m =, when m = 8 nd = y (v) Trnspose ech of the following formuls to mke the pronumerl shown in brckets the subject. A = πrh (r) b I = Prt 100 (r) c p = m( + n) (n) d d = + b c () e V = πr h (r > 0) (r) f P = v (v > 0) R g S = πrh + πr (h) h A = (p + q) (p) i l T = π g PROBLEM-SOLVING AND REASONING (g) j A + B = C (A) 5 The formul s = d gives the speed s km/h of cr tht hs trvelled distnce of d km in t hours. t Find the speed of cr tht hs trvelled 00 km in.5 hours. Round to deciml plces. b i Trnspose the formul s = d to mke d the subject. t ii Find the distnce covered if cr trvels t 75 km/h for.8 hours. 5, 6, 9 5 7, 9, , 9(½), 10

36 H Using formuls The formul F = 9 C + converts degrees Celsius, C, to degrees Fhrenheit, F. 5 Find wht ech of the following tempertures is in degrees Fhrenheit. i 100 C ii 8 C b Trnspose the formul to mke C the subject. c where necessry. i 1 F ii 98 F Clculte wht ech of the following tempertures is in degrees Celsius. Round to 1 deciml plce 7 The velocity, v m/s, of n object is described by the rule v = u + t, where u is the initil velocity in m/s, is the ccelertion in m/s nd t is the time in seconds. Find the velocity fter seconds if the initil velocity is 5 m/s nd the ccelertion is 10 m/s. b Find the time tken for body to rech velocity of 0 m/s if its ccelertion is m/s nd its initil velocity is 1 m/s. 8 The volume of wter ( V litres) in tnk is given by V = t, where t is the time in seconds fter tp is turned on. Over time, does the wter volume increse or decrese ccording to the formul? b Find the volume fter minutes. c Find the time it tkes for the volume to rech 1500 litres. Round to the nerest minute. d How long, to the nerest minute, does it tke to completely empty the tnk? 9 Write formul for the following situtions. Mke the first listed pronumerl the subject. $D given c cents b d cm given e metres c The discounted price $D tht is 0% off the mrked price $M d The vlue of n investment $V tht is 15% more thn the initil mount $P e The cost $C of hiring cr t $50 upfront plus $18 per hour for t hours f The distnce d km remining in km mrthon fter t hours if the running speed is 1 km/h g The cost $C of bottle of soft drink if b bottles cost $c 10 Write formul for the vlue of in these digrms. b d Perimeter = P b e b b c c f b Are = A

37 116 Chpter Epressions, equtions nd inequlities ENRICHMENT 11 Bsketbll formuls 11 The formul T = + y + f cn be used to clculte the totl number of points mde in bsketbll gme where: = number of three-point gols y = number of two-point gols f = number of free throws mde T =totl number of points Find the totl number of points for gme where 1 three-point gols, 15 two-point gols nd 7 free throws were mde. b Find the number of three-point gols mde if the totl number of points ws 6 with 5 two-point gols mde nd 5 free throws mde. c The formul V = p + r s + + ( + b 1.5t + f + m o cn be used to ) g clculte the vlue, V, of bsketbll plyer where: p =points erned r =number of rebounds =number of ssists s =number of stels b =number of blocks t =number of turnovers f = number of personl fouls m = number of missed shots o = number of offensive rebounds g = number of gmes plyed Clculte the vlue of plyer with 50 points erned, rebounds, 1 ssists, 5 stels, blocks, 8 turnovers, 1 personl fouls, missed shots, offensive rebounds nd 10 gmes.

38 I Liner simultneous equtions: substitution 117 I Liner simultneous equtions: substitution A liner eqution with one unknown usully hs one unique solution. For emple, = is the only vlue of tht mkes the eqution + = 7 true. The liner eqution + y = 1 hs two unknowns nd it hs n infinite number of solutions. Ech solution is pir of nd y vlues tht mkes the eqution true, for emple = 0 nd y = or = nd y = or = 1 nd y = 1. However, if we re told tht + y = 1 nd lso tht y = 1, we cn find single solution tht stisfies both equtions. Equtions like this re clled simultneous liner equtions, becuse we cn find pir of nd y vlues tht stisfy both equtions t the sme time (simultneously). Let s strt: Multiple solutions A shre trder emining computer models of finncil dt, which cn involve finding vlues tht stisfy two equtions simultneously. There is more thn one pir of numbers nd y tht stisfies the eqution y = 5. Write down t lest 5 pirs (, y) tht mke the eqution true. A second eqution is y = 8. Do ny of your pirs tht mke the first eqution true lso mke the second eqution true? If not, cn you find the specil pir of numbers tht stisfies both equtions simultneously? An lgebric method clled substitution cn be used to solve simultneous equtions. It is used when t lest one of the equtions hs single pronumerl s the subject. For emple, y is the subject in the eqution y = + 1. To solve simultneous equtions using substitution: 1 Substitute one eqution into the other. Solve for the remining pronumerl. Substitute to find the vlue of the second pronumerl. + y = 8 nd y = ( + 1) = = = 8 5 = 5 = 1 y = = Stge 5.# Key ides

39 118 Chpter Epressions, equtions nd inequlities Emple 19 Solving using substitution Solve ech of the following pirs of simultneous equtions by using substitution. + y = 10 b y = 6 c + y = 19 y = y = y = 8 SOLUTION + y = 10 (1) y = () + () = 10 5 = 10 = From () y = = = 8 Solution: =, y = 8 Check: + 8 = 10 nd 8 = b y = 6 (1) y = () ( ) = 6 + = 6 + = 6 = = 1 From () y = = 1 = Solution: = 1, y = Check: + = 6 nd = 1 c + y = 19 (1) y = 8 () + ( 8) = = = 19 7 = 5 = 5 From () y = 8 = 5 8 = Solution: = 5, y = Check: 5 + = 19 nd = 5 8 EXPLANATION Number the equtions for reference. Substitute y = into (1). Combine like terms nd solve for. Substitute = into () to find the vlue of y. Check your nswer by substituting = nd y = 8 into (1) nd (). Substitute y = into (1) using brckets. Use the distributive lw nd solve for. Substitute = 1 into () to find the vlue of y. Check: substitute = 1 nd y = into (1) nd (). Substitute y = 8 into (1). Use the distributive lw nd solve for. Substitute = 5 into () to find the vlue of y. Check: substitute = 5 nd y = into (1) nd ().

40 I Liner simultneous equtions: substitution 119 Emple 19 Emple 19b Emple 19c Eercise I UNDERSTANDING AND FLUENCY 1 Find the vlue of or y by substituting the known vlue. y = ( = ) b = 5 y (y = ) c + y = 8 (y = ) 1, 5(½), 5(½), 6 6(½) Choose the correct option. When substituting y = 1 into + y = 5 the second eqution becomes: A + ( 1) = 5 B ( 1) + y = 5 C + y = 1 b When substituting = 1 y into 5 y = 6 the second eqution becomes: A 1 y = 6 B 5(1 y) = 6 C 5(1 y) y = 6 Check whether = nd y = is solution to ech of the following pirs of simultneous equtions. + y = 0 nd y = b y = 6 nd + y = 0 c + y = nd = y d + y = nd = y 10 Solve ech of the following pirs of simultneous equtions by using substitution. + y = b + y = 6 c + 5y = 8 y = = 5y y = d 5y = e + y = 18 f + y = 15 = y y = y = 5 Solve ech of the following pirs of simultneous equtions by using substitution. + y = 1 y = + 6 b + y = 1 y = + c 5 + y = 5 y = 1 d y = 7 y = + 5 e y = 9 y = 1 f + y = 6 = 9 y g y = 1 = y h + y = y = i y = 1 y = Solve ech of the following pirs of simultneous equtions by using substitution. + y = 8 y = 7 b + y = 11 y = + 1 c + y = = y 8 d + 5y = y = 5 e y = 5 = 5 y f + y = 5 y =

41 10 Chpter Epressions, equtions nd inequlities PROBLEM-SOLVING AND REASONING 7, 8, 10 7, 8, 10 8, 9, 10, 11 7 The sum of two numbers is 8 nd the lrger number is 1 more thn the smller number. Write two equtions nd solve them to find the two numbers. 8 The combined mss of two trucks is 9 tonnes. The hevier truck is 1 tonne less thn twice the mss of the smller truck. Write two equtions nd solve them to find the mss of ech truck. 9 The perimeter of rectngle is 11 cm nd the length is cm more thn hlf the width. Find the dimensions of the rectngle. 10 One of the common errors when pplying the method of substitution is mde in this working. Find the error nd describe how to void it. Solve y = 1 nd y = 7. 1 = 7 substituting y = 1 into y = 7 1 = 7 = 8 = 11 If both equtions hve the sme pronumerl s the subject, substitution is still possible. For emple, solve y = 1... (1) nd y =... () Substitute (1) into () 1 = = = nd y = 5 Use this method to solve these simultneous equtions. y = + 1 y = b y = y = + 8 ENRICHMENT Literlly chllenging c y = 1 + y = Use substitution to solve ech of the following pirs of simultneous equtions for nd y in terms of nd b. + y = b y = b b + by = b = by c + y = = y b d by = y = e y = y = b + f by = = y b

42 J Liner simultneous equtions: elimintion 11 J Liner simultneous equtions: elimintion Another method used to solve simultneous liner equtions is clled elimintion. This involves the ddition or subtrction of the two equtions to eliminte one of the pronumerls. We cn then solve for the remining pronumerl nd substitute to find the vlue of the second pronumerl. Let s strt: To dd or subtrct? To use the method of elimintion you need to decide if using ddition or using subtrction will eliminte one of the pronumerls. Decide if the terms in these pirs should be dded or subtrcted to give the result of 0. nd y nd y nd 7y nd 7y Describe under wht circumstnces ddition or subtrction should be used to eliminte pir of terms. Elimintion involves the ddition or subtrction of two equtions to remove one pronumerl. Elimintion is often used when both equtions re of the form + by = d or + by + c = 0. Add equtions to eliminte terms of opposite sign: + y = 5 + y = 8 = 8 Subtrct equtions to eliminte terms of the sme sign: + y = 6 5y = 7 8y = 1 If terms cnnot be eliminted just by using ddition or subtrction, first multiply one or both equtions to form mtching pir. For emple: mtching pir 1 y = 1 + y = 7 y = 5y = 6 y = 1 + y = 6 (Multiply both sides by ) mtching pir 8 8y = 1 8 5y = (Multiply both sides by ) (Multiply both sides by 7 ) Stge 5.# Key ides

43 1 Chpter Epressions, equtions nd inequlities Emple 0 Solving simultneous equtions using elimintion Solve the following pirs of simultneous equtions by using elimintion. y = 1 + 5y = b y = 5 5 y = 11 c 5 + y = 7 + 7y = 5 d + y = 18 y = 5 SOLUTION y = 1 (1) + 5y = () (1) + () y = y = 1 From (1) y = 1 (1) = 1 = 1 = =, y = 1 Check (1) = 1 + 5(1) = b y = 5 (1) 5 y = 11 () () (1) = 6 = From (1) y = 5 () y = 5 y = y = =, y = Check () () = 5 (5) () = 11 c 5 + y = 7 (1) + 7y = 5 () 5 () 5 + 5y = 15 () 5 + y = 7 (1) () (1) y = 1 y = From () + 7y = () = = 5 = =, y = EXPLANATION Add the two equtions to eliminte since + ( ) = 0. Then solve for y. Substitute y = 1 into eqution (1) to find. Substitute = nd y = 1 into the originl equtions to check. Subtrct the two equtions to eliminte y since they re the sme sign, i.e. y ( y) = y + y = 0. Alterntively, you could do (1) () but () (1) voids negtive coefficients. Solve for. Substitute = into eqution (1) to find y. Substitute = nd y = into the originl equtions to check. There re different numbers of nd y in ech eqution so multiply eqution () by 5 to mke the coefficient of equl in size to (1). Subtrct the equtions to eliminte. Substitute y = in eqution () to find. Substitute = nd y = into the originl equtions to check.

44 J Liner simultneous equtions: elimintion 1 Emple 0 Emple 0b d + y = 18 (1) y = 5 () (1) 8 + 6y = 6 () () 9 6y = 15 () () + () 17 = 51 = From (1) + y = 18 () + y = y = 18 y = 6 y = =, y = Eercise J UNDERSTANDING AND FLUENCY 1 Add these equtions to eliminte or y. i + y = 11 y = 5 ii y = 0 + y = b Subtrct these equtions to eliminte or y. i + 5y = 11 + y = 7 ii + 5y = y = 7 Multiply eqution (1) by nd eqution () by to mke the coefficients of y equl in size but opposite in sign. Add the equtions to eliminte y. Substitute = into eqution (1) to find y. Substitute = nd y = into the originl equtions to check. iii + y = 7 + y = 11 iii + y = + y = Decide if ddition or subtrction should be chosen to eliminte the pronumerl in these simultneous equtions. i + y = 5y = ii y = 9 + y = 11 iii y = 0 y = 8 b Decide if ddition or subtrction will eliminte the pronumerl y in these simultneous equtions. i y = 6 + y = ii 7 y = 5 y = 5 iii 10y + = 1 10y = Solve these simultneous equtions by first dding the equtions. + y = + y = b y = + 6y = d y = e y = + y = 5 + y = 1,, (½), 5(, b), 6(½), (½), 5(, b), 6 7(½) (½), 5(c), 6 7(½) c + y = 1 y = 7 f + y = 5 5y = Solve these simultneous equtions by first subtrcting the equtions. + y = 10 + y = 6 b + 7y = 9 + 5y = 11 c + y = 1 y = 10 5 Solve these simultneous equtions by first subtrcting the equtions. 5 y = y = b 5 + y = y = c 9 y = y = 9

45 1 Chpter Epressions, equtions nd inequlities Emple 0c Emple 0d 6 Solve the following pirs of simultneous liner equtions by using elimintion. + y = 8 y = 17 b y = 5 + y = 1 c + y = 8y = d + y = 0 + y = 5 e + y = 1 + y = f y = 1 6 5y = 10 g y = 5 7 y = 0 h y = 1 5 y = 19 i 5 y = 7 y = 9 7 Solve the following pirs of simultneous liner equtions by using elimintion. + y = 1 + y = b 7 + y = 8 5y = 1 c 6 5y = y = d y = y = 7 e 7 + y = 1 + y = 8 f 5 + 7y = 1 + 5y = 1 g 5 + y = y = 5 h 7y = 8 y = i y = 1 + y = 8 j 7y = y = 7 k + 5y = y = l y = y = 11 PROBLEM-SOLVING AND REASONING 8 The sum of two numbers is 0 nd their difference is 1. Write two equtions nd find the numbers. 9 Two supplementry ngles differ by. Write two equtions nd find the two ngles. 10 The perimeter of rectngulr city block is 800 metres nd the difference between the length nd width is 1 metres. Wht re the dimensions of the city block? 11 A techer collects totl of 17 mobile phones nd ipods before group of students heds off on bushwlk. From second group of students, 0 phones nd ipods re collected. The second group hd twice the number of phones nd times the number of ipods thn the first group. How mny phones nd ipods did the first group hve? 8, 9, , 1, , 1, 1 1 Consider the pir of simultneous equtions + y = 5 (1) 5 + y = 11 () Solve the equtions by first subtrcting eqution () from eqution (1), i.e. (1) (). b Now solve the equtions by first subtrcting eqution (1) from eqution (), i.e. () (1). c Which method or b is preferble nd why?

46 J Liner simultneous equtions: elimintion 15 1 To solve ny of the pirs of simultneous equtions in this section using the method of substitution, wht would need to be done before the substitution is mde? Try these using substitution. + y = 5 y = 7 b y = + y = 5 1 Find the solution to these pirs of simultneous equtions. Wht do you notice? + y = nd + y = 1 b 7 1y = nd y = ENRICHMENT Literl elimintion Use elimintion to solve the following pirs of simultneous equtions to find the vlue of nd y in terms of the other pronumerls. + y = y = b b + y = 0 y = b c by = by = d + y = b + y = b e b + 5y = b b + y = b f + y = 1 y = 10 g + y = b y = b h y = b + y = b i + y = b y = b j + y = c + y = c k y = 1 by = 1 l + by = + y = 1 m + by = c + y = d n by = + y = o + by = b y = p by = b c y = q + by = c d by = f r + by = c d + by = f

47 16 Chpter Epressions, equtions nd inequlities Key ides K Using liner simultneous equtions to solve problems Mny problems cn be described mthemticlly using pir of simultneous liner equtions from which solution cn be obtined lgebriclly. Let s strt: The tyre store In one prticulr week totl of 8 crs nd motorcycles re checked into grge to hve their tyres chnged. Ech motorcycle hs tyres chnged nd ech cr hs tyres chnged. The totl number of tyres sold in the week is 8. If you hve to find the number of motorcycles nd the number of crs tht hve their tyres chnged in the week: wht two vribles should you define? wht two equtions cn you write? which method (substitution or elimintion) would you use to solve the equtions? FPO wht is the solution to the simultneous equtions? how would you nswer the question in words? To solve worded problems with simultneous equtions: Define two pronumerls by writing down wht they represent. For emple: Let $C be the cost of Let be the number of Write pir of simultneous equtions from the given informtion using your two pronumerls. Solve the equtions simultneously using substitution or elimintion. Check the solution by substituting into the originl equtions. Epress the nswer in words. Stge 5.#

48 K Using liner simultneous equtions to solve problems 17 Emple 1 Solving word problems with simultneous equtions Andre bought two continers of ice-crem nd three bottles of mple syrup for totl of $. At the sme shop, Bettin bought one continer of ice-crem nd two bottles of mple syrup for $1. How much does ech continer of ice-crem nd ech bottle of mple syrup cost? SOLUTION Let: $ be the cost of continer of ice-crem $y be the cost of bottle of mple syrup + y = (1) + y = 1 () () + y = 6 () + y = (1) () (1) y = From () + y = 1 + () = = 1 = 5 The cost of one continer of ice-crem is $5 nd the cost of one bottle of mple syrup is $. Eercise K UNDERSTANDING AND FLUENCY EXPLANATION Define the unknowns. Ask yourself wht you re being sked to find. continers of ice-crem nd bottles of mple syrup for totl of $ 1 continer of ice-crem nd bottles of mple syrup for $1. Choose the method of elimintion to solve. Multiply () by to obtin mtching pir. Subtrct eqution (1) from (). Substitute y = into (). Solve for. Substitute y = nd = 5 into originl equtions to check. Answer the question in sentence. 1,, 7, 8, 6, 8, 9 1 The sum of two numbers is nd their difference is 6. Find the two numbers nd y by completing the following steps. Write pir of simultneous equtions relting nd y. b Solve the pir of equtions using substitution or elimintion. c Write your nswer in words. The length l cm of rectngle is 5 cm longer thn its bredth b cm. If the perimeter is 8 cm, find the dimensions of the rectngle by completing the following steps. Write pir of simultneous equtions relting l nd b. b Solve the pir of equtions using substitution or elimintion. c Write your nswer in words. A rectngulr block of lnd hs perimeter of 10 metres nd the length l m of the block is three times the bredth b m. Find the dimensions of the block of lnd by completing the following steps. Write pir of simultneous equtions relting l nd b. b Solve the pir of equtions using substitution or elimintion. c Write your nswer in words.

49 18 Chpter Epressions, equtions nd inequlities Emple 1 Ml bought bottles of milk nd bgs of chips for totl of $17. At the sme shop, Brbr bought 1 bottle of milk nd 5 bgs of chips for $1. Find how much ech bottle of milk nd ech bg of chips cost by: defining two pronumerls to represent the problem b writing pir of simultneous equtions relting the two pronumerls c solving the pir of equtions using substitution or elimintion d writing your nswer in words 5 Leonie bought seven lip glosses nd two eye shdows for totl of $69 nd Chrissie bought four lip glosses nd three eye shdows for totl of $5. Find how much ech lip gloss nd ech eye shdow costs by completing the following steps. Define two pronumerls to represent the problem. b Write pir of simultneous equtions relting the two pronumerls. c Solve the pir of equtions using substitution or elimintion. d Write your nswer in words. 6 Steve bought five cricket blls nd fourteen tennis blls for $10. Ben bought eight cricket blls nd nine tennis blls for $11. Find the cost of cricket bll nd the cost of tennis bll. 7 At birthdy prty for 0 people ech person could order hot dog or chips. If there were four times s mny hot dogs s chips ordered, clculte how mny hot dogs nd how mny chips were bought. 8 The entry fee for fun run is $10 for dults nd $ for children. A totl of $60 ws collected from the 0 competitors. Find the number of dults running nd the number of children running. 9 Mil plnts 80 hectres of pottoes nd corn. To mimise his profit he plnts 10 hectres more of pottoes thn of corn. How mny hectres of ech does he plnt? PROBLEM-SOLVING AND REASONING 10 Crrie hs 7 coins in her purse. All the coins re 5-cent or 0 -cent coins. If the totl vlue of the coins is $.75, how mny of ech type does she hve? 11 Michel is 0 yers older thn his dughter. In five yers time Michel will be times s old s his dughter. How old is Michel now? 1 Jenny hs twice s much money s Kristy. If I give Kristy $50, she will hve three times s much s Jenny. How much did ech of them hve originlly? 10, 11, , 1, , 15, 16

50 K Using liner simultneous equtions to solve problems 19 1 At prticulr cinem the cost of n dult movie ticket is $15 nd the cost of child s ticket is $10. The seting cpcity of the cinem is 0. For one movie session ll sets re sold nd $00 is collected from the sle of tickets. How mny dult nd how mny children s tickets were sold? 1 Wilfred nd Wendy hve long distnce bike rce. Wilfred rides t 0 km/h nd hs hour hed strt. Wendy trvels t 8 km/h. How long does it tke for Wendy to ctch up to Wilfred? Use distnce = speed time. 15 Andrew trvelled distnce of 9 km by jogging for hours nd cycling for hours. He could hve trvelled the sme distnce by jogging for 7 hours nd cycling for hours. Find the speed t which he ws jogging nd the speed t which he ws cycling. 16 Mlcolm s mother is 7 yers older thn he is nd their ges re both two-digit numbers. If Mlcolm swps the digits in his ge, he gets his mother s ge. How old is Mlcolm if the sum of the digits in his ge is 5? b Wht is the reltionship between the digits in Mlcolm s ge if the sum of the digits is unknown. c If the sum of the digits in Mlcolm s two-digit ge is unknown, how mny possible ges could he be? Wht re these ges? ENRICHMENT Digit swp 17, The digits of two-digit number sum to 10. If the digits swp plces, the number is 6 more thn the originl number. Wht is the originl number? Cn you show n lgebric solution? 18 The difference between the two digits of two-digit number is. If the digits swp plces, the number is 18 less thn the originl number. Wht is the originl number? Cn you show n lgebric solution?

51 10 Chpter Epressions, equtions nd inequlities Key ides L Qudrtic equtions of the form = c In liner eqution, if the highest power of the vrible is, the eqution is clled qudrtic eqution. Since = 16 nd ( ) ( ) = 16, we would sy tht the eqution = 16 hs two possible solutions = nd =. In this section we will solve some simple qudrtic equtions. Qudrtic equtions my rise in res such s the pth of projectile, flling object or mesurement clcultion. Let s strt: More thn one solution? Is it possible to hve more thn one solution to n eqution? Consider the following. Wht do you notice when you evlute:? ( )? ( 5)? 5? From the bove results, wht possible vlue(s) of mke the following equtions true? = 9 = 5 Cn you find vlue of to mke the eqution = 9 true? Why or why not? A qudrtic eqution is one in which the highest power of the pronumerl is. In this section you will solve qudrtic equtions of the form = c, such s these: = 0 = 9 = 18 6 = 7 To solve equtions of the form = c, mke the subject, then tke the squre root of both sides of the eqution. This process my produce two solutions, one solution or no rel solutions. If = 18, then = 9. This eqution hs two solutions. These re found by tking the squre root of both sides of the eqution. We know tht = 9 nd lso ( ) = 9, so there re two solutions: = nd =. This cn be bbrevited to = ±. If = 0, then = 0. This eqution only hs one solution, = 0, becuse the squre root of zero is zero. If = 18, then = 9. This eqution is sid to hve no rel solutions becuse the squre root of 9 hs no vlue in the set of rel numbers. In future yers you my study topic clled Comple Numbers, in which equtions like = 9 cn be solved. Some qudrtic equtions hve non-integer solutions. These cn be epressed using surds. For emple, the solution of = 10 is = ± 10. These re the ect solutions, but they could be epressed s = ±. (to 1 d.p.), for emple. In prcticl problems such s those involving mesurement, the negtive solution is usully rejected. For emple, if the re of squre is 10 squre metres, the ect side length is 10 metres, not 10 metres or ± 10 metres. Stge 5.#

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