8 Algebr 8A Using vribles 8B Substitution 8C Working with brckets 8D Substituting positive nd negtive numbers 8E Number lws nd vribles 8F Simplifying expressions 8G Multiplying nd dividing expressions with vribles 8H Expnding brckets 8I Fctorising WHAT DO YOU KNOW? 1 List wht you know bout lgebr. Crete concept mp to show your list. Shre wht you know with prtner nd then with smll group. As clss, crete lrge concept mp tht shows your clss s knowledge of lgebr. Digitl doc Hungry brin ctivity Chpter 8 doc-691 OPENING QUESTION This flooded creek hs ctchment of 70 km. Find n expression to clculte the volume of wter tht potentilly runs into the creek fter x mm of rin.
Are you redy? Try the questions below. If you hve difficulty with ny of them, extr help cn be obtined by completing the mtching SkillSHEET locted on your ebookplus. Digitl doc SkillSHEET 8.1 doc-69 Alterntive expressions used to describe the four opertions 1 Write expressions for the following: the difference between M nd C M - C b the mount of money erned by selling B lmingtons for $ ech c the product of X nd Y XY d 1 more thn H H + 1 e the cost of 10 ornges if ech ornge costs D cents. 10D cents $B Digitl doc SkillSHEET 8. doc-69 Order of opertions II Find the vlue of ech of the following using the order of opertions rule. 1 + 18 ó 6 15 b 1-16 ì ó 8 6 c 7 ì - 5 ì 1-9 Digitl doc SkillSHEET 8. doc-69 Order of opertions with brckets Find the vlue of ech of the following using the order of opertions rule. 5 + (10 ì ) - 1 1 b ( ì 1) + (18 ó 6) - ( + 7) 18 Digitl doc SkillSHEET 8. doc-695 Opertions with directed numbers Perform ech of the clcultions. -8 + -10-18 b 8 - -17 5 c 6 ì - -5 d - ó - 8 Digitl doc SkillSHEET 8.5 doc-696 Combining like terms 5 Simplify the following expressions by dding or subtrcting like terms. g + g 7g b y + y + y 6y c 6gy - yg gy d 0x - 19x + 11 x + 11 e 7g + 8g + 8 + 9 15g + 17 f 7h + t - h h + t Digitl doc SkillSHEET 8.6 doc-697 Simplifying frctions 6 Simplify the following. 7 1 1 b 1 0 c 15 5 0 Digitl doc SkillSHEET 8.7 doc-698 Highest common fctor 7 Find the highest common fctor for ech of the following pirs of numbers. 8, 8 b 1, 5 7 c 18, 7 9 18 Mths Quest 8 for the Austrlin Curriculum
8A elesson Using vribles eles-00 Using vribles A vrible (or pronumerl) is letter or symbol tht represents vlue in n lgebric expression. In lgebric expressions such s + b, the vribles represent ny number. In lgebric equtions such s x + 1 = 9, vribles re referred to s unknowns becuse the vrible represents specific vlue tht is not yet known. When we write expressions with vribles, the multipliction sign is omitted. For exmple, 8n mens 8 ì n nd 1b mens 1 ì ì b. The division sign is rrely used. For exmple, y ó 6 is usully written s y 6. WORKED EXAMPLE 1 Suppose we use b to represent the number of nts in nest. Write n expression for the number of nts in the nest if 5 nts died. b Write n expression for the number of nts in the nest if the originl nt popultion doubled. c Write n expression for the number of nts in the nest if the originl popultion incresed by 50. d Wht would it men if we sid tht nerby nest contined b + 100 nts? e Wht would it men if we sid tht nother nest contined b - 1000 nts? f Another nest in very poor soil contins b nts. How much smller thn the originl is this nest? THINK WRITE b The originl number of nts (b) must be reduced by 5. The originl number of nts (b) must be multiplied by. It is not necessry to show the ì sign. b - 5 c 50 must be dded to the originl number of nts (b). c b + 50 d e This expression tells us tht the nerby nest hs 100 more nts. This expression tells us tht the nest hs 1000 fewer nts. b d e b The nerby nest hs 100 more nts. This nest hs 1000 fewer nts. f The expression b mens b ó, so this nest is hlf the size of the originl nest. f This nest is hlf the size of the originl nest. REMEMBER 1. A vrible is letter or symbol tht is used in plce of number.. Vribles my represent single number, or they my be used to show reltionship between two or more numbers.. When writing expressions with vribles, it is importnt to remember the following points: () The multipliction sign is omitted. For exmple: 8n mens 8 ì n nd 1b mens 1 ì ì b. (b) The division sign is rrely used. For exmple, y ó 6 is shown s y 6. Chpter 8 Algebr 18
EXERCISE 8A INDIVIDUAL PATHWAYS Activity 8-A-1 Using vribles doc-60 Activity 8-A- More vribles doc-61 Activity 8-A- Advnced vribles doc-6 Using vribles FLUENCY 1 WE 1 Suppose we use x to represent the number of nts in nest. Write n expression for the number of nts in the nest if 0 nts were born. x + 0 b Write n expression for the number of nts in the nest if the originl nt popultion tripled. x c Write n expression for the number of nts in the nest if the originl nt popultion x - 10 decresed by 10. The nerby nest hs 60 more nts. d Wht would it men if we sid tht nerby nest contined x + 60 nts? e Wht would it men if we sid tht nerby nest contined x - 90 nts? The nerby nest hs 90 f Another nest in very poor soil contins x fewer nts. nts. How much smller thn the originl is this nest? The nest is one qurter of the size of the originl nest. Between 9.00 m nd 9.15 m one Dnish pstry ws sold. In the next hour-nd--hlf, further 11 Dnish pstries were sold. No more Dnish pstries hd been sold t 1.0 pm, but in the next hlf-hour 18 more were sold. No Dnish pstries were sold fter 1.00 pm. Suppose x people re in ttendnce t the strt of footbll mtch. If further y people rrive during the first qurter, write n expression for the number of people t the ground. x + y b Write n expression for the number of people t the ground if further 60 people rrive prior to the second qurter commencing. x + y + 60 c At hlf-time 170 people leve. Write n expression for the number of people t the ground fter they hve left. x + y + 90 d In the finl qurter further 50 people leve. Write n expression for the number of people t the ground fter they hve left. x + y - 60 The cnteen mnger t Browning Industries orders m Dnish pstries ech dy. Write prgrph tht could explin the tble below. Time 9.00 m m Number of Dnish pstries 9.15 m m - 1 10.5 m m - 1 1.0 pm m - 1 1.00 pm m - 0 5.0 pm m - 0 18 Mths Quest 8 for the Austrlin Curriculum
Imgine tht your cutlery drwer contins knives, b forks nd c spoons. Write n expression for the totl number of knives nd forks you hve. + b b Write n expression for the totl number of items in the drwer. + b + c c You put more forks in the drwer. Write n expression for the number of forks now. b + d Write n expression for the number of knives in the drwer fter 6 knives re removed. - 6 5 If y represents certin number, write expressions for the following numbers. A number 7 more thn y. b A number 8 less thn y. y + 7 y - 8 c A number tht is equl to five times y. 5y d The number formed when y is subtrcted from 1. 1 - y y e The number formed when y is divided by. f The number formed when y is multiplied by 8 nd is dded to the result. 8y + 6 Using nd b to represent numbers, write expressions for: the sum of nd b + b b the difference between nd b - b c three times subtrcted from two times b b - d the product of nd b b e twice the product of nd b b f the sum of nd 7b + 7b g multiplied by itself h multiplied by itself nd the result divided by 5. 5 7 If tickets to bsketbll mtch cost $7 for dults nd $1 for children, write n expression for the cost of: y dult tickets $7y b d child tickets $1d c r dult nd h child tickets. $(7r + 1h) UNDERSTANDING 8 Nomi is now t yers old. Write n expression for her ge in yers time. t + b Write n expression for Steve s ge if he is g yers older thn Nomi. t + g c How old ws Nomi 5 yers go? t - 5 d Nomi s fther is twice her ge. How old is he? t 9 Jmes is trvelling into town one prticulr evening nd observes tht there re t pssengers in his crrige. He continues to tke note of the number of people in his crrige ech time the trin deprts from sttion, which occurs every minutes. The tble t the top of the next pge shows the number of pssengers. Chpter 8 Algebr 185
The number of pssengers doubled t the next stop nd continued to increse, more thn qudrupling in the first nine minutes. At 7. pm, 5 people lighted the trin, nd by 7.5 pm the sme number of pssengers were on the trin s there were t the beginning. By 7. pm there were 1 fewer pssengers thn there were t the beginning. Time Number of pssengers 7.10 pm t 7.1 pm t 7.16 pm t + 1 7.19 pm t + 1 7. pm t + 7 7.5 pm t 7.8 pm t + 1 7.1 pm t - 8 7. pm t - 1 Write prgrph explining wht hppened. b When did pssengers first begin to light the trin? 7. pm c At wht time did the crrige hve the most number of pssengers? d At wht time did the crrige hve the lest number of pssengers? REASONING 7.19 pm 7. pm The number of bcteri in ech of these intervls is double the number of bcteri in the previous intervl. 10 A microbiologist plces m bcteri onto n gr plte. She counts the number of bcteri t pproximtely hour intervls. The results re shown in the tble below. Time Number of bcteri 9.00 m m 1 noon m.18 pm m 6.0 pm 8m 9.05 pm 16m 1 midnight m - 10 Explin wht hppens to the number of bcteri in the first 5 intervls. b Wht might be cusing the number of bcteri to increse in this wy? c Wht is different bout the lst bcteri count? d Wht my hve hppened to cuse this? 11 n represents n even number. Is the number n + 1 odd or even? Odd b Is n odd or even? Even c Write expressions for: i the next three even numbers tht re greter thn n n +, n + nd n + 6 ii the even number tht is less thn n. The bcteri could be dividing in two. It is lower thn expected, bsed on the previous pttern of growth. Some of the bcteri my hve died, or filed to divide nd reproduce. n - REFLECTION List some resons for using vribles insted of numbers. 186 Mths Quest 8 for the Austrlin Curriculum
8B Substitution If the vlue of vrible (or vribles) is known, it is possible to evlute (work out the vlue of) n expression by using substitution. The vrible is replced with the number. Substitution cn lso be used with formul or rule. WORKED EXAMPLE Find the vlue of the following expressions if = nd b = 15. b 6 b 7 THINK WRITE 1 Substitute the vrible () with its correct vlue nd replce the multipliction sign. 6 = 6 ì Evlute nd write the nswer. = 18 b 1 Substitute ech vrible with its correct vlue nd replce the multipliction signs. b 7 - b 15 = 7 ì - Perform the first multipliction. = 1 15 Perform the next multipliction. = 1 0 Perform the division. = 1-10 5 Perform the subtrction nd write the nswer. = 11 WORKED EXAMPLE The formul for finding the re (A) of rectngle of length l nd width w is A = l ì w. Use this formul to find the re of the rectngle t right. 70 m m THINK WRITE 1 Write the formul. A = l ì w Substitute ech vrible with its vlue. = 70 ì Perform the multipliction nd stte the correct units. = 860 m REMEMBER Replcing vrible with number is clled substitution. Chpter 8 Algebr 187
EXERCISE 8B INDIVIDUAL PATHWAYS Activity 8-B-1 Substitution doc-6 Activity 8-B- More substitution doc-6 Activity 8-B- Advnced substitution doc-65 Digitl doc Spredsheet Substitution doc-87 Substitution FLUENCY 1 WE Find the vlue of the following expressions, if = nd b = 5. 6 b 7 1 c 6b 0 d 1 e + 7 9 f b - 1 g + b 7 h b - i 5 + b 8 6 j + 9 15 k + b 19 l 5 5 m 5 n b 10 o b 0 p 7b - 0 5 b b 15 7 89 9 q 6b - r s + t 5 b (8 9 ) 9 10 10 b Substitute x = 6 nd y = into the following expressions nd evlute. x y 9 6x + y b + c xy 5 d 1 x y 1 7x e + + y 9 f x - y 15 g.5x 15 h 1 x 1y xy i.x + 1.7y. j 11y - x 1 k x 1 l.8 15 x y m.8x -.5y 18. n 8.7y - x 8.1 o 1.x - 9.6x 16. p 7 9 1 (1) Evlute the following expressions, if d = 5 nd m =. d + m 7 b m + d 7 c m - d - d d - m ( 9 ) 10 10 e m f md 10 g 5dm 50 h md 10 1 i -d -15 j -m - k 6m + 5d 7 l md 15 m 5m - d 0 n 7d 15 1 o dm - 1 19 p m 15 d 1 WE The formul for finding the perimeter (P) of rectngle of length l nd width w is P = l + w. Use this formul to find the perimeter of the rectngulr swimming pool t right. 150 m 5 The formul for the perimeter (P) of squre of side length l is P = l. Use this formul to find the perimeter of squre of side length.5 cm. 10 cm 6 The formul c = 0.1 + is used to clculte the cost in dollrs (c) of renting cr for one dy from Poole s Cr Hire Ltd, where is the number of kilometres trvelled on tht dy. Find the cost of renting cr for one dy if the distnce trvelled is 0 kilometres. C = $6 7 The re (A) of rectngle of length l nd width w cn be found using the formul A = lw. Find the re of the rectngles below: length 1 cm, width cm b length 00 m, width m 8 cm 800 m c length. m, width 10 cm..7 m or 70 cm 188 Mths Quest 8 for the Austrlin Curriculum
UNDERSTANDING 8 The formul F = 9 C + is used to convert tempertures mesured in degrees Celsius to n 5 pproximte Fhrenheit vlue. F represents the temperture in degrees Fhrenheit (èf) nd C the temperture in degrees Celsius (èc). Find F when C = 100 èc. F = 1 èf b Convert 8 èc to Fhrenheit. 8 èc = 8. èf c Wter freezes t 0 èc. Wht is the freezing temperture of wter in Fhrenheit? èf 9 The formul D = 0.6T cn be used to convert dis tnces in kilometres (T) to the pproximte equivlent in miles (D). Use this rule to convert the following distnces to miles: 100 kilometres b 8 kilometres c 1.5 kilometres. REASONING 100 km 60 miles 8 km 18.8 miles 1.5 km 7.5 miles 8C 10 Ben sys tht x = x. Emm sys tht is not x correct if x = 0. Explin Emm s resoning. If x = 0, then the expression becomes 0 0, which is indeterminte. Working with brckets REFLECTION Cn ny vlue be substituted for vrible in every expression? Brckets re grouping symbols. The expression ( + 5) cn be thought of s three groups of ( + 5), or ( + 5) + ( + 5) + ( + 5). When substituting into n expression with brckets, remember to plce multipliction sign (ì) next to the brckets. For exmple, ( + 5) is thought of s ì ( + 5). Following opertion order, evlute the brckets first nd then multiply by the vlue outside of the brckets. WORKED EXAMPLE Substitute r = nd s = 5 into the expression 5(s + r) nd evlute. b Substitute t =, x = nd y = 5 into the expression x(t - y) nd evlute. THINK WRITE 1 Plce the multipliction sign bck into the expression. 5(s + r) = 5 ì (s + r) Substitute the vribles with their correct vlues. = 5 ì (5 + ) Evlute the expression in the pir of brckets first. = 5 ì 9 Perform the multipliction nd write the nswer. = 5 b 1 Plce the multipliction signs bck into the expression. b x(t - y) = ì x ì ( ì t - y) Substitute the vribles with their correct vlues. = ì ì ( ì - 5) Perform the multipliction inside the pir of brckets. = ì ì (1-5) Perform the subtrction inside the pir of brckets. = ì ì 7 5 Perform the multipliction nd write the nswer. = Chpter 8 Algebr 189
REMEMBER 1. Brckets re grouping symbols.. When substituting into n expression with brckets, remember to plce multipliction (ì) sign next to the brckets.. Work out the brckets first. EXERCISE 8C INDIVIDUAL PATHWAYS Activity 8-C-1 Working with brckets doc-66 Activity 8-C- More brckets doc-67 Activity 8-C- Advnced use of brckets doc-68 Working with brckets FLUENCY 1 WE Substitute r = 5 nd s = 7 into the following expressions nd evlute. (r + s) 6 b (s - r) c 7(r + s) 8 d 9(s - r) 18 e s(r + ) 56 f s(r - 5) 5 g r(r + 1) 90 h rs( + s) 50 i 11r(s - 6) 55 j r(s - r) 0 k s( + r) 1 l 7s(r - ) 17 m s(rs + 7) 78 n 5r( - s) 50 o 5sr(sr + s) 9800 p 8r(1 - s) Evlute ech of the expressions below, if x =, y = 5 nd z = 9. 1 xy(z - ) 90 b x ( z y) 16 c 00 z y + x 10 6 d (x + y)(z - y) e (z - )x 7 f zy(17 - xy) 90 y g x 5 ( 7 + ) 7 h (8 - y)(z + x) 6 i 7 1 x y 6 60 j x ( xz + y ) 58 z k ( y + ) 1 l x(xyz - 105) 180 x 7 m 1(y - 1)(z + ) n ( x 7) 7 x + 576 o + 1 6 z ( x ) z -6 p ( y 11) + 8 x The formul for the perimeter (P) of rectngle of length l nd width w is P = l + w. This rule cn lso be written s P = (l + w). Use the rule to find the perimeter of rectngulr comic covers with the following mesurements. l = 0 cm, w = 11 cm 6 cm b l = 7.5 cm, w = 1. cm 97.8 cm 190 Mths Quest 8 for the Austrlin Curriculum
UNDERSTANDING MC When = 8 nd b = 1 re substituted into the expression (15 - b + 9), the expression is equl to 6 A B 16 C 1 1 D E 7 5 A rule for finding the sum of the interior ngles in mny-sided figure such s pentgon is S = 180(n - ), where S represents the sum of the ngles inside the figure nd n represents the number of sides. The digrm t right shows the interior ngles in pentgon. Use the rule to find the sum of the interior ngles for the following figures: hexgon (6 sides) 70è b pentgon 50è c tringle 180è d qudrilterl ( sides) 60è e 0-sided figure. 0è REASONING 6 The dimensions of the figure on the right re given in terms of m nd n. Write, in terms of m nd n, n expression for: the length of CD CD = m + n b the length of BC BC = m + n c the perimeter of the figure. Perimeter = 8m + 18n A m + n F m + n B C m + 5n D n - m E REFLECTION Is opertion order followed when substituting vlues for vribles? 8D Substituting positive nd negtive numbers If the vrible you re substituting hs negtive vlue, simply remember the following rules for directed numbers: 1. For ddition nd subtrction, signs tht occur together cn be combined. Sme signs positive for exmple, 7 + + = 7 + nd 7 - - = 7 + Different signs negtive for exmple, 7 - + = 7 - nd 7 + - = 7 -. For multipliction nd division. Sme signs positive for exmple, +7 ì + = +1 nd -7 ì - = +1 Different signs negtive for exmple, +7 ì - = -1 nd -7 ì + = -1 Chpter 8 Algebr 191
WORKED EXAMPLE 5 Substitute m = 5 nd n = - into the expression m - n nd evlute. b Substitute m = - nd n = -1 into the expression n - m nd evlute. c Substitute = nd b = - into the expression 5b - 1 b THINK nd evlute. WRITE 1 Substitute the vribles with their correct vlue. m - n = 5 - - Combine the two negtive signs nd dd. = 5 + Write the nswer. = 8 b 1 Replce the multipliction sign. b n - m = ì n - m Substitute the vribles with their correct vlues. = ì -1 - - Perform the multipliction. = - - - Combine the two negtive signs nd dd. = - + 5 Write the nswer. = 0 c 1 Replce the multipliction signs. c 5b - 1 b = 5 ì ì b - 1 b Substitute the vribles with their correct vlues. = 5 ì ì - - 1 Perform the multiplictions. = -60-1 Perform the division. = -60 - - 5 Combine the two negtive signs nd dd. = -60 + 6 Write the nswer. = -56 REMEMBER When substituting, if the vrible you re replcing hs negtive vlue, simply remember the rules for directed numbers: 1. For ddition nd subtrction, signs tht occur together cn be combined. Sme signs positive for exmple, 7 + + = 7 + nd 7 - - = 7 + Different signs negtive for exmple, 7 - + = 7 - nd 7 + - = 7 -. For multipliction nd division. Sme signs positive for exmple, +7 ì + = +1 nd -7 ì - = +1 Different signs negtive for exmple, +7 ì - = -1 nd -7 ì + = -1 19 Mths Quest 8 for the Austrlin Curriculum
EXERCISE 8D INDIVIDUAL PATHWAYS Activity 8-D-1 Wht is the word? A doc-69 Activity 8-D- Wht is the word? B doc-70 Activity 8-D- Wht is the word? C doc-71 If x is negtive then 5x will lso be negtive integer (less thn or equl to 5). Subtrcting this number is equivlent to dding positive integer. The result will be positive. Substituting positive nd negtive numbers FLUENCY 1 WE5 Substitute m = 6 nd n = - into the following expressions nd evlute. m + n b m - n 9 c n - m -9 d n + m e n -9 f -m -1 g n - m -1 h n + 5 m i m + n - 5 j 11n + 0-1 k -5n - m 9 l m mn m m 1 - n - o -8 p - 9 n 5 n n 9 m n q + 0 r 6mn - 1-109 s + 15. 6 t 1 mn 16 n 9 WE5b Substitute x = 8 nd y = - into the following expressions nd evlute. (x - ) 18 b x(7 + y) c 5y(x - 7) -15 d ( - y) 1 e (y + 5)x 16 f xy(7 - x) g ( + x)(5 + y) h 5(7 - xy) 155 x i y ( 5 ) x y j 1 6 + 9 y k ( 6 x) 6 y l ( x 1) + 1 WE5c Substitute = - nd b = -5 into the following expressions nd evlute. + b -9 b - b 1 c b - d b 0 e 1 - b -8 f -(b - ) g - b - - h (b + ) 1 i b - 5 j 8 16 6b - k -1 l -6 5 b m 5 + b 15 n 8b - b 175 o + -5 p.5b -1.5 5 q 11 + 6b -7 r ( - 5)(8 - b) -117 s (9 - )(b - ) -10 t 1.5b + -15.5 UNDERSTANDING If p = - nd q = -, evlute ( pq p). q+ p 0 7 or 7 REASONING Digitl doc WorkSHEET 8.1 doc-90 8E 5 Consider the expression 1-5x. If x is negtive integer, explin why the expression will hve positive vlue. Number lws nd vribles When deling with ny type of number, we must obey prticulr rules. Commuttive Lw REFLECTION Wht cn you sy bout the sign of x? The Commuttive Lw refers to the order in which two numbers my be dded, subtrcted, multiplied or divided. The Commuttive Lw holds true for ddition nd multipliction becuse the order in which two numbers re dded or multiplied does not ffect the result. + = + ì = ì Chpter 8 Algebr 19
Since vribles tke the plce of numbers, the Commuttive Lw holds true for the ddition nd multipliction of vribles. x + y = y + x x ì y = y ì x The Commuttive Lw does not hold true for subtrction or division becuse the results obtined re different. - - Since vribles tke the plce of numbers, the Commuttive Lw does not hold true for the subtrction nd division of vribles. x - y y - x x y y x WORKED EXAMPLE 6 Find the vlue of the following expressions if x = nd y = 7. Comment on the results obtined. i x + y ii y + x b i x - y ii y - x c i x ì y ii y ì x d i x ó y ii y ó x THINK WRITE i 1 Substitute ech vrible with its correct vlue. i x + y = + 7 Evlute nd write the nswer. = 11 ii 1 Substitute ech vrible with its correct vlue. ii y + x = 7 + Evlute nd write the nswer. = 11 Compre the result with the nswer obtined in prt i. b i 1 Substitute ech vrible with its correct vlue. b i x - y = - 7 Evlute nd write the nswer. = - ii 1 Substitute ech vrible with its correct vlue. ii y - x = 7 - Evlute nd write the nswer. = Compre the result with the nswer obtined in prt b i. c i 1 Substitute ech vrible with its correct vlue. c i x ì y = ì 7 Evlute nd write the nswer. = 8 ii 1 Substitute ech vrible with its correct vlue. ii y ì x = 7 ì Evlute nd write the nswer. = 8 Compre the result with the nswer obtined in prt c i. d i 1 Substitute ech vrible with its correct vlue. d i x ó y = ó 7 The sme result is obtined; therefore, order is not importnt when dding two terms. Two different results re obtined; therefore, order is importnt when subtrcting two terms. The sme result is obtined; therefore, order is not importnt when multiplying two terms. Evlute nd write the nswer. = 7 (ö0.57) 19 Mths Quest 8 for the Austrlin Curriculum
ii 1 Substitute ech vrible with its correct vlue. ii y ó x = 7 ó Evlute nd write the nswer. = 7 (1.75) Compre the result with the nswer obtined in prt d i. Two different results re obtined; therefore, order is importnt when dividing two terms. Interctivity The Associtive Lw int-70 Associtive Lw The Associtive Lw refers to the order in which three numbers my be dded, subtrcted, multiplied or divided, tking two t time. Note: The Associtive Lw refers to the order in which the ddition (or other opertion) is performed, nd this order is indicted by the use of brckets. The order in which the vribles re written does not chnge. Like the Commuttive Lw, the Associtive Lw holds true for ddition nd multipliction of numbers. 5 + (10 + ) = (5 + 10) + 5 ì (10 ì ) = (5 ì 10) ì Since vribles tke the plce of numbers, the Associtive Lw holds true for the ddition nd multipliction of vribles. x + (y + z) = (x + y) + z x ì (y ì z) = (x ì y) ì z Like the Commuttive Lw, the Associtive Lw does not hold for subtrction nd division of numbers. 5 - (10 - ) ò (5-10) - 5 ó (10 ó ) ò (5 ó 10) ó Since vribles tke the plce of numbers, the Associtive lw does not hold true for the subtrction nd division of vribles. x - (y - z) ò (x - y) - z x ó (y ó z) ò (x ó y) ó z WORKED EXAMPLE 7 Find the vlue of the following expressions if x = 1, y = 6 nd z =. Comment on the results obtined. i x + (y + z) ii (x + y) + z b i x - (y - z) ii (x - y) - z c i x ì (y ì z) ii (x ì y) ì z d i x ó (y ó z) ii (x ó y) ó z THINK WRITE i 1 Substitute ech vrible with its correct vlue. i x + (y + z) = 1 + (6 + ) Evlute the expression in the pir of brckets. = 1 + 8 Perform the ddition nd write the nswer. = 0 ii 1 Substitute ech vrible with its correct vlue. ii (x + y) + z = (1 + 6) + Evlute the expression in the pir of brckets. = 18 + Perform the ddition nd write the nswer. = 0 Compre the result with the nswer obtined in prt i. The sme result is obtined; therefore, order is not importnt when dding terms. b i 1 Substitute ech vrible with its correct vlue. b i x - (y - z) = 1 - (6 - ) Evlute the expression in the pir of brckets. = 1 - Perform the subtrction nd write the nswer. = 8 Chpter 8 Algebr 195
ii 1 Substitute ech vrible with its correct vlue. ii (x - y) - z = (1-6) - Evlute the expression in the pir of brckets. = 6 - Perform the subtrction nd write the nswer. = Compre the result with the nswer obtined in prt b i. c i 1 Substitute ech vrible with its correct vlue. 196 Mths Quest 8 for the Austrlin Curriculum Two different results re obtined; therefore, order is importnt when subtrcting terms. c i x ì (y ì z) = 1 ì (6 ì ) Evlute the expression in the pir of brckets. = 1 ì 1 Perform the multipliction nd write the nswer. = 1 ii 1 Substitute ech vrible with its correct vlue. ii (x ì y) ì z = (1 ì 6) ì Evlute the expression in the pir of brckets. = 7 ì Perform the multipliction nd write the nswer. = 1 Compre the result with the nswer obtined in prt c i. The sme result is obtined; therefore, order is not importnt when multiplying terms. d i 1 Substitute ech vrible with its correct vlue. d i x ó (y ó z) = 1 ó (6 ó ) Evlute the expression in the pir of brckets. = 1 ó Perform the division nd write the nswer. = ii 1 Substitute ech vrible with its correct vlue. ii (x ó y) ó z = (1 ó 6) ó Evlute the expression in the pir of brckets. = ó Perform the division nd write the nswer. = 1 Compre the result with the nswer obtined in prt d i. Identity Lw Two different results re obtined; therefore, order is importnt when dividing terms. The Identity Lw for ddition sttes tht when zero is dded to ny number, the originl number remins unchnged. For exmple, 5 + 0 = 0 + 5 = 5. The Identity Lw for multipliction sttes tht when ny number is multiplied by one, the originl number remins unchnged. For exmple, ì 1 = 1 ì = 1. Since vribles tke the plce of numbers: x + 0 = 0 + x = x Inverse Lw x ì 1 = 1 ì x = x The Inverse Lw for ddition sttes tht when number is dded to its opposite, the result is zero. For exmple, 5 + -5 = 0. The Inverse Lw for multipliction sttes tht when number is multiplied by its reciprocl, the result is one. For exmple, ì 1 = 1. Since vribles tke the plce of numbers: x + x = x + x = 0 1 1 x = x = 1 x x
REMEMBER 1. When deling with numbers nd vribles, prticulr rules must be obeyed.. The Commuttive Lw holds true for ddition (nd multipliction) becuse the order in which two numbers or vribles re dded (or multiplied) does not ffect the result. Therefore, in generl, () x + y = y + x (b) x - y ò y - x (c) x ì y = y ì x (d) x ó y ò y ó x. The Associtive Lw holds true for ddition (nd multipliction) becuse the order in which three numbers or vribles, tking two t time, re dded (or multiplied) does not ffect the result. Therefore, in generl, () x + (y + z) = (x + y) + z (b) x - (y - z) ò (x - y) - z (c) x ì (y ì z) = (x ì y) ì z (d) x ó (y ó z) ò (x ó y) ó z. The Identity Lw sttes tht, in generl, x + 0 = 0 + x = x x ì 1 = 1 ì x = x 5. The Inverse Lw sttes tht, in generl, x + -x = -x + x = 0 1 1 x = x = 1 x x EXERCISE 8E INDIVIDUAL PATHWAYS Activity 8-E-1 Number lws nd vribles doc-7 Activity 8-E- More number lws nd vribles doc-7 Activity 8-E- Advnced number lws nd vribles doc-7 Number lws nd vribles FLUENCY 1 WE 6,b Find the vlue of the following expressions if x = nd y = 8. Comment on the results obtined. i x + y 11 ii y + x 11 Sme b i x + y 5 ii y + x 5 Sme c i 5x + y 1 ii y + 5x 1 Sme d i 8x + y ii y + 8x Sme e i x - y -5 ii y - x 5 Different f i x - y -18 ii y - x 18 Different g i x - 5y -8 ii 5y - x 8 Different h i x - y 1 ii y - x -1 Different WE 6c,d Find the vlue of the following expressions if x = - nd y = 5. Comment on the results obtined. i x ì y -10 ii y ì x -10 Sme b i 6x ì y -180 ii y ì 6x -180 Sme c i x ì y -0 ii y ì x -0 Sme d i 7x ì 5y -50 ii 5y ì 7x -50 Sme e i x ó y - ii y ó x 5-5 Different f i 10x ó y -1 ii y ó 10x -1 Sme g i 6x ó y - ii y ó 6x - 5 Different 5 h i 7x ó 9y - 1 ii 9y ó 7x Different 5-5 1 Chpter 8 Algebr 197
Indicte whether ech of the following is true or flse for ll vlues of the vribles. + 5b = 5b + True b 6x - y = y - 6x Flse c 7c + d = -d + 7c Flse d 5 ì x ì x = 10x True e x ì -y = -y ì x True f ì x ì x = 1x ì x True 5p r g = Flse h -7i - j = j + 7i Flse r 5p i -y ó x = x ó -y Flse j -c + d = d - c True k 0 s = l x x Flse 15 = 15 s 0 WE 7,b Find the vlue of the following expressions if x =, y = 8 nd z =. Comment on the results obtined. i x + (y + z) 1 ii (x + y) + z 1 Sme b i x + (y + 5z) ii (x + y) + 5z Sme c i 6x + (y + z) 0 ii (6x + y) + z 0 Sme d i x - (y - z) - ii (x - y) - z -7 Different e i x - (7y - 9z) -5 ii (x - 7y) - 9z -71 Different f i x - (8y - 6z) - ii (x - 8y) - 6z -67 Different 5 WE 7c,d Find the vlue of the following expressions if x = 8, y = nd z = -. Comment on the results obtined. i x ì (y ì z) -6 ii (x ì y) ì z -6 Sme b i x ì (-y ì z) 768 ii (x ì -y) ì z 768 Sme True c i x ì (y ì z) -156 ii (x ì y) ì z -156 Sme d i x ó (y ó z) - ii (x ó y) ó z -1 Different e i x ó (y ó z) -6 ii (x ó y) ó z - 1 Different 6 8 f i -x ó (5y ó z) ii (-x ó 5y) ó z Different 5 1 10 UNDERSTANDING 6 Indicte whether ech of the following is true or flse for ll vlues of the vribles. - 0 = 0 Flse b ì 1 000 000 = 0 Flse c 1 15t = 1 d 1 Flse d = 1 True 15t d 8x 8x e = 1 f 11t True = 0 Flse 9y 9y 0 7 MC The vlue of the expression x ì (-y ì z) when x =, y = nd z = - is: A 108 B - C D 11 E -108 8 MC The vlue of the expression (x - 8y) - 10z when x = 6, y = 5 nd z = - is: A -7 B 7 C -6 D 6 E -6 REFLECTION The Commuttive Lw does not hold for subtrction. Wht cn you sy bout the results of x - nd - x? 198 Mths Quest 8 for the Austrlin Curriculum
8F Simplifying expressions Expressions cn often be written in more simple form by collecting (dding or subtrcting) like terms. Like terms re terms tht contin exctly the sme vribles, rised to the sme power. To understnd why + cn be dded but + b cn not be dded, consider the following identicl bgs of lollies, ech contining lollies. lollies lollies + lollies lollies lollies So, + = 5. Then consider the following bgs contining lollies nd bgs contining ì b lollies. lollies lollies + ì b lollies ì b lollies ì b lollies So + b cnnot be dded s they re not identicl nd we do not hve ny further informtion. So, + b = + b. For exmple: x nd x re like terms. x nd y re not like terms. b nd 7b re like terms. 7b nd 8 re not like terms. bc nd cb re like terms. 8 nd re not like terms. g nd 5g re like terms. WORKED EXAMPLE 8 Simplify the following expressions. + 5 b 7b - - b c c - 6 + c + 15 THINK WRITE 1 Write the expression nd check tht the two terms re like terms, tht is, they contin the sme vribles. + 5 Add the like terms nd write the nswer. = 8 b 1 Write the expression nd check for like terms. b 7b - - b Rerrnge the terms so tht the like terms re together. Remember to keep the correct sign in front of ech term. = 7b - b - Subtrct the like terms nd write the nswer. = b - c 1 Write the expression nd check for like terms. c c - 6 + c + 15 Rerrnge the terms so tht the like terms re together. Remember to keep the correct sign in front of ech term. Simplify by collecting like terms nd write the nswer. = c + c - 6 + 15 = 6c + 9 Chpter 8 Algebr 199
REMEMBER 1. When simplifying expressions, we cn collect (dd or subtrct) only like terms.. Like terms re terms tht contin the sme vrible prts. EXERCISE 8F INDIVIDUAL PATHWAYS Activity 8-F-1 Simplifying expressions doc-75 Activity 8-F- More simplifying expressions doc-76 Activity 8-F- Advnced simplifying expressions doc-77 Simplifying expressions FLUENCY 1 WE8 Simplify the following expressions. c + c 6c b c - 5c -c c + 5 - d 6q - 5q q e -h - h -h f 7x - 5x x g - 7 - -6 h -f + 7f f i p - 7p -p j -h + h h k 11b + b + 5b 18b l 7t - 8t + t t m 9m + 5m - m 1m n x - x -x o 7z + 1z 0z p 5p + p + p 10p q 9g + 1g - g 17g r 18b - b - 11b b s 1t - t + 5t 1t t -11j + j -7j u -1l + l - 5l -15l v 1m - m - m + m 8m w m + m - m 0 x t + t - t + 8t 10t WE8b,c Simplify the following expressions. x + 7x - y 10x - y b x + x - 1 7x - 1 c 11 + 5f - 7f 11 - f d u - u + 6 6 - u e m + p + 5m 7m + p f -h + r - h r - 5h g 11-5b + 6 17-5b h 9t - 7 + 5 9t - i 1 - g + 5 17 - g j 6m + m - n + n 10m - n k 5k - 5 + k - 7 7k - 1 l n - + n - 5 n - 9 m b - 6 - b + 18 1 - b n 11-1h + 9 0-1h o 1y - y - 7g + 5g - 6 9y - g - 6 p 8h - 6 + h - 11h - 8 q 11s - 6t + t - 7s s - t r m + 1l - 7m + l 1l - 5m k - 1h + 7 s h + k - 16h - k + 7 t 1 + 5t - 9t - 8 5 - t u g + 5 + 5g - 7 7g - v 17f - k + f - 7k 19f - 10k UNDERSTANDING Simplify the following expressions. x + x x b y + y 5y c + d d + 6d 7d e 7g - 8g -g f y + 7y 10y g b + 5b 7b h - i g - g -g j + + + 5 + 9 k 11x - 6 + 1x + 6 x l 1s - + 7 - s 11s + 8 + 5 m + + 5 + n 11b - b + b + 1b o 6t - 6g - 5t + g - 7 t - g - 7 8g - g + p 11g + 17 - g + 5 - g q 1b + + 6b 18b + r 1xy + xy - xy - 5xy 11xy fg + s s fg + s - fg + s t 11b + b - 5 1b - 5 u 18b - c + b - 10c b + b 0b - 1c REASONING Rose owns n rt gllery nd sells items supplied to her by vrious rtists. She receives commission for ll items sold. She uses the following method to keep trck of the money she owes the rtists when their items re sold. Check with your techer. Ask the rtist how much they wnt for the item. Add 50% to tht price, then mrk the item for sle t this new price. When the item sells, tke one-third of the sle price s commission, then return the blnce to the rtist. 00 Mths Quest 8 for the Austrlin Curriculum
Use lgebr to show tht this method does return the correct mount to the rtist. 5 Explin, using mthemticl resoning nd with digrms if necessry, why the expression x + x cnnot be simplified. Check with your techer. REFLECTION Wht do you need to remember when checking for like terms? 8G Multiplying nd dividing expressions with vribles Multiplying vribles When we multiply vribles (s lredy stted) the Commuttive Lw holds, so order is not importnt. For exmple: ì 6 = 6 ì 6 ì w = w ì 6 ì b = b ì The multipliction sign (ì) is usully omitted for resons of convention. ì g ì h = gh ì x ì y = x y Although order is not importnt, conventionlly the vribles in ech term re written in lphbeticl order. For exmple, ì b ì ì c = b c WORKED EXAMPLE 9 Simplify the following. 5 ì g b -d ì 6b ì 7 THINK 1 Write the expression nd replce the hidden multipliction signs. WRITE 5 ì g = 5 ì ì g Multiply the numbers. = 0 ì g Remove the multipliction sign. = 0g b 1 Write the expression nd replce the hidden multipliction signs. b -d ì 6b ì 7 = - ì d ì 6 ì ì b ì 7 Plce the numbers t the front. = - ì 6 ì 7 ì d ì ì b Multiply the numbers. = -16 ì d ì ì b Remove the multipliction signs nd plce the vribles in lphbeticl order. = -16bd Dividing expressions with vribles When dividing expressions with vribles, rewrite the expression s frction nd simplify by cncelling. Remember tht when the sme vrible ppers s fctor on both the numertor nd denomintor, it my be cncelled. Chpter 8 Algebr 01
WORKED EXAMPLE 10 Simplify 16 f. b Simplify 15n ó n. THINK WRITE 1 Write the expression. 16 f Simplify the frction by cncelling 16 with (divide both by ). = = f 1 16 f 1 No need to write the denomintor since we re dividing by 1. = f b 1 Write the expression nd then rewrite it s frction. b 15n ó n = 15 n n = 5 1 15n n Simplify the frction by cncelling 15 with nd n with n. No need to write the denomintor since we re dividing by 1. = 5 1 = 5 WORKED EXAMPLE 11 Simplify -1xy ó 7y. THINK WRITE 1xy 1 Write the expression nd then rewrite it s frction. 1xy + 7y = 7y Simplify the frction by cncelling 1 with 7 (divide both by ) nd y with y. = 9 = x 9 1xy 7y WORKED EXAMPLE 1 Simplify the following. m ì m b 5p 10 ì p c 6x 7 1x 6 y y d 1y 8 THINK WRITE 1 Write the problem. m ì m 0 Mths Quest 8 for the Austrlin Curriculum
The order is not importnt when multiplying, so plce the numbers first. = ì ì m ì m Multiply the numbers. = 6 ì m ì m Check to see if the bses re the sme. They re both m. 5 Simplify by dding the indices. = 6 ì m + 1 = 6m b 1 Write the problem. b 5p 10 ì p The order is not importnt when multiplying, so plce the numbers first. = 5 ì ì p 10 ì p Multiply the numbers. = 15 ì p 10 ì p Check to see if the bses re the sme. They re both p. 5 Simplify by dding the indices. = 15 ì p 10 + = 15p 1 c 1 Write the problem nd express it s frction. c 6x 7 1x = 6 x 1x Divide the numbers. = x 7 x Check to see if the bses re the sme. They re both x. Subtrct the powers. = x d 1 Write the problem. d 6 y y 1y 7 8 Multiply the numbers in the numertor. Simplify the numbers in index form in the numertor. = y 1y Divide the numbers nd subtrct the powers. = y 7 11 REMEMBER 1. When multiplying vribles: () the order is not importnt. For exmple, d ì e = e ì d. (b) plce the numbers t the front of the expression nd leve out the ì sign.. When dividing vribles, rewrite the expression s frction nd simplify it by cncelling.. When the sme vrible ppers on both the top nd bottom lines of the frction, it my be cncelled. Chpter 8 Algebr 0
EXERCISE 8G INDIVIDUAL PATHWAYS Activity 8-G-1 Who invented lgebr? doc-78 Activity 8-G- Who is the fther of lgebr? doc-79 Activity 8-G- Mthemticin's riddle doc-80 Multiplying nd dividing expressions with vribles FLUENCY 1 WE9 Simplify the following. ì g 1g b 7 ì h 1h c d ì 6 d d z ì 5 15z e 6 ì 5r 0r f 5t ì 7 5t g ì u 1u h 7 ì 6p p i 7gy ì 1gy j ì 11ht ht k x ì 6g gx l 10 ì 7h 70h m 9m ì d 6dm n c ì 5h 15ch o 9g ì x 18gx p.5t ì 5b 1.5bt q 1m ì 1n 156mn r 6 ì 1d 7d s b ì c 6bc t f ì gh 1fgh u ì 8w ì x 8wx v 11b ì d ì 7 w 16xy ì 1.5 xy x.5x ì y 10.5xy y 11q ì s ì 1qs z ì b ì c 1bd WE10 Simplify the following. bc 8 f 6h 15x f b h c 5x d 9g ó g 16m e 10r ó 5 r f x ó x g 8r ó r h 8m i 1q ó 1q j m 8 f f q 7h ó h x 6x k 1h 1h l 50g ó 75g n 5x ó 70x o m ó 6m p y ó y r 0d 8d Simplify the following. 15fg 5fg b 1cd ó cd c e 11xy 9pq p y f 11x 18q g i 5 jk 5r 5 j 55rt ó 77t kj 7 k m 1xy ó x q 1 1mnp 60np 16cd 1y n 0cd 5 r 11d 66d s 6q q 8xy 1 1b 8b 10mxy 5mx o 1bc ó 7bc d s 18dg ó 5g t 5 ) t 81l ó 7l 11 d cg ó h l 9dg 1g 6bc ó 7c p gh ó 6h Simplify the following. ì -5f -15f b -6 ì -d 1d c 11 ì -g -g d -9t ì -g 7gt e -5t ì -dh 0dht f 6 ì -st -18st g - ì -w ì 7d dw h - ì -b ì c ì e bce i 11b ì -f -bf j s ì -b ì -x 18bsx k -5h ì -5t ì -q -75hqt l ì -w ì - ì 6p 1pw m -7 ì b ì g -1bg n 17b ì -gh -51bgh o -.5g ì h ì 7-9gh p 5h ì 8j ì -k -0hjk q 75x ì 1.5y 11.5xy r 1rt ì -z ì p -1prtz s b ì c ì 5 0bc t -w ì x ì -08wx u -b ì -5cd ì -6e -90 bcde 5 WE11 Simplify the following. 8 9 11m 5 b 1 5 1 1 6 1 11b b 6 7 16 11 (1 5 xy y 7 bh 7h c 60jk ó -5k b 7 d -1j 1 cg g b 0 Mths Quest 8 for the Austrlin Curriculum
d -h ó -6dh 1 g e d 0gl 5l f -1xy ó 8y x g 1b 6 fgh f 6 h i -xyz ó 6yz x 7 1b 0ghj 5 j j rt 1 k -5mn ó 0n l -1st ó -8 6 m st 6rt m b ó -17b - n b b 7dg o d 5gh 5h p -60mn ó 55mnp 1 8def 1ef q r -7xyz ó 8yz 18 x 11p 18d 9 7 s 5pq t 11 oc 11 6pqr r 1oct 1t UNDERSTANDING 6 WE 1 Simplify the following. ì b -5p ì -5p 5p c -5 ì x ì x -0x d b ì 7 7 b e b ì cd 6b cd f -5xy ì ì 8x -160x y g 7pq ì p ì q p q h 5m ì n ì 6nt ì -t -0mn t i - ì xyz ì -z ì -y -18xy z j -7 ì -b ì -c -bc k mn ì - ì n ì 0 0 l w x ì -9z ì xy -18w x y z m ì 7 6 11 n x y ì x x 5 y o 0m 1 ó m 10m 9 p 5p q 15p 8q 1 7 10 5 8x 7y z b b 5p q xyz b q r 6 6x 1y 5b 5 7 Simplify the following. 6 5b b 10b b c w 5 5 w w d rk 6st s 5rt 15gt 10 g f ht 1hk 9k 16h g 5 g 5t 5 dk 9dt 9d g 5t 1 5t 9th tg h 7 h gn g n g 6h g REFLECTION i xy x 8y How is multipliction nd division 10 f 5 j of expressions with vribles 7wz 1z w w 9wz similr to multipliction nd 6fz division of numbers? 8H Expnding brckets The Distributive Lw The Distributive Lw is the nme given to the following process. (5 + 8) = ì 5 + ì 8 This is becuse the number out in front is distributed to ech of the terms in the brcket. Since vribles tke the plce of numbers, the Distributive Lw lso holds true for lgebric expressions. (b + c) = b + c Chpter 8 Algebr 05
WORKED EXAMPLE 1 The Distributive Lw cn be demonstrted using the concept of re. As cn be seen in the digrm t right, ( + b) = + b We cn think of ( + b) = ( + b) + ( + b) + ( + b) Collecting like terms, ( + b) = + + + b + b + b = + b An expression contining brcket multiplied by number cn be written in expnded or fctorised form. Fctorised form Expnded form ( + b) = + b Expnding nd fctorising re the inverse of ech other. The Distributive Lw cn be used when the terms inside the brckets re either dded or subtrcted. (b - c) = b - c The Distributive Lw is not used when the terms inside the brckets re multiplied or divided. You cn see this with numbers ( ì 5) = ì ì 5; not ( ì ) ì ( ì 5). When simplifying expressions, we cn leve the result in either fctorised form or expnded form, but not combintion of both. Use the Distributive Lw to expnd the following expressions. ( + ) b -x(x - 5) + b b THINK WRITE 1 Write the expression. ( + ) = ( + ) Use the Distributive Lw to expnd the brckets. = ì + ì Simplify by multiplying. = + 6 b 1 Write the expression. b x(x - 5) = x(x - 5) Use the Distributive Lw to expnd the brckets. = x ì x + x ì -5 Simplify by multiplying. = x - 5x WORKED EXAMPLE 1 Some expressions cn be simplified further by collecting like terms fter ny brckets hve been expnded. Expnd the expressions below nd then simplify by collecting ny like terms. (x - 5) + b (x + ) + 7x + 1 c x(y + ) + x(y + 1) d x(x - 1) - (x - 1) THINK WRITE 1 Write the expression. (x - 5) + Expnd the brckets. = ì x + ì -5 + = x - 15 + Collect the like terms (-15 nd ). = x - 11 b 1 Write the expression. b (x + ) + 7x + 1 Expnd the brckets. = ì x + ì + 7x + 1 = 1x + 16 + 7x + 1 06 Mths Quest 8 for the Austrlin Curriculum
Rerrnge so tht the like terms re together. (Optionl) = 1x + 7x + 16 + 1 Collect the like terms. = 19x + 8 c 1 Write the expression. c x(y + ) + x(y + 1) Expnd the brckets. = x ì y + x ì + x ì y + x ì 1 = 6xy + 6x + xy + x Rerrnge so tht the like terms re together. (Optionl) Simplify by collecting the like terms. = 9xy + 9x = 6xy + xy + 6x + x d 1 Write the expression. d x(x - 1) - (x - 1) Expnd the brckets. Tke cre with negtive terms. = x ì x + x ì -1 - ì x - ì -1 = 8x - x - 6x + Simplify by collecting the like terms. = 8x - 10x + REMEMBER 1. Brckets re grouping symbols.. Removing brckets from n expression is clled expnding the expression.. When expnding brckets, put the ì sign before the brcket.. The rule tht is used to expnd brckets is clled the Distributive Lw. 5. After expnding brckets, collect ny like terms. EXERCISE 8H INDIVIDUAL PATHWAYS Activity 8-H-1 Snp doc-81 Activity 8-H- More snp doc-8 Activity 8-H- Advnced snp doc-8 0b + 1c 8m + 10m r 6k - 15k Expnding brckets FLUENCY 1 WE1 Use the Distributive Lw to expnd the following expressions. (d + ) d + 1 b ( + 5) + 10 c (x + ) x + 8 d 5(r + 7) 5r + 5 e 6(g + 6) 6g + 6 f (t + ) t + 6 g 7(d + 8) 7d + 56 h 9(x + 6) 18x + 5 i 1( + c) 8 + 1c j 7(6 + x) + 1x k 5(g + ) 90g + 15 l 1.5(t + 6) 1.5t + 9 m 11(t - ) 11t - n (t - 6) 6t - 18 o t(t + ) t + t p x(x + ) x + x q g(g + 7) g + 7g r g(g + 5) g + 10g s f (g + ) fg + 9f t 6m(n - m) 6mn - 1m Expnd the following. 15xy - 5y (x - ) 9x - 6 b x(x - 6y) x - 18xy c 5y(x - 9y) d 50(y - 5) 100y - 50 e -(c + ) -c - 9 f -5(x + ) -15x - 0 g -5x(x + 6) -5x - 0x h -y(6 + y) -1y - y i -6(t - ) -6t + 18 j -f(5 - f) -0f + 8f k 9x(y - ) 7xy - 18x l -h(b - 6h) -6bh + 18h m (5b + c) n -(g - 7) o 5(b + 6c) p -w(9w - 5z) q 1m(m + 10) r -k(-k + 5) -6g + 1 15b + 0c -18w + 10wz WE1 Expnd the expressions below nd then simplify by collecting ny like terms. 7(5x + ) + 1 5x + 9 b (c - ) + c - c c(5 - c) + 1c c - c d 6(v + ) + 6 6v + 0 e d(d - ) + d 5d - 1d f y + (y + ) 11y + 1 g r + r( + r) 6r + r h 5 - g + 6(g - 7) 9g - 7 i (f - g) + f - 7 11f - 1g - 7 j (x - ) + 1 9x k -(k + 5) - k -5k - 10 l x( + r) + 9x - 6xr 18x + 6rx m 1 + 5(r - 5) + r 8r - 1 n 1gh + g(h - 9) + g o (t + 8) + 5t - 11t + 1 p + r( - r) - r + 5r + 11r - 11r 18gh - g Chpter 8 Algebr 07
Digitl doc Spredsheet Expnding brckets doc-88 Expnd the following nd then simplify by collecting like terms. (x + ) + (x + 1) 5x + 8 b 5(x + ) + (x + ) 9x + c (y + 1) + (y + 6) 6y + 6 d (d + 7) - (d + ) d + e 6(h + 1) + (h - ) 1h f (m + ) + (6m - 5) 1m - g 9(f + ) - (f + 7) 8f - 1 h ( + ) - 5( + 7) i ( - t ) + t(t + 1) 6 - t + t j m(n + ) - mn + m 7m - - 5 UNDERSTANDING 6 6 b y x Are = 5 ì Are required = 5 ì 6-5 ì = 0 5 Are = ì y Are required = x - y 8I 5 Simplify the following expressions by removing the brckets nd then collecting like terms. h(k + 7) + k(h + 5) b 6n(y + 7) - n(8y + 9) c g(5m + 6) - 6(gm + ) d 11b( + 5) + b( - 5) e 5( - 7) - 5( + 7) f 7c(f - ) + c(8 - f ) g 7x( - y) + xy - 9 h 11v(w + 5) - (8-5vw) i x( - y) + 6x(y - 9) j 8m(7n - ) + n( + 7m) REASONING WORKED EXAMPLE 15 6 Using the concept of re s shown bove, explin with digrms nd mthemticl resoning why 5(6 ) = 5 ì 6 5 ì. b Using the concept of re s shown bove, explin with digrms nd mthemticl resoning why (x - y) = ì x - ì y. 7 Expressions of the form ( + b)(c + d) cn be expnded by using the Distributive Lw twice. Distribute one of the fctors over the other; for exmple, ( + b)(c + d). The expression cn then be fully expnded following Worked exmple 1. (x + 1)(x + ) b ( + )( + ) c (c + )(c - ) d (y + )(y - ) e (u - )(u - ) f (k - 5)(k - ) Fctorising Fctorising is the opposite process to expnding. Fctorising involves identifying the highest common fctors of the lgebric terms. To find the highest common fctor of the lgebric terms: 1. Find the highest common fctor of the number prts.. Find the highest common fctor of the vrible prts.. Multiply these together. Find the highest common fctor of 6x nd 10. THINK 1 Find the highest common fctor of the number prts. Brek 6 down into fctors. Brek 10 down into fctors. The highest common fctor is. Find the highest common fctor of the vrible prts. There isn t one, becuse only the first term hs vrible prt. WRITE 6 = ì 10 = 5 ì HCF = REFLECTION 7 x + x + b + 7 + 1 c c - c - 6 d y - 16 e u - 5u + 6 f k - 7k + 10 Why doesn t the Distributive Lw pply when there is multipliction sign inside the brckets, tht is for (b ì c)? The HCF of 6x nd 10 is. 5 10hk + 1h + 0k b 15n - 6ny c 8gm + g - 18 d 18b + 67b e 5-5 - 5 f 11cf + c g 8x - 5xy - 9 08 Mths Quest 8 for the Austrlin Curriculum
WORKED EXAMPLE 16 Find the highest common fctor of 1fg nd 1gh. THINK WRITE 1 Find the highest common fctor of the number prts. Brek 1 down into fctors. Brek 1 down into fctors. The highest common fctor is 7. Find the highest common fctor of the vrible prts. Brek fg down into fctors. Brek gh down into fctors. Both contin fctor of g. 1 = 7 ì 1 = 7 ì HCF = 7 fg = f ì g gh = g ì h HCF = g Multiply these together. The HCF of 1 fg nd 1gh is 7g. To fctorise n expression, plce the highest common fctor of the terms outside the brckets nd the remining fctors for ech term inside the brckets. WORKED EXAMPLE 17 Fctorise the expression x + 6. THINK WRITE 1 Brek down ech term into its fctors. x + 6 = ì x + ì Write the highest common fctor outside the brckets. = ì (x + ) Write the other fctors inside the brckets. Remove the multipliction sign. = (x + ) WORKED EXAMPLE 18 Fctorise 1gh - 8g. THINK WRITE 1 Brek down ech term into its fctors. 1gh - 8g = ì ì g ì h - ì ì g Write the highest common fctor outside the brckets. = ì g ì ( ì h - ) Write the other fctors inside the brckets. Remove the multipliction signs. = g(h - ) REMEMBER 1. Fctorising is the opposite process to expnding.. Fctorising number or expression involves breking it down into smller fctors.. To find the highest common fctor (HCF) of lgebric terms, follow these steps. () Find the highest common fctor of the number prts. (b) Find the highest common fctor of the vrible prts. (c) Multiply these together.. To fctorise n expression we plce the highest common fctor of the terms outside the brckets nd the remining fctors for ech term inside the brckets. Chpter 8 Algebr 09
EXERCISE 8I INDIVIDUAL PATHWAYS Activity 8-I-1 Fctorising doc-8 Activity 8-I- More fctorising doc-85 Activity 8-I- Tricky fctorising doc-86 mn c(1 - + d) Digitl doc Spredsheet Finding the HCF doc-89 Fctorising FLUENCY 1 WE15 Find the highest common fctor of the following. nd 6 b 6 nd 9 c 1 nd 18 6 d 1 nd 6 1 e 1 nd 1 7 f x nd g x nd 9 h 1 nd 16 WE16 Find the highest common fctor of the following. gh nd 6g g b mn nd 6mp m c 11 nd b 11 d m nd 6m m e 1b nd 1c f fg nd 6gh 1g g 0dg nd 18ghq g h 11gl nd lp 11l i 16mnp nd 0mn j 8bc nd 1c c k c nd 1cd c l x nd xz x WE17 Fctorise the following expressions. x + 6 (x + ) b y + (y + ) c 5g + 10 5(g + ) d 8x + 1 (x + ) e 6f + 9 (f + ) f 1c + 0 (c + 5) g d + 8 (d + ) h x - (x - ) i 1g - 18 6(g - ) j 11h + 11 11(h + 11) k s - 16 (s - ) l 8x - 0 (x - 5) m 1g - 1(g - ) n 1 - b (7 - b) o 16 + 6 16( + ) p 8-1q 1( - q) q 16 + 8f 8( + f ) r 1-1d 1(1 - d ) WE18 Fctorise the following. gh + 1 (gh + ) b xy + 6y y(x + ) c 1pq + p p(q + 1) d 1g - 7gh 7g( - h) e 16jk - k k(8j - 1) f 1eg + g g(6e + 1) g 1k + 16 (k + ) h 7mn + 6m m(7n + 6) i 1b + 7b 7b( + 1) j 5-15bc 5(1 - bc) k 8r + 1rt r( + 7t) l mb + 1b 1b(m + 1) m b - 6b b( - ) n 1fg - 16gh g(f - h) o b - bc b( - c) p 1x - 1xy 7x( - y) q 11jk + k k(11j + ) r p + 7pq p(1 + 9q) s 1c - c + dc t g + 8gh - 16 (g + h - ) u 8s + 1st 1s( + t) v 15uv + 7vw v(5u + 9w) UNDERSTANDING 5 Find the highest common fctor of b, 6 b nd 1 b. b 6 Find the lowest common multiple of b, bc nd 6 b. 1 b c x 15x + 15 6x 1 7 Simplify:. 5 10x 0 x 0x + 0 8 Simplify: x + 6 1x + x 1 6( x + 1). ( x + 1) ( x 1) 9 Fctorise nd hence simplify: xy 6 xy + 8b x 6y. + b xy 7 REASONING 8 or 91 Digitl doc WorkSHEET 8. doc-91 10 Simplify (5x y - 6bxy + x y - bxy) ó (x - bx). 7y REFLECTION Wht strtegies will you use to find the highest common fctor? 10 Mths Quest 8 for the Austrlin Curriculum
Summry Using vribles A vrible (or pronumerl) is letter or symbol tht is used in plce of number. Vribles my represent single number, or they my be used to show reltionship between two or more numbers. When writing expressions with vribles, it is importnt to remember the following points: The multipliction sign is omitted. For exmple: 8n mens 8 ì n nd 1b mens 1 ì ì b. The division sign is rrely used. For exmple, y ó 6 is shown s y 6. Substitution Replcing vrible with number is clled substitution. Working with brckets Brckets re grouping symbols. When substituting into n expression with brckets, remember to plce multipliction (ì) sign next to the brckets. Work out the brckets first. Substituting positive nd negtive numbers When substituting, if the vrible you re replcing hs negtive vlue, simply remember the rules for directed numbers: For ddition nd subtrction, signs tht occur together cn be combined. Sme signs positive for exmple, 7 + + = 7 + nd 7 - - = 7 + Different signs negtive for exmple, 7 - + = 7 - nd 7 + - = 7 - For multipliction nd division. Sme signs positive for exmple, +7 ì + = +1 nd -7 ì - = +1 Different signs negtive for exmple, +7 ì - = -1 nd -7 ì + = -1 Number lws nd vribles When deling with numbers nd vribles, prticulr rules must be obeyed. The Commuttive Lw holds true for ddition (nd multipliction) becuse the order in which two numbers or vribles re dded (or multiplied) does not ffect the result. Therefore, in generl, x + y = y + x x - y ò y - x x ì y = y ì x x ó y ò y ó x The Associtive Lw holds true for ddition (nd multipliction) becuse the order in which three numbers or vribles, tking two t time, re dded (or multiplied) does not ffect the result. Therefore, in generl, x + (y + z) = (x + y) + z x - (y - z) ò (x - y) - z x ì (y ì z) = (x ì y) ì z x ó (y ó z) ò (x ó y) ó z Chpter 8 Algebr 11
The Identity Lw sttes tht, in generl, x + 0 = 0 + x = x x ì 1 = 1 ì x = x The Inverse Lw sttes tht, in generl, x + -x = -x + x = 0 1 1 x = x = 1 x x Simplifying expressions When simplifying expressions, we cn collect (dd or subtrct) only like terms. Like terms re terms tht contin the sme vrible prts. Multiplying nd dividing expressions with vribles When multiplying vribles: () the order is not importnt. For exmple, d ì e = e ì d. (b) plce the numbers t the front of the expression nd leve out the ì sign. When dividing vribles, rewrite the expression s frction nd simplify it by cncelling. When the sme vrible ppers on both the top nd bottom lines of the frction, it my be cncelled. Expnding brckets Brckets re grouping symbols. Removing brckets from n expression is clled expnding the expression. When expnding brckets, put the ì sign before the brcket. The rule tht is used to expnd brckets is clled the Distributive Lw. After expnding brckets, collect ny like terms. Fctorising Fctorising is the opposite process to expnding. Fctorising number or expression involves breking it down into smller fctors. To find the highest common fctor (HCF) of lgebric terms, follow these steps. () Find the highest common fctor of the number prts. (b) Find the highest common fctor of the vrible prts. (c) Multiply these together. To fctorise n expression we plce the highest common fctor of the terms outside the brckets nd the remining fctors for ech term inside the brckets. Homework Book MAPPING YOUR UNDERSTANDING Using terms from the summry, nd other terms if you wish, construct concept mp tht illustrtes your understnding of the key concepts covered in this chpter. Compre this concept mp with the one tht you creted in Wht do you know? on pge 181. Hve you completed the two Homework sheets, the Rich tsk nd the two Code puzzles in Mths Quest 8 Homework Book? 1 Mths Quest 8 for the Austrlin Curriculum
Chpter review FLUENCY 1 Using x nd y to represent numbers, write expressions for: the sum of x nd y x + y b the difference between y nd x y - x c five times y subtrcted from three times x d the product of 5 nd x 5x e twice the product of x nd y xy f the sum of 6x nd 7y 6x + 7y g y multiplied by itself y h twice number is decresed by 7 x - 7 i the sum of p nd q is tripled. (p + q) If tickets to the school ply cost $15 for dults nd $9 for children, write n expression for the cost of: x dult tickets 15x b y child tickets 9y c k dult tickets nd m child tickets. 15k + 9m Jke is now m yers old. Write n expression for his ge in 5 yers time. b Write n expression for Jo s ge if she is p yers younger thn Jke. m - p c Jke s mother is 5 times his ge. How old is she? Find the vlue of the following expressions if = nd b = 6. 5m b 6 1 c 5b 0 d 1 e + 8 10 f b - g + b 8 h b - i 5 + b 8 j + 7 1 k + b l 0 10 m b - 1 n b 1 o b p -b -6 q 5 5 b r b 9 5 The formul C =.k + cn be used to clculte the cost in dollrs, C, of trvelling by txi for distnce of k kilometres. Find the cost of trvelling.5 km by txi. $1.90 6 The re (A) of rectngle of length l nd width w cn be found using the formul A = lw. Find the width of rectngle if A = 65 cm nd l = 1 cm. x - 5y 7 Substitute r = nd s = 5 into the following expressions nd evlute. 0 (r + s) 16 b (s - r) c 5(r + s) 0 d 8(s - r) 16 e s(r + ) 5 f s(r - ) 15 m + 5 5 cm g r(r + 1) h rs(7 + s) 180 i r (5 r) 18 j s (s + 15) 500 k r(s + r) 96 l 1r(r s) -7 8 Find the vlue of the following expressions if = nd b = -5. + b - b b + - c b -10 d b 5 - e b -0 f 5 - g 1 - b h - i ( + ) 1 j b( - ) 10 k 1 - (b - ) 8 l 5 + 6b -0 9 Indicte whether ech of the following is true or flse for ll vlues of the vribles. + 5b = 5b + b 7x - 10y = 10y - 7x Flse Flse c 8c + d = d + 8c True d 16 ì x ì x = x e 9x ì -y = -y ì 9x True True f - ì x ì x = 1x ì x Flse g 11p 5r = 5r 11p Flse h 7i + j = j + 7i True i -y ó 7x = 7x ó -y j -8c + 5d = 5d - 8c k 0 5k = 5k 0 7x 7x l 1 = 1 10 Simplify the following by collecting like terms. d + d 7d b c - 5c -c c d + 5 - d + d 6g - g g e x + 11 - x x + 11 f g + 5 - g - 6 g - 1 g xy + 7xy 9xy h 1t + t + t - t 15t + t 11 Simplify the following. ì 7g 1g b 6 ì y 18y c 7d ì 6 d d -z ì 8 -z 1 Simplify the following. 8 b 11b b c 6rt ó -t -r d -gh ó -6g e g t 0stv 1b 1b 5sv Flse - 6 7 Flse True True f -6xy ó -1y h 5egh 0ghj 1 e 6 j h x Chpter 8 Algebr 1
Check with your techer. b -10 ì 8 + -10 ì -6 = -10(8 + -6) or -10 ì 8 - -10 ì -6 = -10(8 - -6) d - ì 7 + 1 ì 7 = -(1 + 7) or - ì 7-1 ì 7 = -(1-7) NUMBER AND ALGEBRA 1 Use the Distributive Lw to expnd the following expressions. (x + ) x + 6 b 5(x - 1) 10x - 5 c -( f + 7) -f - 1 d m(b - m) bm - m e -y(7 - y) y - 1y f 9b(c - ) 9bc - 18b 1 Expnd the following nd then simplify by collecting like terms. (v + 5) - 15 b 6t + 5(t - 7) 1v 16t - 5 c + 5(p - ) + p d (x + 5) + 5(x + 1) 17p + 7x + 15 e g(g - 6) + g(g - 7) 5g - g f (t - ) - 6(t - 9) - t P (points), G (gols) nd B (behinds) A builder hs 10-cm section of wood nd wnts to cut it into two unequl pieces. If the length of one of the sections is cm, wht is the length of the other piece? He hs second section of wood tht is b cm long, nd he cuts it into four unequl pieces: one is 6 cm, nother 60 cm nd nother is 1 cm. (Hint: Drw digrm.) 10 cm b Wht is the length of the fourth prt of this second section of wood? b 1 96 cm c If the totl length of four of the six sections of wood is (96 + 1) cm, wht is the totl length of the other two pieces in terms of nd b? The bse of box hs length of (x + ) cm nd width of x cm. Drw digrm, lbelling the length nd the width. b Write n expression for the re of the bse of the box. x(x + ) cm c Expnd prt b. x + x d If x =, wht is the re of the bse of the box? 1 cm e If the height of the box is x cm, find n expression for the volume of the box. f Find the volume of the box if x is cm. Aussie Rules Footbll is plyed in mny Austrlin sttes. The scoring for the gme is in Gols (G) nd Behinds (B). Ech Gol (G) scores six points nd ech behind (B) scores one point. To clculte the totl number of points (P) scored by tem use the following rule: P = 6G + B Nme the vribles in the rule. b Stte the expression in the rule. 6G + B c A tem scored 11 gols nd 10 behinds. How mny points is this? 76 points d A second tem scored nine gols nd 18 behinds. How mny points is this? 15 Fctorise the following expressions. 7 points g + 1 (g + ) b xy + 5y y(x + 5) e How mny gols nd behinds might tem hve c 5n - 0 5(n - ) d 1mn + pn n(m + p) e 1g - 6gh f 1xy - 6yz scored if its totl points score ws 87 points nd 6g( - h) 1y(x - z) the tem scored more thn six gols? 16 Show tht the Distributive Lw holds for the following. 5 Stephnie bought some clothes from Trget during 10(16 6) = 10 ì 16 10 ì 6 their nnul sle. She spent $79.00. She bought PROBLEM SOLVING skirt, T-shirt nd pir of shorts. She pid $9 more for the T-shirt thn for the shorts, nd 1 Using only +, -, ì, ( ), complete the following $7 more for the skirt thn for the T-shirt. How equtions to demonstrte the Distributive Lw. much did the skirt cost her? $ 1 = 1 ( + 1) = ì + ì 1 or ( - 1) = ì - ì 1 6 If the pttern below continues, how mny brrels b -10 8-6 -10 = -10 8-6 re needed to mke 8 lyers? Note: The top brrel is c 8 6 5 = 8 5 6 8 8(6 + 5 ) = 8 ì 5 + 8 ì 5 or 8(6-5 ) = 8 ì 5-8 ì 5 sitting on the next two nd the lyers go stright up. d - 7 1 7 = - 1 7 e x y = x y (x + y) = ì x + ì y or (x - y) = ì x - ì y Lyer 1 6 brrels f b = b ì ( + b) = ì ì + ì ì b Brrels 1 6 b 1 + cm b If the pttern below continues, how mny brrels re needed to mke 8 lyers? Note: Ech lyer forms tringle. 10 brrels (x + ) cm Lyer 1 Brrels 1 10 x cm Volume = x + x cm 6 cm 1 Mths Quest 8 for the Austrlin Curriculum
7 Bobby the pinter hs two prtilly used 10-L tins of pint, A nd B. There is more pint in Tin A thn in Tin B. He mixes the pint in the following fshion. He pours pint from Tin A into Tin B until the volume of pint in Tin B is doubled. He pours pint from Tin B into Tin A until the volume of pint in Tin A is doubled. He pours pint from Tin A into Tin B until the volume of pint in Tin B is doubled. When he s finished, the tins both contin L of pint. How much pint ws in the tins before Bobby begn mixing?.5 L in Tin B nd 5.5 L in Tin A 8 Two numbers hve sum of 7 nd product of 1. Find the sum of their reciprocls.. 9 Given + b = ( b), find the vlue of b 10 If you dd the first nd lst of ny three consecutive integers together, cn you find reltionship to the middle number? The sum is twice the middle number. 11 The Flesch-Kincid Grde Level formul is used to determine the redbility of piece of text. It produces 0 to 100 score tht cn be used to determine the number of yers of eduction generlly required to understnd prticulr piece of text. The formul is s follows: Totl words 09. Totl sentences Totl syllbles + 11. 8 Totl words 15. 59 7 1 15 or 17 7 7 Text suitble for Yer 8 student should hve vlue of roughly 8. A pssge of text contins 0 sentences, with 500 words nd 70 syllbles. Would it be suitble for Yer 8 student? Explin your nswer. 1 Consider the expression x x x ( ) ( + + 1 ) Yes. This is clled power tower. Wht is the vlue of the lst digit of this expression when x =? Note: You will hve to look t ptterns to determine the nswer, s clcultor will not give you n exct nswer to the power. The finl digit of 81 is. Interctivities Test yourself Chpter 8 int-71 Word serch Chpter 8 int-69 Crossword Chpter 8 int-60 Chpter 8 Algebr 15