algebra - Maths Study Sheet 11 Introduction Part 1 algebra...

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Unformatted text preview: Maths Study Sheet 11 Introduction Part 1 algebra and the use of symbols — This UNl'l” censists at 2 parts. Part 1; Yeu are intrecluced te the basic ideas ef algebra. When ynu have werked Lhreugh Part 1, yeu will II have met algebraic espressiens censisting ef numerals and letters " lmew how be interpret an algebraic expressicm * be able te distinguish “like” terms in an espressien and simplify by adding er subtracting these terms . be able te simplify espressiens invnlving multiplicatiens and divisien. Prerequisites: An understanding ef the use at the rules at basic aritlunetic. Algebra is the branch at mathematics in which espressiuns invulving letters and numbers are cembitied with the arithmetic eperatiens +, —, :s, + and brackets etc. 2 J' t For example, tt2h+ed , ‘l q , Algebra is frequently referred in as generalised arithmetic arithmetic because the rules at arithmetic are ebeyed; generalised because a variety ef numbers may be substituted fer the letters in the espressien, and then the espressien can be evaluated. lts usefulness lies in the [act that, if we can give an espressitm as the answer te a general prehlem, then this can be used to find answers te several specific preblems {if the same type. Example: The Beat Prublem A beat travels H miles upstream at 3: mph and then returns dewnstream at y mph. What is the fetal time taken? It may help te substitute snme numbers [er the letters in erder te decide hew te tackle the prebiem- Cheese very simple enes at this stage. Fer instanee, suppese e = 10, a: = 2,, y = 5 then te gs 10 miles at 2 mph takes 10 If] - - — 5 hears, and ten 3e 10 miles at 5 mph takes 2 5 fielding, the tntal time is ,7 he-urs. —= E henrs- Ask yourself ”what tiperatiens did I use?”. New gt] ten the general prublern. at mph means I miles in '| hnnr | i er 1 mile in — hetlrs er H miles in H K — hunts I I r: This ean he 1written as — hears. .r n . Similarly; it ta kes — herurs te travel H. miles at tf mph. .1: Fin espressienn fer the tetal time. is thereiere H. H — + — henrs. .‘r f; It is new pessible te answer this same prehlem fur a variety til dillerent “mines fur the speeds .‘t’ and l!” and a variety eat different tlistanees trrwelleel. if H. = 8, .‘t‘ = 3, y : 5 the tetal time taken is _,, ’j. , —-£+fi—E=4ihenrs 3 5 15 15 15 15 ie apprise-timateljir 4.3 hnurs IUI preeiselj; 4 hr 16 mins}- Interpreting 'i'he algebraie expressien expressuens q r. rrb + 4r.“ —- t" +j If.“ eensists ell-4t terms eenneeteei h}: ”+” and ”—" ste'mhels”. An impertanl differenee trem arithmetie is that the mnltiplieatinn sembnl "e” is rarely written explleitly'. It is just implied. The first term at: meters a at l1 and the seeeuti term 4r! meet-rs 4 s: d. a (7 e means e ”s: r: and — means e + d. n’ am.r er. ressitii s vein - i: F e ' ' " . fit 1 ‘ " as t s; '* M E p 1 E meet [111} [3 51m lined ‘1 time exam lL Ji thi ne given in the nest twu seetiens. Adding] When terms are ef the same type they may be added er subtraeteel. Subtracting - m “le3 Terms” E“ P'Es (i) :r + :1“ = 2.17 {ii} m + m + m = 3m {iii} es +- 5m — fies {iv} 5d -l 3e! — 2d 2 [id {y} fill — tile‘ 1 2d 2 5d + 2d wflri = —d This last example demenstrates twe things: {I} Heme answers may be negative, net pesitiye. {ll} sea, the erder ef the terms may he ehanged. Remember [hat the sign in lrent ef a term belengs te it, and the first term is assumed te be “plus” it it is net stated eaplieitly. {yi} 3e l t1-—e +4tr: 3e —e I. {1+ ills -.: 2.51 + 5b {yiijl 4.1:}; + Er y + Fairy — 4y 2 —3.ry + Exy + 2.1? — y — sly = 21y + 2:: - 5y Netiee 1lyl'ren te step! Terms in J; are net lLke terms in .ry. {yiill 3r2 + .r ".- fir— 'x‘ri = "4!: 4-0: 7: [Remember 1” means t H t] MUltipliCfltiOH Seme espressiens myrtle-i115 multiplieatien and diyisien may be simplified as and division these examples illustrate. til As meted abeye :1 is l? 2 El: t'he erder ef terms in a prednet may be reyersed henre l1 :s: 5 = 5 is: E? = 5b The numerals in a multiplieatien term are usually written first, and ether letters are written in alphabetical errl er. .. '} {n} rrxrrze“ rm} exhshth3 {iv} r: K d >< e’ = rrri2 Exercise Solutions T'we important examples te remember are (e) and {vi} {V} {vi} {vii} {viii} lUnHEHHb21Ux2HeHe§b= 20:12}: UK) (31} {3'11} {xii} 13e~e HEU=U .3, . I‘lvixj =I><x X IEXXX= .1," 12pq+3pqz 2' 1.? H12: 'H 0 [Bab e be“ +15eed = Simplify the. fullnwing: 1. Lu} 11+ 10. 51': + 4x2 :13: + be -— 33):: I 4:1?) 3:35} — 4M) - Ee + 3b :13 7x: {:5 4172}: x 3x2}: 3.1}‘K xzje2 + gxyz 9.1:2 2 lfinwy 5 as e. U 11. i T D but 4 calmet be evaluated- 6 1 l MMKIJXEJKXXL‘ _f';-EJEL' MMx/xd 5d 5 l l 10. 12. 1, 3? + H 512+?! 9mm; + way + 4:111:11: .1 1 7y“ — 519*)?“ + 4_1-‘— 2 ...
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