1. Solving simple equations 2. Evaluation and transposition of formulae

Transcription

1 Algebra Matheatics Worksheet This is oe of a series of worksheets desiged to hel you icrease your cofidece i hadlig Matheatics. This worksheet cotais both theory ad eercises which cover:-. Solvig sile equatios. Evaluatio ad trasositio of forulae There are ofte differet ways of doig thigs i Matheatics ad the ethods suggested i the worksheets ay ot be the oes you were taught. If you are successful ad hay with the ethods you use it ay ot be ecessary for you to chage the. If you have robles or eed hel i ay art of the work the there are a uber of ways you ca get hel. For studets at the Uiversity of Hull Ask your lecturers. You ca cotact a Matheatics Tutor fro the Skills Tea o the eail show below. Access ore Maths Skills Guides ad resources at the website below. Look at oe of the ay tetbooks i the library. Web: htt://libguides.hull.ac.uk/skills Eail: skills@hull.ac.uk

2 . Solvig sile equatios A equatio is a atheatical way of writig a stateet about the equality of two quatities. It always uses the = sig (read is equal to or equals ) to searate the two equal quatities. For eale, the stateet i four years tie, I will be thirtytwo years old ca be writte as A, where A is y curret age i years. To solve a equatio eas to fid the value (or the set of all of the values, if there is ore tha oe) of the ukow quatity, which fits (or satisfies) the equatio, or fid the uber (or set of ubers) with ake the stateet true. I this eale there is oly oe solutio, aely A = 8 (i.e. y reset age is 8 years). A basic ricile i solvig all equatios is to start with the give equatio which is true ad to kee it true by doig the sae thig to both sides. I this resect you ca thik of a equatio as beig a balace where you have to do the sae thig i each balace a to kee it level. You reeat this actio as ofte as ecessary util the ukow quatity stads o its ow. I our eale, startig fro A A + subtract fro (or add to) both sides [iverse of +] A Hece A = 8 A A 8 Differet equatios use differet letters for the ukow (occasioally associated with the eaig of the ukow - like A for age i years ) but doig the sae thig to both sides is essetial for correct solutios. May eole fid the idea of the balace helful ad use it util their cofidece grows. I solvig ay equatio, o atter how difficult, you ca always check that you have a solutio by substitutig back ito the origial equatio. That is, take the solutio ad substitute it for the ukow i each side of the origial equatio, do the arithetic, ad check that both sides rereset the sae uber. I our eale, we ca easily check that 8 is the solutio. The left-had side of the origial equatio is A +, ad whe we substitute A = 8, this gives 8 +, which is ad this equals the value o the right had side of the origial equatio. age

3 Eales (a) Solve the equatio to ove the fro the left add to both sides [iverse of -] Four ties what we wat is divide both sides by [iverse of ], 0 Check by substitutio Left had side (LHS) 0 = Right had side (RHS) (b) Solve the equatio. (This could be writte as.) Multily both sides by, the iverse of : Check LHS = RHS ote i ultilicatio the iverse of, say, is as. This also shows that the iverse of is as (c) Solve the equatio 9 Ukows aear o both sides of the equatio so first we eed to tidy u the equatio by ultilyig the brackets out ad collectig ters. Multily out brackets Subtract fro both sides [iverse of +] Subtract fro both sides [iverse of ] Divide both sides by 0 [iverse of 0 ] Solutio Check i the origial equatio: age

4 LHS RHS LHS = RHS Note we could have added 6 to both sides at lie ad the subtracted fro both sides to give 0 0 c (d) Solve the equatio c 6 Multily both sides by [iverse of ] (reeber to ultily everythig by ) c c 6 6c c Add c to both sides [iverse of c ] 8c Subtract fro both sides [iverse of +] Divide both sides by 8 [iverse of 8] 8c 8c 8 8 c It ay ot always be ecessary but it is iortat to be able to check your aswer: Checkig (a calculator with fractios could be useful for this) c LHS c RHS= LHS = RHS so our solutio is correct. 0 (e) Solve the equatio 0 Multily both sides by 0 Subtract fro both sides 6 6 Divide both sides by This silifies to 8 8 How you write your fial aswer deeds o the accuracy you require. The eact aswer is but, deedig o the cotet, a aswer of 0. 8 (to decial laces) or 09. (to decial laces) ay be aroriate. If you wat to check your aswer fully you ust use the eact value. For eale usig as 0.9, ad givig the aswer to d.. gives LHS 0. 8, RHS age

5 To check roerly LHS RHS LHS = RHS (eactly). 0 (f) Solve the equatio This looks colicated but if you follow the rules you ca do it, though you ll robably ever eet aythig as bad! With ractice soe stes ay be cobied. Multily through by [iverse of ] Multily through by [iverse of ] subtract 6 fro both sides [iverse of + 6 ] divide both sides by [iverse of ] or. (to sig figs) Checkig is ot easy!! Usig the decial value we have LHS =.. (to sig. figs.) RHS (to sig. figs.) Hece LHS RHS, the solutio is correct well to sigificat figures! With ractice you could ultily through by ad i oe go but you do eed to be very careful. Eercise Solve the followig equatios: age

6 c 9 8 f d d 8 b b f d d d ( y ) (y ) b b s s 8 ( A ) 9 ( ) ( ). s s 8. c c.. 9. d d ( A ) 8d A Evaluatio ad Trasositio of Forulae A forula is a equatio that describes a relatioshi betwee a uber of differet quatities - for eale, the volue (V ) i c (cubic cetietres) of a rectagular bo, with legth l c, width w c ad height h c is give by the forula V lwh. As show i the leaflet Algebra, you ca work out the value of V, give values for l, w ad h, by substitutig those ubers ito the forula ad calculatig: e.g. if l, w 8, h 0, the V lwh i.e. the volue of a bo c by 8c by 0c is 00c. I the forula above, V is the subject - that is V (o its ow) is eressed i ters of the other quatities. This akes fidig the value of V, give the other values, fairly easy. If you eed to fid the value of, say, the height, the you ay wat to fid the forula that gives the height i ters of the volue, width ad legth - that is you eed to ake h the subject of the forula. The rules for rearragig (or trasosig) a forula are the sae as those for solvig a equatio - do the sae thigs to both sides. For eale, usig the forula V lwh l is ultilyig wh so divide both sides by l V lwh wh [iverse of l l is l ] givig l l V wh dividig both sides by w [iverse of w ] givig h lw w V ad this is the sae as h lw This fial lie has h as the subject of the forula. Eales (a) The surface area of a cylider of height h ad base radius r is give by the forula A rr ( h). If a cylider has surface area 8.6 c ad the radius of the base is c fid the height of the cylider (take as.). age

7 Substitute the values i silify the right had side Divide by 8.8 [iverse of 8.8] Height of cylider c ( h) 8.8 ( h) 8.8 h 8.8 h h R (b) If V fid the value of R give V, r. R r Substitutig the values i R R Multily both sides by ( R ) [iverse of ( R )] R R Multily out bracket R R Add to both sides [iverse of ] Subtract R fro both sides [iverse of R ] Dividig by gives R R R R R R This could have bee doe by chagig the subject of the forula ad the substitutig values i, though the stes are alost idetical. (see et eale). (c) Make R the subject of the forula V R R r Multily both sides by ( R r) [iverse of ( R r) ] Ead the bracket o the LHS Add Vr to both sides [iverse of Vr ] Subtract R fro both sides [iverse of R] Factorise the LHS Divide by ( V ) [iverse of ( V )] Note substitutig the values fro eale (b) gives R V R r V R r R VR Vr R VR R Vr VR R Vr R V Vr Vr R V R Vr as above ( V ) (d) Make h the subject of the forula d hr It is first ecessary to deal with the ad as the iverse of is square square both sides d hr Divide both sides by r age 6

8 This is the sae as d h r d h r (e) Make t the subject of the forula s ut ultily both sides by s ut divide both sides by u s s t or t u u take the square root [iverse of square ] s t u Notice the sig, it is iortat! but The root sig eas take the ositive value of the square root. Eercise. Make the subject i each of the followig a) d) g) j) 0 b) e) h) k) r c) f ) s r t i) l). Give the forula b a T (a) fid the value of T whe a, b. (b) ake T the subject of the forula.. Give the forula a b B (a) fid the value of B whe a, b (b) ake B the subject of the forula. d. Give the forula D Q (a) fid the value of Q whe d,. D 6. (b) ake Q the subject of the forula.. The volue V (i c ) of a coe with height h c ad base radius r c is give by the forula V r h. (a) Calculate the radius of the base of coe with height c ad volue c, takig as.. Give your aswer correct to decial laces. age

9 (b) ake r the subject of the forula a 6. Make the subject of the forula C =. Make t the subject i the forula v u at. 8. Make h the subject of the forula S ar( r h). 9. Make b the subject of the forula b a. b c 0. The forula v at 6t is used i echaics. Rewrite the forula to ake a the subject ad fid its value whe v 00, t.. Give the forula y (a) fid the value of whe forula. y,. (b) ake the subject of the age 8

10 ANSWERS Eercise Eercise. ( a) ( e) () i ( b) ( f ) rs ( t) t ( j) ( c) ( g) ( ) ( k) ( d) ( ) ( h) r r () l. T ; T ab d D. Q ; Q D 6. C a S ar S 8. h r ar ar v 6t v 0. a 6 t; t t. B 9; B a b. r.; r V h. v u t a 9. a c a c b a a. ( ) 0.8 to decial l ; ( b) y y a We would areciate your coets o this worksheet, esecially if you ve foud ay errors, so that we ca irove it for future use. Please cotact the Maths tutor by eail at skills@hull.ac.uk Udated d Jue 0 The iforatio i this leaflet ca be ade available i a alterative forat o request. Telehoe age 9