SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra


 Abigayle Lamb
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1 SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting Formule nd Solving Equtions 7. Qudrtic Equtions A Reminder. Simultneous Equtions A Reminder Tutoril Eercises Dr Derek Hodson
2 Bsic Alger Opertions nd Epressions You will lred e fmilir with vrious forms of mthemticl epressions. It is importnt tht ou re confident in hndling nd mnipulting such epressions, so this section should refresh our memor regrding some sic lgeric concepts nd techniques. The Bsic Arithmetic Opertions In mthemticl epressions, numers nd / or vriles (i.e. unknowns re comined using the rithmetic opertions of ddition, sutrction, multipliction, division nd eponentition (i.e. squring, cuing, etc., long with rckets to group or seprte terms. If n epression contins onl sums nd differences, or contins onl multiplictions nd divisions, then the opertions cn e delt with from left to right. For emple, 0 ;. 0 For more involved epressions, we use the BODMAS principle to determine the order of evlution: Brckets ] first priorit Order (i.e. powers ] second priorit Division Multipliction Addition Sutrction third priorit fourth priorit. For emple: (B (O (M (A (
3 Another emple: (B (O (D (S ( Indices (Powers The simple use of powers or indices to represent repeted multipliction should e fmilir, for emple n... [ n terms], ut ou must lso e cler on the wider interprettion of indices. The lines elow collte the importnt spects of indices: 0 n n nd n n nd m m n m n m n m n m n m ( m ( m n m m m m m These results will e needed in oth the epnsion nd simplifiction of lgeric epressions.
4 Emple ( Epnd ( 7 (. 7 ( 7 ( ( 7 ( ( Epnd (. ( ( ( ( ( ( ( (c Simplif ( z z. ( z z z z z z See the Tutoril Eercises for prctice.
5 Common Mistkes Here re just few common mistkes to void when mnipulting lgeric epressions: ( : ( ( : ( d c d c : d c d c d c c c ( : c c ( n m n m : n m n m : Division of Algeric Epressions Providing the denomintor (i.e. the ottom it of division contins no s or s, the division itself is usull strightforwrd to crr out. Emple ( ( When the denomintor does contin s nd / or s, we m require polnomil division. Note:,, re emples of polnomils. 0 0 (
6 Eponentil Functions nd Logrithms An eponent is nother nme for power or inde. Eponents cn e used to crete eponentil functions:, ( > 0 : 0,. In this contet, is clled the se of the eponentil function. Commonl used ses re 0 nd the eponentil constnt e. 7 : 0 ; e. The function with the eponentil constnt e s its se is so importnt in mthemtics, science nd engineering tht it is referred to s the eponentil function, ll others eing suordinte. Relted to eponentil functions re logrithms. If we epress numer N s power of, i.e. N, then the power is defined to e the (se logrithm of N : log N ( log of N to the se Note tht ecuse of the importnce of the eponentil constnt, logrithms to se e re given the specil nme of nturl logrithms nd denoted ln (. Emple ( ( (c 00 0 log 0 ( log 0 (000 log ( The definition cn e etended to frctionl powers with logrithms to se 0 nd se e otinle from clcultors: Emple ( log 0 ( (to deciml plces ( ln (. 77 (to 7 deciml plces
7 Becuse logs re definition indices, we cn use the rules for comining indices to determine the soclled lws of logrithms: log ( R S log R log S [L] R log log R log S S [L] n log ( R n log R. [L] We cn use these to epnd or contrct epressions involving logrithms: Emple ( Epnd log 0 z. log 0 z log 0 ( log 0 ( z [L] log 0 ( log 0 ( log 0 ( z [L] log 0 ( log 0 ( log 0 ( z [L] ( Write ln ( ln ( z ln ( s single logrithm. ln ( ln ( z ln ( ln ( ln[( z ] ln ( [L] ln[ ( z ] ln ( [L] ( ln z [L]
8 Opertions nd their Inverses Ech sic rithmetic opertion tht we m encounter (with few eceptions hs ssocited with it corresponding inverse opertion. An opertion nd its inverse, when pplied in sequence, effectivel cncel out one nd other. For emple, suppose we strt off with. If we now dd nd then sutrct, we re ck to gin. Tht is,. We cn s tht the inverse of dding is sutrcting (nd lso vice vers!. Similrl, if we multipl, then divide, we re gin ck to :. So the inverse of multipling is dividing (nd vice vers. The tle elow shows opertions nd their inverses, including the eponentils nd logs from the previous section, nd some others ou m lred e fmilir with: This is specil cse of the entr ove. Multipling the sme s multipling nd dividing. Hence the inverse opertion is multipling. is or ( ( or ( or ( ( or ( n n n n or ( ( n n or n n ( log 0 0 log 0 (0 ln e ln ( e sin (sin sin (sin cos (cos cos(cos tn ( tn tn ( tn 7
9 Mnipulting Formule nd Solving Equtions The opertion / inverse opertion effect descried in the previous section provides the ke to mnipulting formule nd solving equtions. A formul is n eqution tht epresses reltionship etween quntities. In prticulr, it epresses how to determine the vlue of one quntit from the vlues of one or more other quntities. Below re some emples of formule tht ou m hve seen efore: C ( F V r π d u t t v u t v i R In ech of these formule, the quntit on the left is clled the suject of the formul. Often, when working with formul, we wnt to chnge the suject to one of the other quntities. For emple: v i R v i  i is now the suject. R This is ver simple emple. However, the mnipultion of formule is n spect of lger tht cn pose difficulties for students. Consider the formul in the ove list tht reltes temperture in degrees Celsius to temperture in degrees Fhrenheit: C ( F. It is quite es to mke F the suject of this formul. How ou would tckle this would prol depend on methods rought from school. If ou re thinking something like, move terms from one side of the eqution to the other to get F on its own, then plese think gin. Although the method of moving terms will prol get ou the correct nswer in this cse, it is not mthemticll correct w of thinking nd cn cuse prolems in more complicted formule. Let us now look t the correct w to mnipulte formul or eqution. This will let ou see wht is rell hppening when terms pper to move round n eqution. Wrning: Ignore the following t our peril!
10 First, let us look t the formul s given nd the mthemticl opertions within it: C ( F. Given vlue of F, (following BODMAS we would determine the corresponding vlue of C : sutrcting multipling. A formul (or, in fct, n eqution is like lnced set of scles: C ( F When working with n eqution, we must not upset the lnce. This mens tht if we do something to one side of the eqution, then we must do the ect sme thing to the other side. This is the one nd onl rule of lgeric mnipultion. [Note: We cn, of course, swp the sides round, ut tht is just like turning the scles round; it does not ffect the lnce.] To chnge the suject of formul, we ppl crefull chosen opertions to oth sides of the eqution whose net effect is to isolte the new suject. These opertions re the inverse opertions of the those within the originl formul. Tht is: sutrct multipl re reversed nd inverted to give multipl dd. These opertions re now pplied, in turn, to oth sides of the eqution.
11 The process in full is s follows: C ( F Multipl oth sides : C ( F C ( F C F Add to oth sides: C F C F Swp sides: F C... nd with pictures: Formul: C ( F Multipl oth sides : C ( F C F Add to oth sides: C F C F Swp sides: F C 0
12 In prctice, we needn t put in s much detil. For such simple formul we might simpl set down the following lines: C ( F Multipl oth sides : C F Add to oth sides: C F Swp sides: F C. This revited version m give the impression tht terms re moving round the eqution, ut the re not. You must lws keep in mind wht is trul hppening in the ckground. Alws think BALANCE. Further Emples Note: In the following emples, full detils re shown. In prctice, the level of detil ou show in ou own working will depend on our own confidence nd ilit; s these grow, ou will nturll displ less. Emple For the formul mke the suject. [Note: This is the sme s sing, solve for.] Anlse opertions. Given, how is clculted? squre multipl dd Reverse nd invert opertions: sutrct divide squre root
13 Appl inverse opertions to formul: Formul: Sutrct : Divide : At this point it is useful to swp sides Squre root: Shortened version (for the more confident: Formul: Sutrct : Swp sides nd divide : Squre root:
14 Emple 7 For the formul V, mke r the suject. r π Anlse opertions. Given r, how is V clculted? cue (or rise to the power π multipl Reverse nd invert opertions: multipl π cue root (or rise to the power Appl inverse opertions to formul: Formul: V r π V π r Multipl π : π π V π r V π r Swp sides: r V π Rise to the power r V π : ( ( r V ( π Shortened version: Formul: V π r Swp sides nd multipl π r V : π Rise to the power r V π : (
15 Emple For the eqution e, solve for. Note: For BODMAS purposes, ou hve to imgine rckets grouping the power ( terms together, i.e. e Anlse opertions. Given, how is clculted? multipl e to the power ( i.e. multipl ( e Reverse nd invert opertions: divide tke nturl log ( i.e. ln ( divide Appl inverse opertions to formul: Eqution: e Divide : e e Nturl log: ln ln ( e ln Swp sides: ln Divide : ln ln
16 Shortened version: Eqution: e Divide : e Swp sides nd tke nturl log: ln Divide : ln Emples (with less detil: Emple Determine t when 0. t. 0 This is not formul with suject, so we cnnot write down sequence of opertions. It is, however, n eqution (with n unknown nd the concept of lnce still pplies. In mnipulting this eqution, we wnt to end up with the form t (. Clerl we must mnipulte t down to the level of the equls sign. One of the lws of logrithms will help here: Eqution: 0. t 0 0.t Tke log (se 0 of oth sides: ( 0 log ( log t log0 ( Multipl oth sides : 0.t log ( 0 t t log0.0 (
17 Emple 0 t Determine t when e 0.. t Eqution: e 0. t Tke nturl log of oth sides: ln ( ln (0. e t ln (0. Divide oth sides : t ln (0. (sme s multipling t 0.7 Emple Determine when. Eqution: Tke log (se 0 of oth sides: ( log ( log 0 0 log0 ( log0 ( Divide oth sides log0 ( : log log 0 0 ( ( 0. Note: You could use nturl logs nd get the sme nswer.
18 7 Qudrtic Equtions A Reminder One tpe of eqution tht crops up quite frequentl is the qudrtic eqution: c 0. This tpe of eqution is solved either fctoristion (which isn t lws possile or use of the qudrtic formul ± c. Emple ( Solve 0 fctoristion. 0 ( ( 0 ( ( or 0 0 or ( Solve 0 the qudrtic formul. 0,, c ± c ± ( ± ± or or Note: Qudrtic equtions m hve two rel solutions, one rel solution or no rel solutions, depending on the vlue of the discriminnt c. 7
19 Simultneous Equtions A Reminder Equtions cn contin more thn one unknown. For emple, the eqution hs two unknowns. A solution of this eqution is mde up of n vlue nd vlue tht together stisf the eqution. We could hve ( 0, or (, or n one of n infinite numer of solutions. When plotted on Crtesin es sstem, ll possile solutions of this eqution lie on stright line (see the hndout Coordinte Geometr The Bsics. If we hve second eqution, s, this lso hs n infinite numer of solutions, which lso lie on stright line. However, there is one solution tht is common to oth equtions. Grphicll, this common solution is given the coordintes of the point of intersection of the two strightline grphs. To find this solution, we could drw the grphs nd red off the vlues of nd. This would e fine for the ove equtions which, s it turns out, hve integer vlues in the solution. For more generl method of solving equtions simultneousl, we require n lgeric pproch. There re vrious ws of setting this out. Method Elimintion Sustitution Tke either eqution nd epress one unknown in terms of the other:. Sustitute this into the other eqution, there eliminting one of the unknowns nd leving n eqution with single unknown tht is esil solved: (. This vlue is then sustituted into to give. Solution:,
20 Method Elimintion the Addition or Sutrction of Equtions Line up the equtions one ove the other:. If necessr, multipl up one or other or oth equtions to otin common coefficients on one of the unknowns: or. Net, eliminte the unknown with the common coefficient dding or sutrcting corresponding sides of the equtions: Add : or Sutrct : Solve : Solve :. Solution:, Whichever method ou use, lws check our nswers sustituting the vlues into the originl equtions: st eqution : LHS RHS ( nd eqution : LHS ( RHS.
21 Tutoril Eercises ( Epnsion of Algeric Epressions Contining Brckets (Revision (. Epnd the following epressions involving products nd powers: (i ( ( (ii ( ( (iii ( (7 (iv ( ( 7 (v ( ( 7 (vi ( ( (vii ( ( (viii ( ( (i ( ( ( ( ( ( (. (. Open out the following rcketed epressions: (i ( ( (ii ( ( 7 (iii ( ( (iv ( ( (v ( (vi ( (vii ( (viii ( ( (i ( ( 7 ( ( ( (. (. Fctorise the following qudrtic epressions: (i (ii (iii (iv (v (vi (vii (viii (i (. (. Show tht ( ( nd use the result to (i epnd: ( ( ; ( ( (ii fctorise: ;. 0
22 ( Division of Algeric Epressions (. Perform the following divisions: (i (ii (iii 7c d (iv c d (v (vi π r h π r π r h h. ( Eponentils nd Logrithms The Bsics (. Use our clcultor functions, nmel log (which is ctull log 0 nd 0, to evlute the following epressions. Comment on the results for prts (v (viii. (i log 0 (000 (ii log 0 (. (iii. 0 (iv 0... (v log0 (0 (vi log0 (0 (vii log 0 (. 0 (viii log 0 ( (. Use our clcultor functions, nmel ln (which is ctull log e nd e, to evlute the following epressions. Comment on the results for prts (v (viii. (i ln (. (ii ln (.7 (iii 0. e (iv. e 0.. (v ln ( e (vi ln ( e (vii ln (. e (viii ln ( 0.7 e.
23 (. Write ech of the following epressions s sums nd differences of logrithms (where possile, without using powers: (i log 0 (ii ln (iii log 0 00 (iv ln e (v log (vi 0 ln ( ( (. (. Write ech of the following s single logrithm: (i (ii (iii (iv log0 log0 log0 z log0 ( log0 z ln ln ln ( ln z.
24 ( Chnging the Suject of Formul (. For ech of the following formule, chnge the suject to the quntit indicted in rckets: (i (ii (iii (iv (v v u t ( t s ( t t ( u s u t s u t t ( t v u s ( s (vi (vii v u s ( u ( (viii t i e ( t (i i t e ( t ( (i ( 0 ( ( 0 ( (ii ( t. k t c0 0 e (. For ech of the following formule, chnge the suject to the quntit indicted in rckets: (i ( (ii ( (iii z ( (iv C C C C C ( C.
25 ( Solution of Equtions (. Solve the following equtions: (i (ii 0 (iii (iv (v (vi 0 0 (vii e 0. 7 (viii e 0. (i e 7 ( e (i ln ( (ii ln ( 0 (iii ln ( 0 (iv ln ( 0. (v 0. (vi 0. 7 (vii log 0 ( 0. (viii log 0 ( 0. (i ( (. Solve the following qudrtic equtions fctoristion, then repet using the qudrtic formul: (i 0 (ii 0 0 (iii 0 0 (iv 7 0.
26 (. The following qudrtic equtions do not hve n rel solutions. Tr solving them the qudrtic formul nd see wht hppens: (i 0 (ii 0. (. Solve the following sets of simultneous equtions: (i (ii (iii 7 0 (iv 0...
27 Answers (. (i (iii (v (vii (i (ii (iv (vi 0 (viii ( 7 (. (i (ii (iii (iv 00 0 (v (vi (vii (viii 0 (i ( 7 0 (. (i ( ( (ii ( ( (iii (iv ( ( ( (v ( ( (vi ( ( (vii ( ( (viii ( ( (i ( ( ( ( ( (. (i ( ( ; ( ( (ii ( ( ; ( ( (. (i (ii (iii (v 7 (iv c d (vi h r
28 (. (i (ii (iii.77 (iv (v. (vi. (vii. (viii 0. (. (i (ii.000 (iii.00 (iv 0.00 (v 0. (vi. (vii. (viii 0.7 (. (i log0 log0 log0 (ii ln ln ln (iii log0 ( (iv ln ( (v log ( 0 (vi [ ln ( ln ( ln ( ] (. (i log 0 z (ii log 0 ( z (iii ( ln (iv ln z 7
29 (. (i t v u (ii t s (iii u s t ( s (iv t t u t (v s v u (vi u v s (vii (i (viii t ln i t ln ( [ log0 ( ] i (i log ( ] (ii [ 0 t k ln 0 c 0 (. (i (ii (iii z (iv z C C C C C (. (i (ii (iii (iv (v, (vi (vii ln ( (viii ln ( (i ln (. 0 ( ln ( (i e 0. 0 (ii e (iii e 7. 0 (iv ( e 0. (v log 0 ( (vi log (.7 ] [ (vii (viii [0 ] (i / Over the pge...
30 (i log0 log [could lso use nturl logs] ( log0 log0. [could lso use nturl logs] (. (i, (ii, (iii [repeted root] (iv, (. Negtive vlues under the squre root sign indicte no rel solutions. Solutions onl possile moving into comple numers. (. (i, (ii, (iii, (iv.,.
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