Math From Scratch Lesson 34: Isolating Variables

Transcription

1 Mah From Scrach Lesson 34: Isolaing Variables W. Blaine Dowler July 25, 2013 Conens 1 Order of Operaions Muliplicaion and Addiion Division and Subracion Exponens Brackes The Resuling Acronym Isolaing Variables 4 3 Nex Lesson 5 1 Order of Operaions 1.1 Muliplicaion and Addiion The rules for algebra ha we have laid ou are very flexible, and allow for a variey of operaions and very complicaed expressions and equaions. When we use hese equaions o represen he physical world, as is he foundaion of much of science, hen we may need o deermine he value of a single variable a any ime. To do so, we ll need a means o perform calculaions clearly. Afer all, expressions like can be ambiguous: if we perform addiion firs, hen = 3 3 = 9, bu performing muliplicaion firs resuls in = = 7. Since a single expression is useless if i canno be consisenly calculaed 1, we need o deermine he bes way o perform hese operaions. 1 In some cases, his is unavoidable, bu here will be sric regulaions dealing wih hese ambiguiies 1

2 We know ha addiion and muliplicaion are boh commuaive. We wish o look for he mos consisen way o calculae , , , , and Le us see wha he resuls are for each arrangemen of numbers wih each operaion. 2 Expression Addiion Firs Muliplicaion Firs Doing addiion firs resuls in hree differen resuls, appearing wice each. Doing muliplicaion firs is much more consisen, wih he resul 7 appearing four imes and he resul 5 appearing wice. Furhermore, all arrangemens ha resul in he answer 5 are only valid rearrangemens if we calculae addiion before muliplicaion. Thus, if muliplicaion is calculaed before addiion, he las wo rows of he able are invalid arrangemens, and all resuls are a consisen value of 7. Therefore, if muliplicaion mus be performed before addiion, hen we have a wholly consisen sysem. 1.2 Division and Subracion We mus also find places o pu he operaions of division and subracion in his sequence. Since hey are formally defined in relaion o muliplicaion and addiion, his is no difficul: division ges he same prioriy as muliplicaion, and subracion ges he same prioriy as addiion. 1.3 Exponens The final major operaion o dae is exponeniaion. Again, we can have a seemingly ambiguous case wih expressions such as If we perform muliplicaion firs, we ge = 6 4 = 1296, bu if we do he exponeniaion firs, we ge = 2 81 = 162, which is a very differen resul. In his case, we resolve he issue hrough he definiion of he exponen: wriing his ou in oaliy, we find ha = = 162. Therefore, i is clear ha 2 One may objec o some of hese arrangemens, allowing only four of hese arrangemens insead of six. Tha is a side effec of an insincive grasp of he order of operaions ha we have been rained o apply; here is no inrinsic meaning o hese sysems ha makes he answer clear. 2

3 exponens mus be calculaed before muliplicaion, since ha is he consisen resul. 1.4 Brackes There are imes in which we would prefer o prioriize he laer operaions. For example, if we wan o deermine he oal revenue colleced from sales in Dollar Sore, i is preferable o add up he iems sold firs, and hen muliply by he revenue per iem second, avoiding a much lenghier calculaion if each iem s price is muliplied in advance. We can use brackes as he signal for his, as we did wih he disribuive propery: a (b + c) = a b + a c Any calculaions wrien in brackes are o be performed before calculaions wrien ouside of he brackes. Here, on he lef, we compue b + c before muliplying anyhing. 1.5 The Resuling Acronym The resuling acronym commonly augh as a mnemonic for his process is expressed in wo ways, depending upon he region of insrucion: PEMDAS - This sands for parenheses, exponens, muliplicaion, division, addiion, subracion. BEDMAS - This sands for brackes, exponens, division, muliplicaion, addiion, subracion. In boh cases, he final resul is he same, as muliplicaion and division are inerchangeable (by virually of really being he same operaion) and parenheses is a synonym for brackes. This is he order we use o calculae hese expressions. For example, in d = (v f v i ) we compue he value of d by firs calculaing he brackeed v f v i, and hen dividing he resul by. I is cusomary o imply brackes using fracion noaion: d = v f v i We know ha he v f v i ges compued before he division because he line in a fracion (named he vinculum) exends beneah he enire fracion. In general, we should always read he fracion a b as (a) (b) = (a) (b) 1 o avoid ambiguiy. 3

4 2 Isolaing Variables Calculaing d in d = v f v i is all well and good, bu wha if he value of d is known and we need o find he value of v i? Tha requires rearranging variables o isolae v i. The rick o his is o perform our order of operaions (BEDMAS/PEMDAS) in reverse order. Expressing his as d = (v f v i ) gives us he firs sep in he process. Le us colour code wha we are looking for: d = (v f v i ) I is currenly conained wihin brackes, so our order of operaions for isolaing variables (SAMDEB/SADMEP) needs o deal wih he porion firs. We deal wih i by finding a way o cancel i ou, using wha we know of cancelable elemens from our earlier lessons. If we muliply by, his will eliminae he porion. If we muliply by on he righ, we mus also do so on he lef; failing o do so would mean he wo sides are no equal any longer. Thus, we have d = (v f v i ) d = (v f v i ) d = v f v i Now ha we are down o he inerior of he brackes, we run hrough (SAMDEB/SADMEP) again. The v f erm is being added, so we eliminae i by subracing v f again, which is he same as adding v f o boh sides. Thus, we have d = v f v i v f + d = v f + v f v i d v f = v i Our final sep is o muliply by 1 o compleely isolae v i. 1 (d v f ) = 1 ( v i ) v f d = v i Thus, v i = v f d and he variable is isolaed. We will run ino he aforemenioned ambiguiies when we deal wih solving for variables ha appear wih exponens. As i urns ou, here are no ambiguous cases, bu raher are cases ha legiimaely have more han one correc answer. 4

5 3 Nex Lesson In our nex lesson, we begin o examine polynomials and heir roos, which will evenually lead us o he realizaion ha he real numbers alone are no adequae o describing he human experience. 5