FundamentalConceptsofAlgebraSupposewehavetheequality
15 = 10+ 5Usuallywehavesomethingthatlookslike
15 = 𝑥 + 5andourjobistosolveit.Aslongasweperformthesameoperationtobothsidesofourequation,wepreserveequality.Forourexample,willalwaysbeabletofind‘10’asthevalueof𝑥.Ifweinadvertentlydosomethingdifferenttoeachsidewewilllose‘10’astheanswerandwillendupwithsomeothernumberthatdoesn’tmakesenseintheoriginalequation.15 = 10+ 5;not(anythingelse)+5.Legitimateprocessespreservetheoriginalequality.Herearesomeexamplesoflegitimateprocesses.Foreachprocess,makesurethat‘10’makessenseas‘x’inthefinalline.Addthesamethingtobothsides,forexample,7:
15 = 𝑥 + 5
15+ 7 = 𝑥 + 5+ 7
Simplifiesto:
22 = 𝑥 + 12
Takethesamethingfrombothsidesforexample,3:
15 = 𝑥 + 5
15− 3 = 𝑥 + 5− 3
Simplifiesto:
12 = 𝑥 + 2
Multiplybothsidesbythesamenumber,forexample2:
15 = 𝑥 + 5
2 15 = 2(𝑥 + 5)
Expandsto:
30 = 2𝑥 + 10
Noticethatalltermsonbothsidesneedtobemultipliedby2.
Dividebothsidesbythesamenumber,forexample,5:
15 = 𝑥 + 515 (15) =
15 (𝑥 + 5)
Expandsto:
3 =𝑥5 + 1
Thesecondlinecouldalsobewrittenas:155 =
𝑥 + 55
Whichsimplifiesto:
3 =𝑥5 +
55
Whichinturnsimplifiesto:
3 =𝑥5 + 1
Raiseasanexponentwiththesamebase:
15 = 𝑥 + 5
2!" = 2(!!!)
Takealogarithmwiththesamebase
15 = 𝑥 + 5
log! 15 = log!(𝑥 + 5)
Applyanyotherfunction:
15 = 𝑥 + 5
𝑓 15 = 𝑓(𝑥 + 5)
N.B.Formostfunctions,suchas‘squaring’,𝑓 𝑥 + 5 ≠ 𝑓 𝑥 + 𝑓 5 . Eg,15! = 225,
but10! + 5! = 125,not225.Functionsdonotgenerallydistribute.Whatis
illustratedhereisthatifyouputthesamevalueintoafunction,yougetthesame
valueout.Withoutthedetailsofthefunction𝑓(eg,𝑓(𝑥) = 𝑥!),wecannotsimplify
therighthandside.
Sometimes,weworkonlyoneside.Thatiswhenwewritetheexpressiononone
sideinanequivalentformat.Forexample:
2𝑥 + 5 ! − 𝑥 + 3 ! = 456
2𝑥 + 5 2𝑥 + 5 − 𝑥 + 3 𝑥 + 3 = 456
4𝑥! + 10𝑥 + 10𝑥 + 25− 𝑥! + 3𝑥 + 3𝑥 + 9 = 456
4𝑥! − 𝑥! + 20𝑥 − 6𝑥 + 25− 9 = 456
3𝑥! + 14𝑥 + 16 = 456
Inthisexample,wemanipulatethelefthandsideonly.Ifyouknowhowtosolvea
quadratic,youwouldprobablynowsubtract456frombothsides,andifyou’re
havingagooddayyoumightfactorthetrinomial.Ifyourobservationissharp,you
might’venoticedthat‘10’isoneofthetwosolutionsforthisequation.
