Overview
Alsoknownastheauxiliaryunknownmethod,alsoknownasthevariablesubstitutionmethod.Animportantmethodforsolvingequations.Itisacommonlyusedmethod,anditsgeneralmeaningistoexpressapartofamathematicalexpressioncomposedofoneorseveralvariableswithnewvariablestofacilitatethesolutionoftheproblem.Here,onlytheequations(groupsof)Applicationinreconciliationinequalities(groups).
Itcanturnhigh-orderintolow-order,fractionalexpressionsintointegralexpressions,irrationalexpressionsintorationalexpressions,andtranscendentalexpressionsintoalgebraicexpressions.Itcanbeusedinthestudyofequations,inequalities,functions,sequencesofnumbers,triangles,etc.Widerangeofapplications.
Classification
Thesubstitutionmethodreferstotheintroductionofoneorseveralnewvariablestoreplacesomeoftheoriginalvariablestofindtheresult,andthenreturntofindtheresultoftheoriginalvariable.Themeta-methodlinksthescatteredconditionsbyintroducingnewelements,orrevealstheimplicitconditions,orlinkstheconditionswiththeconclusion,orbecomesafamiliarproblem.Itstheoreticalbasisisequivalentsubstitution.
Therearetwomaintypesofexchangemethodsinhighschoolmathematics:
(1)Overallexchange:exchange"yuan"for"style".
(2)Triangularexchangeofyuan,with"style"for"yuan".
(3)Inaddition,therearesymmetricexchange,meanexchange,universalexchange,etc.Theexchangemethodiswidelyused.Suchassolvingequations,solvinginequalities,provinginequalities,findingtherangeoffunctions,findingthegeneraltermandsumofasequenceofnumbers,etc.Inaddition,ithasawiderangeofapplicationsinanalyticgeometry.
Applicationskills
Whenweusethesubstitutionmethod,wemustfollowtheprinciplesofoperationandstandardization.Afterthesubstitution,wemustpayattentiontotheselectionofthenewvariablerange.Thevaluerangeofthevariablecorrespondstothevaluerangeoftheoriginalvariable,andcannotbereducedorexpanded.Asintheaboveexamples,t>0andsinα∈[-1,1].
Youcanobservetheformulafirst,andyoucanfindthattheformulathatneedstobereplacedalwayscontainsthesameformula,andthenreplacethemwithalettertodeducetheanswer,andthenifthereisthisletterintheanswer,Thatistosay,bringthisformulaintoit,andthenyoucancalculateit.
Factoringafactor
Sometimeswhenfactoringapolynomial,youcanchoosetoreplacethesamepartofthepolynomialwithanotherunknownnumber,thenfactorize,andfinallyconvertitback.Thismethodiscalledtheexchangemethod.
Relatedsamplequestions
Examplequestion1
Note:Don'tforgettoreturntheyuanafterchangingtheyuan.
[Example]Whendecomposing(x²+x+1)(x²+x+2)-12,youcansety=x²+x,thentheoriginalformula=(y+1)(y+2)-12=y²+3y+2-12=y²+3y-10=(y+5)(y-2)=(x²+x+5)(x²+x-2)=(x²+x+5)(x+2)(x-1).
Example2,(x+5)+(y-4)=8
(x+5)-(y-4)=4
Letx+5=m,y-4=n
Theoriginalequationcanbewrittenas
Thesolutionism=6,n=2
Sox+5=6,y-4=2
So
Feature:Bothequationscontainthesamealgebraicformula,asinthetitleInx+5,y-4,etc.,theequationcanbesimplifiedafterchangingtheelement.
Solvinghigher-orderequations
Sometimeswhensolvingequations,youcanchoosetoreplacethesamepartoftheequationwithanotherunknownnumbertoachievethepurposeofreducingtheorder,Andthenperformanewequationtofindanewunknown,andfinallyconvertitbacktofindtheoriginalunknown.Thismethodiscalledsubstitutionmethod.
Example2
Note:Donotforgettoreturntheyuanafterchangingtheyuan.
[Example]Solvetheequation(x²-2x)²-3(x²-2x)-4=0
Solution:Setx²-2x=y,thenTheoriginalequationbecomesy²-3y-4=0
(y-4)(y+1)=0
y-4=0ory+1=0
p>y1=4y2=-1
Wheny=4,x²-2x=4solvesx1=1+√5x2=1-√5
Wheny=-1,x²-2x=-1solvesx1=x2=1
So,therootoftheoriginalequationisx1=1+√5x2=1-√5x3=1