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Exchange method



Synonymsubstitutionmethodgenerallyreferstothesubstitutionmethod

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

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