Solutionsofsimultaneousequations：联立方程组的解

Solutionsofsimultaneousequations

TheHELMnotesuseCramer’sruletosolvesystemsoflinearequationsbutthatmethodiscompu-

tationallyveryinefficient,withmanyrepeatedsubcalculations.Thetechniquepresentedhere-Gauss

elimination-usuallyisalotfaster,especiallyforlargesystems.

Considerthesimultaneousequations

x+y=5

2x+3y=13

Theny=5−xandso,

2x+3(5−x)=13

2x+15−3x=13

−x=13−15=−2

Thereforex=2andy=5−2=3.

However,thisapproachbecomesverymessywhenwehavethreeormorevariables.And,evenworse,

youcaneasily“lose”informationandgetthewronganswer.Hereisthesortofthingthatcangowrong:

Considerthesystemofequations

2x+y+3z=1

2x+2y+5z=4

x+y+z=1

x+2y+2z=3

ItmightbenaturaltotakeEqu2minusEqu1andthenEqu4minusEqu3togivethetwoequations

y+2z=3

y+z=2

Solvingtheseequationsgivesy=1andz=1.Pluggingthisbackintothefirstequationgives2x=

1−y−z=1−1−3andhence

x=−32.

IsthisOK?NO!-thesevalues,x=−32,y=1,z=1don’tsatisfythethirdequation.So,whathas

gonewrong?Theproblemisthatwehavenotkeptalltheinformationfromall4equations.So,weneed

amethodthatreliablyensuresthatwedon’tloseinformation.WeuseamethodknownasGaussian

elimination.

1Gaussianelimination

Wearegivenasystemofequations

ax+by+cz+···=d

ex+fy+gz+···=k

···

Herea,b,c,d,e,···areconstantsandx,y,z,···areunknownsthatwewanttosolvefor.

Ateachstepoftheprocedureweareallowedtodooneofthefollowing3operations:

(a)Addmultiplesofoneequationtotheothers

(b)swoptwoequations

(c)multiplyanequationbya(nonzero)number.

andthen,crucially,

Writedownalltheresultingequations.

Thislaststepisalittletediousbutitdoesstoperrorsliketheonewehadabove.

IndoingthisyoucanalwaysreducethesystemtoEchelonForm,whereeachequationstartstotheright

oftheoneabove,followedperhapsbyseveralequationsoftheform0=0.

ax+by+cz+···=d

f󰀁y+g󰀁z+···=k󰀁

mz+nw+···=p

······

Notethatitispossibletogetanequationoftheform0=−1orsimilarrubbish(thatmeansthatthe

systemofequationsisinconsistent-there’snosolution).Atthisstage,wecaneasilysolvethesystemby

backsubstitutionwherewestartfromthebottomequationandworkupwards.Thereare3thingsthat

canhappen:

•Youhaveanequationoftheform0=−12.InthiscasethereisNoSolution.

•Youhaveasmany(nonzero)equationsasunknowns.InthiscaseyouwillhaveaUniqueSolution

whichyoucanprettymuchwritedown(seetheexamplesbelow).

•Youhavefewerequationsthanunknowns.InthiscaseyouhaveInfinitelyManySolutions.I

willexplainbelowhowyoufindthemall.

2Letstrythiswiththesystemwehadbefore:

2x+y+3z=1

2x+2y+5z=4

x+y+z=1

x+2y+2z=3

SinceIdon’tlikefractions,Iamgoingtofirstswopthefirstandfourthequation:

x+2y+2z=3

2x+2y+5z=4

x+y+z=1

2x+y+3z=1

NowsubtracttwiceRow1fromtheRows2and4andsubtractonecopyofrowonefromRow3togive:

x+2y+2z=3

−2y+z=−2

−y−z=−2

−3y−z=−5

Nowusingthesecondequationtoeliminatethey’sfromthelast2equationsgives:

x+2y+2z=3

−2y+z=−2

−32z=−1

−52z=−2

And,finallytakingequation4minus5/3timesequation3gives

x+2y+2z=3

−2y+z=−2

−32z=−1

0=−13

So,thelastequationisimpossibleandexplainswhythereisNosolution.

3Let’sdothiswithafewmoreexamples.First,considerthesystemofequations

x+2y+3z=1

4x+5y+6z=1

2x+5y+7z=1

BeforegoingthroughtheGaussianelimination,Iamgoingtointroducesomeconvenientnotation.We

shallusetheshorthandnotationR1,R2,...torepresentrows1,2,...andwriteforexampleR4−R1as

shorthandfor“ReplaceRow4byRow4minusRow1.”Secondly,weonlyneedthenumbers1,2,...,7,1

sowewillwritedowntheAugmentedmatrixforthesystem;thisconsistsofallthenumbers,witha

verticallineinplaceoftheequalssign.Here,thoughitisveryimportanttoputinazeroifsomevariable

doesnotoccur.Thisgives:1231

4561

25

71

Proceedbyeliminatingxfromthesecondandthirdequationsusingrowoperation(a).1231

4561

25

71R2−4R1

R3−2R1∼1231

0−3−6−3

01

1−1.

Nextwedividerow2by(-3)toproduce:1231

0−3−6−3

01

1−1−13R2∼1231

0121

01

1−1.

Finally,weget1231

0121

01

1−1R3−R2∼1231

0121

00−

1−2

Revertingtoasystemofequationsweseethat

x+4y+2z=3

y+2z=1

−z=−2

4Solvingfromthebottomupthisgives

z=2

y=−1−z=−3

x=1−2y−3z=1+6−6=1

ThenextthingIshouldexplainiswhattodowhenyouendupwithfewerequationsthanunknownsin

echelonform.Forexample,onemighthave:

x+y+z+w=2

z+3w=5

Inthiscase,ifIaddintwoequationsy=27andw=34(oranyothernumbers)thenIwouldhavea

systemofequationsinechelonformwiththesamenumberofequationsasunknowns,andIcouldsolve

uniquely.So,wedosomethingsimilar.Theruleis:

•Ifyouareinechelonformandhavefewerequationsthanunknowns,foreachvariable

thatdoesnotappearatthebeginningofanequation,putthatequationequaltoan

arbitraryconstantandthensolve(uniquely)fortheothers.

So,inourexamplethisgives

w=a

z=5−3w=5−3a

y=b

x=2−y−z−w=2−b−(5−3a)−a=−3−b+2a

Letsdoonemoreexample:

−2x+z+w=−3

x+y+z+w=2

x+y−2z−2w=5

−3x+y+4z+4w=−5

Wewriteinechelonformthenswoprowsoneandtwotoget:−2011−3

11112

11−2−25

−314

4−5∼11112

−2011−3

11−2−25

−314

4−5

5