MVR5 I/context/cfpushR6$%"aG'%"LG<$%%nameG%&stackG6#%"tG6#%inCopyright~(c)~1997~by~W aterloo~Maple~Inc.~All~rights~reserved.G6"C%-%'ASSERTG6#-%)assignedG6#F*@$4-%%t ypeG6$9%F*-%'assignG6$F<-&F*6#.%$newGF/-&F*6#.%%pushG6$9$F6$-%*substringG6$FK;"""!"#.Fgn-%'RETURNG6#Fgn>8$-%'indetsGFX@%0Fbo<"- %$mapG6$9!Fbo%%NULLGF/F/F/6", I0context/cfparseR6#'%#CFG%)CF_CLASSG6=%(arglistG%(ActionsG%+ActionNameG%+Actio nKeysG%)AncestorG%-ConditionalsG%0ConditionalKeysG%%DataG%,DescriptionG%%fileG% %headG%+HelpStringG%(IfBlockG%%MenuG%$KeyG%&LocCFG%+MenuStringG%,MenuEntriesG%+ RecognizerG%-ReplacementsG%)SubMenusG%-SubMenusKeysG%#t0G%%tailG%%TestG%)warnin gsG%"xGF-F/CK@$0%4context/InitializedG%%trueG--FS6#.%-context/initGF/-FS6#%1con text/cmglobalG-FS6#%.context/cfgetG-FS6#%8context/cfparse/contentG-FS6#%6contex t/cfparse/menusG-FS6#%;context/cfparse/recognizerG>83FK>%0_EnvContextDataG-%&ta bleGF/>8%-Fdo6$Fjs.%*CF_ACTIONG>8:-%%timeGF/-%)userinfoG6%F[o<#F^p-%(sprintfG6% 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96~by~Ananda~Gunawardena~&~Elias~DeebaGF/C(>8,&9"6#F[o>F]fm&F\^n6#""#>Fbo-%'mat rixG6$-Feu6#Fj]n,&-%(vectdimG6#F]fmF[oF[oF[o>F`t-Fd^n6$Ff^nF[o?(FiyF[oF[oFf^nFd rC$?(FezF[oF[oFi^nFdr>&Fbo6$FiyFez-%&coeffG6$-%$lhsG6#&Fj]nFa\l&F]fm6#Fez>&Fbo6 $FiyFh^n-%$rhsGFj_nFboF/F/F/F/, I?context/cfparse/act_LINEARFUNCR6#'%%CodeGFa_lF/6#FjblF/C$@$FbrFer-F_elFb[lF/F /F/6", I0type/CF_SUBMENURFOF/F-F/3-F:6$FK-%)specfuncG6$%)anythingGF\]l/-Fdo6$FKFdtFgoF /F/F/6", I)rrefautoRF/6?%$ansGF_]nF`]nFbdmFd]n%"kG%%flagG%$AUGGFcdm%)typeflagGFc]n%$eqsG %)responseG%#A1G%#b1G%'secmulG%'rowaddG%%leftG%)firstmulG%)firstrowG%'secrowG%% rsetG%%row2G%%row1G%+multiplierG%'norowsG%'rownumG%(tempaugG%%prevGFf]nF/C<>Fez F[^n>Fez-Ff[n6$FezFd^n-%&printG6$%9The~original~matrix~is~:GFez>Fj]n-%'rowdimGF ]`n>8.-%'coldimGF]`n>Fiy-%)chkrowecGF]`n@$/FiyF[oC)-Fjcn6%%;Note~that~the~given ~matrixGFez%<~is~not~in~row~echelon~formG-%'printfG6#%C~~~~~Choose~one~of~the~f ollowing:|+G-F_en6#%[q~~~~~1.~if~you~want~to~observe~the~steps~to~display~the~r ow~echelon~form;~~type:~>rowec;~~or~1;|+G-F_en6#%[q~~~~~2.~If~you~do~not~want~t o~observe~the~steps~of~the~row~echelon~form;~~~type:~>bypass;~or~2;|+G>Fbo-%)re adstatGF/?(F/F[oF[oF/4555/Fbo%&rowecG/Fbo%'bypassG/FboF[o/FboFa^nC&>%'finishGFb o@$55/Fhfn%%exitG/Fhfn%%EXITG/Fhfn%%ExitG-F_oF/-F_en6#%8Type~bypass;~or~rowec;| +G>FboFien@&55F`fn/Fbo%&ROWECGFdfnC'>%(rreflagGF[o-%*gaussautoGF]`n>F^hnFg`l>Fe z-&%'linalgG6#%*gausselimGF]`n?(FhuF[oF[oF^dnFdrC%>8(F[o?(F/F[oF[oF/3/&Fez6$Fhu F[inFg`l2F[inFbdn>F[in,&F[inF[oF[oF[o@$0F_inFg`l>Fez-%'mulrowG6%FezFhu*$F_inFcu 55/Fbo%'BYPASSGFbfnFefnC%>FezFchn>Fj]n-%%rankGF]`n?(FhuF[oF[oF^dnFdrC%>F[inF[o? (F/F[oF[oF/F]in>F[inFcin@$Fein>FezFgin>FhuF^dn?(F/F[oF[oF/3-%'iszeroG6#-%$rowG6 $FezFhu2F[oFhu>Fhu,&FhuF[oFcuF[o>F[inF[o?(F/F[oF[oF/F]in>F[inFcin@$3Fein0F_inF[ o>FezFgin-%&blankGF]^n-Fjcn6#%DThe~matrix~in~ROW-ECHELON~form~is~:G-FjcnF]`n-Fj cn6#%YNow~let~us~put~this~matrix~into~Reduced~Row~echelon~FormG-%)continueGF/@$ Fjfn-F_o6#%*Try~LaterG>8@F[o>Fj]nFbdn>FadnF^dn?(F[inF[oF[oFj]nFdrC%>FdxF^dn?(F/ F[oF[oF/3/&Fez6$FdxF[inFg`l2F[oFdx>Fdx,&FdxF[oFcuF[o@%/FdxF[oF/@$2F^]oFdxC$>F^] oFdx?(FhuF[^oFcuF[oFdrC%-Fjcn6$%>Perform~elimination~on~columnGF[in-Fjcn6#%:*** **********************G@$0Fg]oFg`lC(-Fjcn6*%.multipliy~rowGFdx%.of~the~matrixGF ez%#byG,$F_inFcu%4and~add~that~to~rowGFhu>Fez-%'addrowG6&FezFdxFhuFb_oF^\o-Fjcn 6$%4The~new~matrix~is~:GFezFg\o@$Fjfn-F_o6#%&AdiosG-Fjcn6$%LThe~matrix~in~reduc ed~row~echelon~form~is~:GFezF^\o-Fjcn6#%jnThis~is~the~end~of~demonstration~vers ion~of~the~reduced~echelonG-Fjcn6#%[oform~procedure.~You~may~try~other~examples ~to~learn~the~process.GF/6$FhfnF^hnF/F/, IFbo-&F*6#FDF/-Fio6%&F* 6#FIFKFbo-Fbal6#FboF/F/F/6", I:context/cfparse/joinblockR6#'%'blocksG-F]dm6#%+CM_IFBLOCKG6'Fcq%'KeySetGF_r%( NewCodeG%-continuationG6#F.6"C)-FS6#.Fhhl>8(%&falseG>8'Q!6">8%<"?&8&9$FdrC&>8$- F_s6$Q$KeyFcvFhco@%-F`fmFafm-%&ERRORG6#-F^u6$Q1Repeated~Key:~%sFhboF\do>Feco-%& unionG6$Feco<#F\do>Faco-Fhhl6%Q"~FccoFaco-%F^coFdr@$/F^coFdr>Faco-Fhhl6%F`eoFacoQ$fi;6"-Fa_l6#FhuFhboFhboFhboFhbo, I0context/fullkeyR6$'%%RootG<$F)%'stringG'%(CurrentGFdfoF/Fh`nF/C$@$5-F:6$FKF)/ FKFbco-F_o6#F<-%$catG6%FKQ"_6"FF`tF_jo>F]fm-Fcio6$FjhmF`t-F26#0F ]fmFbco>Fbo-F^u6%Qenif~not~(~%s~)~then~stack[push](~%a~=~NULL,~Exclusions)~fi;6 "-Fhhl6$F`eoF]fmFK@$4-F_hlFiao-Fcdo6$%>unable~to~parse~condition~forGF`t-Fa_lFi aoF/F/F/F[[p, I4help/text/gausselim-%%TEXTG6D%+HELP~FOR:~G%#~~G%`o-~linsys[gausselim]~:~the~p urpose~of~this~function~is~to~To~reduce~a~G%cq~a~matrix~OR~an~augmented~matrix~ representing~a~system~of~linear~equations~Ax=b~to~its~row~echelon~form.G%fn~Gau ss~elimination~is~performed~on~the~rows~of~the~matrix~AG%N~or~on~the~rows~of~th e~augmented~matrix~[A,b]G%en~There~are~THREE~versions~of~the~gauss~elimination~ programGFj[p%]p~VERSION~1~:~~GAUSSAUTO~:~demonstrates~a~step~by~step~process~of ~gauss~eliminationG%[o~~~~~~~~~~~~~~using~the~row~operations~selected~by~the~ma chine.~G%"~G%K~~example~1~:~~A~:=~matrix([[1,2],[3,3]]);G%=~~~~~~~~~~~~~~~gauss auto(A);G%R~~example~2~:~~A~:=~[[1,2,2],[2,2,2]];~b:=[3,43];G%?~~~~~~~~~~~~~~~g aussauto(A,b);GFb\p%T~~Note~that~the~matrices~can~be~entered~in~any~formG%Y~~as ~~A~:=~[[2,3],[3,4]]~~OR~~A:=~matrix([[2,3],[3,4]]);GFb\p%\o~VERSION~2~:~GAUSSM AN~:~This~procedure~performs~gauss~eliminationG%V~~~~~~~~~~~~~using~the~row~ope rations~provided~by~youGFb\p%J~~example~:~~A~:=~matrix([[1,2],[3,43]]);G%:~~~~~ ~~~~~~~~gaussman(A);GFb\p%co~VERSION~3~:~gausselim~:~This~is~the~general~proced ure.~It~allows~you~toG%_o~~~~~~~~~~~~~select~the~row~operations~first.~However~ you~can~switchG%]o~~~~~~~~~~~~~to~auto~mode~any~time~and~observe~the~machine~pe rformG%=~~~~~~~~~~~~~row~operations.GFb\p%I~~example~:~~A~:=~matrix([[1,2],[3,3 ]]);G%;~~~~~~~~~~~~~gausselim(A);GFb\pFj[pF/, I;context/cfparse/recognizerR6$'FbpFd\l'%%DATAGF^t6'F\qFcqFaqFg`n%'IfCodeGFh`nF /C0F]s-FS6#Fgdl-FS6#F_hl-FS6#Fhhl-FS6#Fa_l>F`t-F_s6$F_doFK>F]fm-FgdlFb[l>F[in-F `im6%F:F]fmFbim-F26#/-Feu6#F[inF[o-F26#-F_hl6#-Fa_l6#-FjwF`_p>Fhu-Fa_l6AQ(proc( f)6"QWlocal~RememberFlag,Exclusions,Vars,Nvars,Funcs,NFuncs;F\`pQ`pglobal~`cont ext/CM_GLOBALG`,~ContextData,~`context/Initialized`,~`context/InitProcs`,F\`pQB ~~~~~_EnvContextData,~ContextKey;F\`pQ/option~system;F\`pQ9"ContextMenu~Procedu re";F\`pQ`oif~`context/Initialized`~<>~true~then~readlib('`context/init`')()~fi :F\`p-F^u6$Q2ContextKey~:=~%a;F\`pF`t-F^u6$Q7_EnvContextData~:=~%a;F\`pQ%DATAF\ `pQFContextData~:=~eval(_EnvContextData);F\`p-F^u6$QHif~`context/InitProcs`[%a] ~<>~true~thenF\`pF`t-F^u6$QJreadlib('`context/InitializeProcs`')(%a);F\`pF`t-F^ u6$QA`context/InitProcs`[%a]~:=~true;F\`pF`tFjeoQ0readlib(stack);F\`pQ>Exclusio ns~:=~stack['new']():F\`pQ6RememberFlag~:=~true;F\`pFg_pQTContext~:=~_EnvContex tData[~CM_BUILD(ContextKey)~];F\`pQGif~stack['depth'](Exclusions)~>~0~thenF\`pQ gnExclusions~:=~convert(~Exclusions~,~`context/StackToList`~);F\`pQGContext~:=~ subs(Exclusions~,~Context);F\`pFjeoQFealFaal>%+GlobalCodeGFhu-%( fprintfG6#-Fhhl6$Q"|+F\`pFhu-Fcdo6#Q;Unable~to~parse~RecognizerF\`p>Fhu-F\ilF^x >Fhu-Fax6$<#/Fi`pFaal-FbalF^xF/6%FealFibp%3context/CM_GLOBALGGF/F\`p, I8context/cfparse/topmenuR6&%3SimplificationTypeGFeqF`q'%&AListGF]dm6'%(Content G%&MenusG%+ActionListGF_r%(EntriesG6#Fjbl6"C,-FS6#F*-FS6#Fbs>8%-F^tFidp>&F`ep6# Q%MainFidp-&F*6#FDFidp-Fio6%%9context/cfparse/pushmenuG9'F`ep>8$-Ff[n6$FcepF\\n -F26#-F:6$F^fpF]dm>F^fp-Fio6$R6#F_rFidpFidpFidp@%-F:6$9$F[^l-Fbs6$F]gpT#F]gpFid 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emented~yetGFijp>Fc[q-F\bqFj]qFj_q>Fi\q-Fcw6$7%-Fhn6$F]]q;F\ip!"#Q.sub-selectio nFhhpFg`qFc]qF]cq-F_o6#-%,CONTEXTMENUG6%FgjpFhjp7#Fi\qFhhp6%FealF\tFd[qFhhpFhhp, I+DELETEPLOTR6$'%)positionG%'posintG'%%PlotGFb^qF/F-F/-F_o6#-%'subsopG6$/FKF\pF Fbo-F\\q6#7#-Fjw6#--FS6#.Ffil6#F \^n-F26#/-FeuFiao""$>Fbo-&Ff_q6#.Fi_qFiaoF/F/F/6", I0help/text/topic-Fg[p6)Fi[pFb\p%gn~linsys[topic]~:~Help~is~available~for~the~f ollowing~topics.G%_o~~~~~~~~~~~~~~~~~~~~~~~~~~~gausselim,~backsub,~graph,~rref, ~solveqnsG%:~~~~~~~~~~~~~~~~~~~~~~~~~G%gn~~~~~~~~~~~~~~~~~To~get~more~details~t ype~?topic-name~at~theG%?~~~~~~~~~~~~~~~~~~Maple~promptGF/, I0help/text/graph-Fg[p67Fi[pFb\p%en~linsys[graph]~:~The~purpose~of~this~procedu re~is~to~graphG%K~a~system~of~linear~equations~in~2D~or~3D~GFb\p%H~An~example~o n~how~to~use~this~functionG%hnTo~graph~2~equations~of~3~variables~we~can~type~t he~followingGFb\p%9~>~eq1~:=~x~+~y~+~z~=10;G%9~>~eq2~:=~2*x~-~3*y~=~z;G%4~>~gra ph(eq1,eq2);~GFb\p%fn~or~just~type~at~the~Maple~prompt~>graph();~and~the~progra mG%9~will~prompt~for~input.~GFb\p%]o~In~2D~case~the~equations~will~be~displayed ~along~with~the~graphs.GFb\p%Z~The~graph~will~dispaly~consistency~or~inconsiste ncy~infoGFb\p%Z~Plot~Range~:~The~program~will~select~the~best~plot~rangeG%I~nec essary~to~display~all~intersections.GF/, I4type/CF_CONDITIONALRFOF/F-F/-F:6$FK-Fcan6$FeanFgxF/F/F/6", I3help/text/solveqns-Fg[p64Fi[pFj[p%ao-~solvepack[solveqns]~:~the~purpose~of~th is~function~is~to~To~solve~a~G%^o~a~system~of~linear~equations.~The~set~of~equa tions~must~be~enteredG%T~as~a~list~or~set.~Example~:~>S:=[x-y=3,2*x+3*y=9];GFj[ p%\o~The~set~of~variables~you~are~solving~for~must~also~be~entered~asG%I~a~set~ or~list.~Example~:~>~vars:=[x,y];GFb\p%jn~If~no~arguments~are~entered~the~progr am~will~prompt~for~input.G%9~Then~follow~directions.GFb\p%E~~example~1~:~~S:=[x -y=3,2*x+3*y=9];G%<~~~~~~~~~~~~~~~vars:=[x,y];G%A~~~~~~~~~~~~~~~solveqns(S,vars );GFb\p%<~~example~2~:~~solveqns();~GFj[pF/, I2type/CF_SEPARATORRFOF/F-F/-F:6$FK-Fcan6$Fean.Fc_lF/F/F/6", I2help/text/backsub-Fg[p6-Fi[pFb\p%ao~linsys[backsub]~:~The~purpose~of~this~fun ction~is~to~perform~the~backG%gn~~~~~~~~~~~substitution~algorithm~in~solving~a~ linear~systemG%in~~~~~~~~~~~The~procedure~will~check~the~row~reduced~matrix~for G%W~~~~~~~~~~~consistency~and~apply~the~back~substitutionG%]o~~~~~~~~~~~Matrix~ must~be~in~row~echelon~form~as~shown~by~examplesG%1~~~~~~~~~~~belowGFb\p%K~~exa mple~1~:~~A~:=~matrix([[1,2],[0,1]]);G%;~~~~~~~~~~~~~~~backsub(A);GF/, I9type/context/Unbounded@@RFOF/Fh`nF/3-F:6$FKFb^q53-F:6$-Fjw6$Fg`lFKFb^q-F:6$-F jw6$Fg`lFd[r-.%#@@G6$FeanF)-F:6$Fd[rFj[rF/F/F/6", I(rrefmanRF/6@F]bnF_]nF`]nFbdmFd]nF^bnF_bnF`bnFcdmFabnFc]nFbbnFcbnFdbnFebnFfbnF gbnFhbnFibnFjbnF[cnF\cn%%RsetGF]cnF^cnF_cnF`cnFacnFbcnFccnFf]nF/C2>Fft-Fcho6#71 %#R1G%#R2G%#R3G%#R4G%#R5G%#R6G%#R7G%#R8G%#R9G%$R10G%$R11G%$R12G%$R13G%$R14G%$R1 5G>Fd]l-Fcho6#71%#r1G%#r2G%#r3G%#r4G%#r5G%#r6G%#r7G%#r8G%#r9G%$r10G%$r11G%$r12G %$r13G%$r14G%$r15G>FezF[^n>FjuF^dn-%,rdgaussmenuGF/>80%&BEGING-Fjcn6$%BThe~orig inal~augmented~matrix~is:GFezF^\o>FiyF[o?(F/F[oF[oF/30FiyFg`l0FiyFa^nC)-F_en6#% ^pPlease~enter~an~appropriate~row~operation~from~the~above~menu~to~reduce~the~m atrix|+G>Fa_rFien>FhfnFa_r@$Fjfn-F_o6#%$ByeG@/5/Fa_rF_gn/Fa_rF]gn>FiyFg`l5/Fa_r Fafn/Fa_rF[hnC%>FezFchn?(FhuF[oF[oF^dnFdrC%>F[inF[o?(F/F[oF[oF/F]in>F[inFcin@$F ein>FezFgin-Fjcn6$%DThe~matrix~in~row~echelon~form~is~:GFez5/Fa_r%$recG/Fa_r%$R ECGC$>Fez-&Fehn6#%%rrefGF]`n>FiyFa^n5/Fa_r%%undoG/Fa_r%%UNDOGC$@(/8-F[o>Fez-%(s waprowG6%Fez8<8;/F]crFa^n>Fez-Fhin6%Fez8?*$F^`lFcu/F]crFdfq>Fez-Ff_o6&FezF`w84, $F^alFcu-Fjcn6$%FBefore~your~selection~the~matrix~was~GFez-F:6$-Fb`n6#Fa_r%)mon omialGC$?(FhuF[oF[oFjuFdr@$50-Ff_n6$Fedr&Fd]lF^xFg`l0-Ff_n6$Fedr&FftF^xFg`lC%@$ F\er>F^`lF]er@$F`er>F^`lFaer>FhcrFhu@%/F^`lF[oC*>FccrFhcr>85-Fi_nFfdr?(FhuF[oF[ oFjuFdrC$@$/F_erF_fr>FbcrFhu@$/FcerF_fr>FbcrFhu>FezF_cr-Fjcn6&%0Interchange~row GFbcr%$andGFccr-Fjcn6$%8This~is~the~new~matrix:GFezF^\o>F]crF[oC'>Fez-Fhin6%Fez FhcrF^`l-Fjcn6&%-Multiply~rowGFhcrFa_oF^`l-Fjcn6$%7This~is~the~new~matrixGFezF^ \o>F]crFa^n-F:6$Fedr%(polynomGC+>F`wFg`l>Fi\lFg`l?(FhuF[oF[oFjuFdrC$@$50-Ff_n6$ F`frF_erFg`l0-Ff_n6$F`frFcerFg`l>F^drFhu@$F[er@%/F`wFg`lC$>F`wFhu@&F\er>F^alF]e rF`er>F^alFaerC$>Fi\lFhu@&F\er>FjsF]erF`er>FjsFaer@$0F^alF[oC$>Fez-Ff_o6&FezF`w F^drF^al-Fjcn6(FhgrF`wFa_oF^al%/and~add~to~rowGF^dr@$0FjsF[oC$>Fez-Ff_o6&FezFi\ lF^drFjs-Fjcn6(FhgrFi\lFa_oFjsFcjrF^dr@$3/F^alF[o/FjsF[oC$>FezF_jrFajrFh_oF^\o> F]crFdfq-Fjcn6#%BChoose~a~selection~from~the~menu.G>F^]oF`br@$/-%%normG6$-%&eva lmG6#,&F^]oF[oFezFcuF[oFg`lC$-Fjcn6$%KThe~matrix~in~reduced~row~echelon~form~is :GFez>FiyFa^n@$3/FiyFg`lFj_r-Fjcn6$%Swarning.~Matrix~is~not~in~reduced~row~eche lon~formGFezF^\o@$Fdfn-Fjcn6#%[oTo~solve~the~system~apply~backsubstitution~to~t 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F[in-F]bm6$F[in/.%&colorGFez-F:6$F]fmF]dmC%@$53-F:6$&F]fmF]^nFb`s2F[o-Feu6#-Ffi l6#Ffes3-F:6$FfesFbim2Fa^nFhesF[ds>F[inF]fm@%4-%$hasG6$F[in<$F\es.%'colourGC%>F ezF_ds>FiyFads>F[inFids>Fiy7$Fbds-Fjw6#-Fibm6#7#F[in-Fcdo6#%2unknown~structureG >F`t6$F`tF[inFh`sC%>Fbo--FS6#FacmF_y>%6_EnvSMARTPLOTCoords3DGFbo?&F]fmFhuFdrC$@ 'FjbsC$@$53FacsFjcs3Fhcs2FdfqFdcs-Fcdo6#%3too~many~variablesG>F[in-Fjw6#-%,smar tplot3dGF[_nF^esC$@$53FdesF_fs3F]fs2FdfqFhesFihs>F[inF]fmFdgs>F`tFhgs6%-FjwF_yF `t-FjwFg^nF/FeemF/6", I0CM_GetArrayVarsR6#'FPFaho6$%%valsGF,Fh`nF/C%>FboFgo?&F`t<#F\^nFdr@%-F:6$F`tFa ho>Fbo-Fjdo6$Fbo-Fio6$Fjw<#-F]xFfu>Fbo<$FboF`t-FfilFiaoF/F/F/6", I)continueRF/6#%(defaultGFf]nF/C'@%/FdasF[o>FboF[^n>FboF[o-F_\oFiao-F_en6#%ao<~ type~;~and~press~Enter~to~continue~?topic~for~help~~exit;~to~quit~>|+G>FhfnFien F^\tF/Fe]sF/F/, I1context/cmglobalR6$'%#wwG<$%'symbolGFdfo'FPFean6'%"wG%,NotAssignedG%%exprG%'n ewvarGFbdmF-F/C.>FboFK-F26#-F:6$FboFh\t>F`tRF]asF/Fh`nF/C$@$4Fjfo-F_o6#F_co-Fgg 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?(FhuF[oF[oFiyFdr>&Fadn6$FhuF^ft-Fb`nFfft-F_o6#FadnF/F/F/F/, I&blankR6#Fcdm6#FbdmF/F/?(FboF[oF[oFKFdr-Fjcn6#Fb\pF/F/F/F/, I0type/CM_NOPRINTRF]asF/Fh`nF/-F:6$FK-Fcan6$Fean<$%&PADICG%&CFRACGF/F/F/6", I0type/CM_SUBMENURFOF/F-F/-F:6$FK-Fcan6$Fdfo.F[^lF/F/F/6", I'bksub1RF/6?Fbdm%#i1GFd]nF^bnFc]nFcdm%&soln1G%"uGF`bn%(charsetG%)solnvarsG%)so lnvectG%(tempsumGF_]nF`]nF_rF`et%)solneqnsGF^etFe]n%&eqns1G%%solnG%&rightG%)tye pflagGFbbnF_bn%&countGFa]n%)templistGFf]nF/C5>F]cr-Fcho6#7.&%#_tGF]^n&F[[uF`^n& F[[u6#Fdfq&F[[u6#""%&F[[u6#""&&F[[u6#""'&F[[u6#""(&F[[u6#"")&F[[u6#""*&F[[u6#"# 5&F[[u6#"#6&F[[u6#"#7>Fj]nF[^n>FhcrF_^n>F[in-Fj^n6#-%$colG6$Fj]nF[o>Fdx,&-Fj^n6 #-Fb[oFa]uF[oFcuF[o>Fjs.Fjs>Fjs-FchoF_y?(FboF[oF[o-Fj^n6#FhcrFdr>&FjsFiao&FhcrF iao>F`w-F]\s6#-%#&*G6$-%*submatrixG6%Fj]n;F[oF[in;F[oFdx-Ff[n6$FjsFd^n>Fi\lFb^u >Fft-Fh^u6%Fj]nFj^u;,&FdxF[oF[oF[oFc_u>Fj]n-FdhnFg^n?(FboF[oF[oF[inFdr@$0&Fj]n6 $FboFboFg`l>Fj]n-Fhin6%Fj]nFbo*$Fi_uFcu>F`wFb^u>81Fg^u>8/Fj]u?(FboF[oF[oFdxFdr> &Fc`uFiaoF_^u>F_fr-%)multiplyG6$Fa`uFc`u@%/-Fcjn6#Fa`u-FcjnFg^nC4>FboF[in?(F/F[ oF[oF/-F_[o6#-Fb[o6$Fj]nFbo>Fbo,&FboF[oFcuF[o>F[inF]]u>FdxFc]u@$2FboF[in>F`wFb^ u@%/F[inFdxF/C$-Fjcn6#%JThis~system~has~infinitely~many~solutionsGF^\o>Fd]lFj]u ?(FboF[oF[oFdxFdr>&Fd]lFiaoF_^u>F^`l-Fcho6$FdxFg`l>FjuFdx?(FhuF_auFcuF[oFdrC,>F boF[o?(F/F[oF[oF//&Fj]n6$FhuFboFg`l>Fbo,&FboF[oF[oF[o>&F^`lFiaoF[o?(F]fmFdcuF[o FdxFdr@$0&F^`lF[_nF[oC%>&Fd]lF[_n&F]cr6#,(FdxF[oFjuFcuF[oF[o>FjcuF[o>Fju,&FjuF[ oFcuF[o>Fa_rFg`l?(F]fmFdcuF[oFdxFdr>Fa_r,&Fa_rF[o*&&Fj]n6$FhuF]fmF[oF]duF[oF[o> Fgbu*&,&&Fj]n6$FhuFc_uF[oFa_rFcuF[oFacuFcu@%0FboFdxC(@%/FhuF_au-Fjcn6#%KChoose~ the~free~variable(s)~and~substituteG-Fjcn6#%-substitute~:G?(F]fmFdcuF[oFdxFdr-F jcn6#/&FjsF[_nF]duF^\o-Fjcn6$%/in~equation~:~G/&F_frF^xF^eu-Fjcn6$%5to~find~the ~variableG/F_^uFgbuF^\oC$-Fjcn6#%FBy~using~the~last~equation~we~can~getG-Fjcn6# /&FjsF_y&Fd]lF_yFg\o@$F[gnFbgn-Fjcn6#%STherefore~the~values~of~the~variables~ar e~given~byG>F^]oFj]u?(FboF[oF[oFdxFdr>&F^]oFiaoFhfu>F^drFj]u?(FboF[oF[oFdxFdr?( F`tFboF[oFdxFdr@$/&FhcrFfu-Fi_n6#&F^]oFfu>&F^drFfu/F_hu-Fb`nFahu-Fjcn6#F^drF^\o C%F^\o-Fjcn6#%coIn~the~above~system~we~observe~that~one~or~more~of~the~statemen t(s)~leadG-Fjcn6#%foto~inconsistent~equation(s).~Therefore~the~original~system~ is~Inconsistent.GF/Fe]sF/F/, I'bksub2RF/6?FbdmFgitFd]nF^bnFc]nFcdmFhitFiitF`bnFjitF[jtF\jtF]jtF`etF_]nF`]nF_ rF^jtF^etFe]nF_jtF`jtFajtFabnFbbnF_bnFcjtFa]nFdjtFf]nF/C6>F]crFgjt>Fj]nF[^n>Fhc rF_^n>F[inF]]u>FdxFc]u?(FboF[oF[o-F_dnFg^nFdr?(F]fmF[oF[o-FcdnFg^nFdrF/>F^dr.F^ dr>F^drFj]u?(FboF[oF[oF\^uFdr>&F^drFiaoF`^u>F`w-F]\s6#-Fe^u6$Fg^u-Ff[n6$F^drFd^ n>Fi\lFdju>FftF`_u>Fj]nFe_u?(FboF[oF[oF[inFdr@$Fh_u>Fj]nF\`u>F`wFdju>F]vFg^u>Fc `uFj]u?(FboF[oF[oFdxFdr>Ff`uFbju>F_fr-Fi`u6$F]vFc`u@%/-FcjnFivF_auC7>FboF[in?(F /F[oF[oF/Fcau>FboFhau>Fj]n-Fh^u6%Fj]n;F[oFbo;F[oFc_u>F[inF]]u>FdxFc]u@$F\bu>F`w Fdju@%F_buF/F/F^\o>Fd]lFj]u?(FboF[oF[oFdxFdr>FgbuFbju>F^`lFibu>FjuFdx?(FhuF_auF cuF[oFdrC)>FboF[o?(F/F[oF[oF/F`cu>FboFdcu>FfcuF[o?(F]fmFdcuF[oFdxFdr@$FicuC%>F] duF^du>FjcuF[o>FjuFcdu>Fa_rFg`l?(F]fmFdcuF[oFdxFdr>Fa_rFgdu>FgbuF\eu>F^]oFj]u?( FboF[oF[oFdxFdr>Figu/FbjuFgbu-Fjcn6#%IThe~values~of~the~variables~are~given~byG >F^]oFj]u?(FboF[oF[oFdxFdr>FiguFa^v>Fa`uFj]u?(FboF[oF[oFdxFdr?(F`tFboF[oFdxFdr@ $F^hu>&Fa`uFfuFehu-FjcnF^auC%F^\oFjhuF]iuF/Fe]sF/F/, I8context/InitializeProcsR6#'FcqFdfo6$F_r%%KeysGFh`nF/C&>F`t-F]ho6%F:7#-F_[nFca l7#Fi\t>F`t-Fio6$FjwF`t?&FboF`tFdrC$-F26#-F:6$&F\tFiaoF\cq-F>6$Fbo-Fbal6#Ff`v>& %2Context/InitProcsGFb[lFdrF/6$F\tF]avF/6", I1help/text/linsys-Fg[p6(args)~~ORG%9linsys[] (args)GFb\p%*SYNOPSIS:G%P-~to~use~a~linsys~function,~invoke~the~functionG%D~usi ng~the~form~linsys[].G%$~~~G%@-~The~functions~available~are~:GFabv%C~ ~~~gausselim~~gaussauto~gaussman~GFabv%hn-~for~more~information~on~a~particular ~function~refer~to~highG%E~~level~function~manual~or~just~typeG%3>~?G%2~eg:~>~?gausselimGF/, I=context/cfparse/act_completeRFe`nF/Fh`nF/C%-FS6#.F\ilFjbo-F\il6#-Fa_l6ZQgnpro c(s1::uneval,s2::uneval,s3::uneval,s4::uneval,s5::uneval)6"Q_o~~~~local~Eval,~E valLabel,Parameters,AutoAssignFlag,ParseFlag,Nargs,FccvQO~~~~~~~~F,Labels,Param s,i,ProcBody,CodeString;FccvQ;~~~~global~none,CM_Global;FccvQ@~~~~"Complete~Act ion~Template";FccvQD~~~~readlib(~'`context/cmglobal`');FccvQB~~~~readlib(~'`con text/testeq`');FccvFhcvQQ~~~~Parameters~:=~select(~type~,~[args]~,~`=`~);FccvQj n~~~~AutoAssignFlag~:=~subs(~op(Parameters),~"autoassign"=FAIL~,FccvQ6~~~~~~"au toassign"~);FccvQX~~~~ParseFlag~:=~~subs(~op(Parameters),~"parse"=false~,FccvQ1 ~~~~~~"parse"~);FccvQG~~~~Nargs~:=~nargs~-~nops(Parameters);FccvQap~~~~if~not~t ype([args[1]]~,~list(list)~)~then~ERROR(`list~of~lists~expected`,args)~fi;FccvQ D~~~~Eval~:=~()~->~op(~eval(~subs(~|frFccvQM~~~~~~~~~~~PIECEWISE=`context/fixpi ecewise`,FccvQ:~~~~~~~~~~~MATRIX=matrix,FccvQI~~~~~~~~~~~VECTOR=vector|hr~,~[ar 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xt/cmglobal`('a'),FccvQS~~~~~~~'`CM_Global`(b)'~=~`context/cmglobal`('b'),FccvQ S~~~~~~~'`CM_Global`(x)'~=~`context/cmglobal`('x'),FccvQL~~~~~~~'CM_GLOBALG'~=~ `context/CM_GLOBALG`,FccvQ1~~~~~~~~NULL~|hr,~Fccv-Fhhl6&F`eoQ9proc()~global~CM_ ASSIGN;FccvFa[lFabpQ'~~~~);FccvQ]o~~~~CodeString~:=~readlib('`context/BuildInpu tString`')(ProcBody);FccvQho~~~~CodeString~:=~readlib('`context/autoassign`')(C odeString,AutoAssignFlag);FccvQ>~~~~if~ParseFlag~=~true~then~FccvQP~~~~~~~~Code String~:=~`context/jointext`(~"~"~,FccvQR~~~~~~~~~~~~"if~true~then"~,~CodeStrin g~,~"fi"~);FccvQJ~~~~~~~~~parse(~CodeString~,~statement~);FccvF_evQ4~~~~~~~~Cod eString;FccvFaevFabpF/F/F/Fccv, I6SMARTPLOTFindCoords2DR6#'F[_sF]dm6)Fbdm%"pG%+candidatesG%(nextposG%*newcoords G%"XG%"YGFh`nF/C)>Fdx.%%_NoXG>Fiy.%%_NoYG-FSF6>F[inF@>Fhu-&F*6#.%&depthGF`_p?&F `tFK2FhuFa^nC(@$1FdfqF]hvFeat>F]fm-F`im6%F`fm&F`t6#;Fa^nFdfq<$FdxFiy@$/F]fmF\`l %%nextG-Fio6%FFF]fmF[in?(FboF[oF[o-%$minG6$Fa^nF]hvFdr@&Fdfn>Fdx&F[inFiaoFefn>F 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/-Fj[s6$-F]\s6#,&F^]oF[oFiyFcuF[oFg`lC%@%2FjdwF^`w>F`tFjdw>F`tF^`w?(F/F[oF[oF// &Fiy6$F`tF`tFg`l>F`t,&F`tF[oFcuF[o@$3/Fa_xF[oFdbwC$-Fjcn6$%LYou~obtained~the~ma trix~in~row~echelon~formGFiy>FjsFa^n@$3/FjsFg`lFdbw-Fjcn6$%Kwarning.~Matrix~is~ not~in~row~echelon~formGFiyF^\o@$/F]vF[oF\]sF^\o-Fjcn6#%boThis~is~the~end~of~th e~Interactive~Gauss~procedure.~Try~other~examples~GF/Fe]sF/F/, I1type/CM_MathFuncRFO6%Fc^sF,%#kkG6%%'systemG%)rememberGFjblF/C$@$F^[rC%>FboFd[ r>F`t-Fax6$/F]fmFbo-.F5F[_n@%54F`t-F:6$Fbo%)mathfuncG-F_o6#FdrF[^tF[^tF/F/F/6", IAcontext/cfparse/removeduplicatesR6#'%&clistGF]dm6'%%LocCG%&lastxGFbdmFcdmFddp FhdpFidpC+@$2-Feu6#FK""#-F_oF^cx>8'F]cx>F^fp-Faho6#;"""Fbcx>F`epFdjm?(8&FgcxFgc xFbcxFdr@%/&F]gp6#FjcxF`ep>&F^fpF^dxF\pC$>F`dxF]dx>F`epF]dx>8(7#-Fcw6$F`dx/Fjcx Ffcx?(FidpFgcxFgcxFidp32""!-Feu6#Fedx/&Fedx6#FgcxQ0Fhhp>Fedx&Fed x6#;F_cx!""?(FidpFgcxFgcxFidp3F\ex/&Fedx6#FhexFcex>Fedx&Fedx6#;Fgcx!"#FedxFidpF idpFidpFidp, 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I4context/InitializedFdrF/, I~op(2,args[i]))~thenFcbyQao~~~~~~~~~~~~~ WARNING(sprintf(~"value~of~%s~has~changed|+",~Labels[i]));FcbyF]cyF[evF\evF]evF ^evF_evF`evFaevFbevQ6~~~~~~~'f'=Labels[1],FcbyQ<~~~~~~~'ARGS'~=~op(Labels),Fcby FgevQL~~~~~~~'`CM_Global`(a)'~=~`CM_Global`('a'),FcbyQL~~~~~~~'`CM_Global`(b)'~ =~`CM_Global`('b'),FcbyQL~~~~~~~'`CM_Global`(x)'~=~`CM_Global`('x'),FcbyF\fv-Fh hl6&F`eoQ>proc()~global~locg,CM_ASSIGN;FcbyFa[lFabpF`fvFafvFbfvFcfvFdfvQfn~~~~~ ~~~~~~~~~~~~~~~~~"if~true~then"~,~CodeString~,~"fi"~);FcbyQG~~~~~~parse(~CodeSt ring~,~statement~);FcbyF_evFgfvFaevFabpF/F/F/Fcby, I+CM_GetVarsRF/F]asF\axF/C$>Fbo-%&minusG6$-Fdo6$FfjsF)<$%*constantsG.%*otherwis eG-F]ho6%R6$%$varGF^]tF/6$%)operatorG%&arrowGF/-%(dependsG6$FFa [y-Fc[y6$Fa[yFe^p-F26#-F5F^s-F26#-F5Fc^p>F]fm-F]ho6%F:FKFbim>Fafm-F_s6$7$F_[yFj hmF]fm@$-F:6$F`tFdfo>F`t-Fa_lFfu>Fhu-F]ho6%F:FKFhio>F[in-F]ho6%F:FKFcz@$2Fg`lF_ _p>F[in-Fio6$-FS6#.FgelF[in>Fiy-Fa_l6#Fbco>Fdx-F]ho6%F:FKFd\l@$2Fg`lF^yC$>Fdx-F 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jfnFj\o>Fa_rFg`l?(F[in,&FhuF[oF[oF[oF[oF^]oFdr@$0&Fadn6$F[inFhuFg`l>Fa_rF[o@$0& Fadn6$FhuFhuFg`lC$@$/Fa_rF[oC$-Fjcn6$%A~Perform~elimination~on~column~:GFhu-Fjc n6#%E************************************G@$0F^calF[oC)F^\o-Fjcn6(%+divide~rowG FhuF`_oFadnFa_oF^cal>Fadn-Fhin6%FadnFhu*$F^calFcuF^\oF`balFg\o@$FjfnF\`oF^\o@$F bcal?(F[inFfbalF[oF^]oFdr@$FhbalC(-Fjcn6*%-multiply~rowGFhuF`_oFadn%$~byG,$Fiba lFcuFc_oF[in>Fadn-Ff_o6&FadnFhuF[inF]ealF^\o-Fjcn6$%(to~get:GFadnFg\o@$Fjfn-F_o 6#%.See~You~laterGC->FdxFh^al?(F/F[oF[oF/Ff_al>FdxF[^o-Fjcn6%%fnThe~matrix~can~ be~further~reduced,~if~we~assume~for~exampleGF``al%3is~non~zero.~~ThenG-Fjcn6#% Dthe~matrix~in~row~echelon~form~is~:G>Fc`u-FdhnFjgt>&Fc`uFa`alF[o-Fjcn6#Fc`u>Fj sFc\uFg\o-Fjcn6#%dqIf~~you~need~to~infer~conditions~on~the~parameters~so~that~t he~system~is~consistent,~use~the~reduced~formG-FjcnFjgt>F^]oFf^al>Fa`uFh^al@$F_ `xC'>FhuFh^al?(F/F[oF[oF/3-F_[o6#-Fb[o6$FadnFhuFd[o>FhuFf[o>F[inF[o?(F/F[oF[oF/ 3/&FadnF`inFg`l2F[inFf^al>F[inFcin@$30F^halFg`l0F^halF[oC$-Fjcn6(%(The~rowGFhuF `_oFadn%2~is~multiplied~byG*$F^halFcu>Fadn-Fhin6%FadnFhuFjhalF^\o@$30FjsFc\u0F^ hnF[oC&-Fjcn6#Fhar>FadnFdfalF]galF^\o@$Fdfn-Fjcn6#%\oTo~solve~the~system~apply~ backsubstitution~to~the~reduced~matrix.GF^\o@$FaialC%-Fjcn6#%goThis~is~the~end~ of~gausselim~demonstration~procedure.~It~shows~the~steps~andG-Fjcn6#%eothe~sequ ence~of~elementary~row~operations~needed~to~obtain~the~row~echelonG-Fjcn6#%Nfor m~of~the~matrix.~Now~try~your~own~examplesGF/Fh`oF/F/, I0type/CM_IFBLOCKRF]asF/Fh`nF/-F:6$FK-Fcan6$FeanFbboF/F/F/6", I(CM_CODERF/F+F-F/C%>Fbo-Fdo6$Fbey-Fcan6$Fean<$Fhio.F`_l>Fbo-Fio6$RF]asF/F/F//F KFa[lF/F/F/Fbo-.F[p6#-Fjw6#-FaxF`\zF/F/F/6", I)chkrowecRF/6)F_]nFbdmF^bnFc]nFcdm%&indexG%)zeroflagGFf]nF/C(>FboF[^n>Fhu-F_dn Fiao>F[in-FcdnFiao>FdxF[o?(F`tF[oF[oFhuFdrC)>F]fmF[o?(F/F[oF[oF/3/&Fbo6$F`tF]fm Fg`l2F]fmF[in>F]fmFa^z@$0Fd]blFg`l@$0Fd]blF[o-F_oF]^n@$3/F]fmF[inFc]bl>Fiy%#onG 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I,rdgaussmenuRF/F/F/F/C.-Fjcn6#%en~*******************~~Operations~Menu~~****** *************G-F_en6#%W~1.~To~multiply~row~i~by~c~and~add~it~to~row~j~type:~|+G -Fjcn6#%=Rj=Ri*c+Rj;~~or~~Rj=c*Ri+Rj;G-F_en6#%J~2.~To~interchange~row~i~and~row ~j~type:|+G-Fjcn6#%7Ri<>Rj;~~~~~~or~~Ri=RjG-F_en6#%M~3.~To~multiply~a~row~i~by~ constant~c~type:|+G-Fjcn6#%:Ri=c*Ri;~~~~~or~~Ri=Ri*c;G-F_en6#%N~4.~To~correct~a ~wrong~selection~type:~undo;|+G-F_en6#%H~5.~To~quit~the~procedure~type:~~exit;| +G-F_en6#%jn~6.~To~display~the~row~echelon~form~of~the~matrix~type:~rowec;|+G-F _en6#%ao~7.~To~display~the~reduced~row~echelon~form~of~the~matrix~type:~>rec;|+ G-Fjcn6#%co******************************************************************** ****GF/F/F/F/, I3context/isparsableRFO6$Fc^sF]hmFh`nF/C'Fb^p-FS6#F\il>F`t-%*traperrorG6#-F\il6 #-Fa_l6%F^jyFKFabp@$/F`tFf]sF[^tF_bxF/F/F/Fbbp, I3context/multiparseRFO6$F_r%%TextGF-F/C%>F`tFbco@%-F:6$FK-Fcan6$Fdfo<%Fg[pF`_l Fa_l>F`t-Fjbo6$F`eoFa[l>F`t-Fjbo6$F`eoF\^n-%&parseGFfuF/F/F/Fcco, I*gaussmenuRF/F/F/F/C0F^\o-F_en6#%_oTo~proceed~select~the~appropriate~operation ~from~the~following~menu|+GFg\o-Fjcn6#%T*****************~Operations~Menu~***** ************G-F_en6#%V~1.~To~multiply~row~i~by~c~and~add~it~to~row~j~type:|+G-F jcn6#%>Rj=Ri*c+Rj;~~or~~~Rj=c*Ri+Rj;G-F_en6#%K~2.~To~interchange~row~i~and~row~ j~type~:|+G-Fjcn6#%9~~~~Ri<>Rj;~~or~~~Ri=Rj;G-F_en6#%K~3.~To~multiply~row~i~by~ constant~c~type:|+G-Fjcn6#%;~~~Ri=c*Ri;~~or~~~Ri=Ri*c;GFeiblFhibl-F_en6#%[o~6.~ To~display~the~row~echelon~form~of~the~matrix~type~:~rowec;|+G-Fjcn6#%[o******* *********************************************************GF/F/F/F/, I1type/CF_CODETYPERFOF/F-F/-F:6$FK<)F]dm-%*identicalG6#Fa`z-F__cl6#Fcaz-F__cl6# Ff]q-F__cl6#F_[q-F__cl6#Fb`z-F__cl6#Fg^qF/F/F/F`[q, I&stack=6"F_coE\[l'FDRF/6$FbdmFhhx6#%aoCopyright~(c)~1990~by~the~University~of~ Waterloo.~All~rights~reserved.GF/C%>&F`tF\_blFdas?(FboF[oF[oFdasFdr>&F`t6#,(Fda 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I3context/autoassignR6$'Fg`n<$FdfoFa_l'%+AssignFlagGFj]dl62%&LabelG%)CodeBodyG% *LocStringG%&StatsGFbdmFd]nFcdm%,laststatendG%(statendG%&NamesG%*NamesSubsGFhcb l%%ProcG%)ProcBodyGFfjx%)LastStatGFh`nF/C(-FS6#F``lF[cv>%3type/CM_AssignmentGRF ]asF/Fh`nF/3-F:6$FKFd^t/Fd[r.%*CM_ASSIGNGF/F/F/>Fa_r-Fhjbl6#-Fa_l6%FgcyFKFabp>F c`u--FS6#.FiclFfdr@%5/FFc`u-FhhlFb\cl-F_oFb[l>F]fmFc`u>Fc`uF\p>F[in-%+SearchTextG 6$Q*CM_ASSIGN6"F]fm?(F/F[oF[oF/2Fg`lF[inC(>Fc`u6$Fc`u-Fhn6$F]fm;F[o,&F[inF[oFcu F[o>F]fm-Fhn6$F]fm;F[inFcu>Fdx-F[ebl6$Q#:=FeedlF]fm@$/FdxFg`l-Fcdo6$%3invalid~a ssignmentGF]fm>F]fm-Fhn6$F]fm;,&FdxF[oFa^nF[oFcu>F[in-F[eblFcedl>Fc`u-Fhhl6%F`e oFc`uF]fmC--F26#-F:6$Fc`uFdfo@$1-Fbedl6$FdedlFc`uFg`lC+>FdxFg`l>Fiy-%'lengthGFh fal>FezFg`l>Fj]nFg`l?(F[inF[oF[oFiyFdrC%@$/-Fhn6$Fc`u;F[inFcinFffdlF`iv@$32F[oF [in/-Fhn6$Fc`u;F^fdlF[inQ#::FeedlF`iv@$-F`fm6$-Fhn6$Fc`uF[inFeeblC%>FdxFc_u>Fez Fj]n>Fj]nF[in-F26#32Fg`lFdx/Fj]nFiy@%F]^oC$>F]vF\p>FjsFc`uC%-F26#32Fg`lFez2FezF 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(FboF[oF[oFjuFdrC'>F`\il<$Fd_flFgbfl>8`oFgo?(F`tF[oF[o-Feu6#F`\ilFdrC%@$0-Ff_n6 $&F`\ilFfuF_rFg`l>Fcbil-Fjdo6$FcbilFh]il@$0-Ff_n6$F\cilFi]wFg`l>Fcbil-Fjdo6$Fcb ilF_^il@$0-Ff_n6$F\cilFcaflFg`l>Fcbil-Fjdo6$FcbilFf^il@(-F`fm6$FcaflFcbil>8ao7% F_rFi]w-F\jfl6$F_\flFcafl-F`fm6$Fi]wFcbil>Fbdil7%F_r-Fax6$/FcaflFi]wFidglFi]w-F `fm6$F_rFcbil>Fbdil7%F\dglF_rFi]w>F`\hl-%'plot3dG6'FbdilF`wFi\lFe\hl/%&styleG%* WIREFRAMEG>F`\ilFgo>F\\hl#Fa[uFd[u>&8inF[_n-Fceil6(7%Fe[ilFj[ilF^\ilF`wFi\l/Ffe il%%LINEG/%*thicknessGFdfqFj`hl>F]fmFa^z>F[in<%&Fa\hlF]^n&Fa\hlF`^n&F]filF]^n>F biflFa^n>Fg\gl-Ff[n6$-Fd[gl6$FccrFa^nFdfo>Fb]glFb\p?(F`tF[oF[oFd]glFdrC$>Fh]glF i]gl@$F`^gl>Fb]glFg_gl>Fg\gl-F`go6#Fb]gl>FbiflFg`lF[]hl@'F^`hl>Fa`hl-%+textplot 3dG6%7&Fc\u!#:"#SFe`hlFf`hlFj`hlFeahlC&>Fa`hl-F^hil6%7&Fc\uFahilFbhilF[bhlFf`hl Fj`hl>Fbbhl-F^hil6%7&Fc\uFahilFbhilF`bhlFcahlFj`hl>F\ahl-F^hil6%7&"#IF^e[l!#SFg \glFh_hlFj`hl>8gn-Fceil6(<$7%FijflF^[glFfhhl7%,&FijflF[o$Fa^nF\oF[o,&F^[glF[oFj 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%5Fi[fl55Fg[fl-F:6$F[^nFcho-F:6$F[^nFa^fl@%Fa_jl>Fjs<#F[^n>FjsF[^n-Fcdo6#%ZA~se t~of~equations~expected.~Type~?solveqns~for~more~infoGF[\t-F_o6#%UTwo~arguments ~expected.~type~?solveqns~for~more~infoGC)-F_en6#%[oPlease~enter~the~set~of~equ ations.~For~example~type:~2*x+2*y=1;|+G>FboF[o>Fadn%&beginG?(F/F[oF[oF/0FadnFg` lC&-F_en6#%MPlease~enter~an~equation.~To~finish~type~0;|+G>FadnFien?(F/F[oF[oF/ 34-F:6$FadnFa^flFe`jlC$-F_en6#%AAn~answer~is~expected.~Try~againG>FadnFien@$Fe` jlC$>F_^uFadn>FboFdcu-F_en6#%jnPlease~enter~the~set~of~variables.~For~example:~ vars:=[x,y,z];|+G>F^drFien>F_frFhau>Fjs-Ff[n6$FjsF]dm>F^dr-Ff[n6$F^drF]dm>F^dr- Ff[n6$F^drFcho?(FboF[oF[o-Fj^n6#FjsFdrC'>F`wF_^u>F`w/-Fi_n6#F`w-Fb`nF_cjl>Fd]l< #-Fjw6#F^cjl?(F`tF[oF[o-FeuFh^lFdr@$-F:6$&Fd]lFfuF[`fl>F`w/,&F^cjlF[oFjcjlFcu,& F`cjlF[oFjcjlFcu>F_^uF`w?(F`tF[oF[o-Fj^nFhhuFdrC%>Fa`uFg`l?(FboF[oF[oFhbjlFdrC% >F`wF_^u>Fd]lFbcjl?(F]fmF[oF[oFfcjlFdr@$0-Ff_n6$F]duFdhuFg`l>Fa`uF[o@$/Fa`uFg`l -Fcdo6%%-The~variableGFdhu%Fis~not~found~in~the~list~of~equationsG>Fj]n-F\]n6$F jsF^dr-F_en6#%CSelect~one~of~the~following~modes|+G-F_en6#%<~~~~~1.~Demonstrati on~mode|+G-F_en6#%6~~~~~2.~No-Step~mode|+G>FadnFien@%F[\xC,-Fjcn6#%@The~origina l~system~is~given~byG?(FboF[oF[o-FeuFibjlFdr-Fjcn6#F_^uFg\o@$F[gnFbgn>Fc`uF]]u> F]vFd]u?(FboF[oF[o,&Fc`uF[oFcuF[oFdrC*?(F]fmFboF[oFc`uFdr@$0&Fj]n6$F]fmFboFg`lC $@$0F]fmFboC%>Fj]n-F`cr6%Fj]nF]fmFboF^\o>Fi\lFa^blFeat>F]crFg`l?(F`tFdcuF[oFc`u Fdr@$0&Fj]nFdeblFg`l>F]crF[o@$Fh_uC$@$F\cr@$2Fa^nFhbjlC$-Fjcn6$%_oBy~performing ~~a~set~of~algebraic~operations,~eliminate~the~variableGFbju-Fjcn6#%6to~obtain~ the~system~G@$0Fi_uF[oC%>FftFi_u>Fj]nF\`u@$/FhbjlFa^n@%/Fi\lFa^blC%-Fjcn6&%4div ide~the~equationG&FjsF`^nFa_oFft-Fjcn6$%'to~getG*&F\jjlF[oFftFcuF^\oC%-Fjcn6&F[ jjl&FjsF]^nFa_oFft-Fjcn6$F_jjl*&FdjjlF[oFftFcuF^\o?(F`tFdcuF[oFc`uFdr@$F`hjlC%> Fd]l,$FahjlFcu>Fj]n-Ff_o6&Fj]nFboF`tF\[[m@$Feijl@%FgijlC%-Fjcn6&%2Multiply~equa tionGF`jjlFa_oFd]l-Fjcn6%%4and~add~to~equationGFdjjlF_jjlF^\oC%-Fjcn6&Fe[[mFgjj lFa_oFd]l-Fjcn6%Fh[[mF\jjlF_jjlF^\o-%'ToeqnsG6$-Fh^u6%Fj]n;FdcuFjiu;F[oF\juF^dr Fg\o@$F[gnFbgn?(F`tF[oF[o,&F]vF[oFcuF[oFdr@$-F_[o6#-F`]u6$Fj]nF`t@$4-F_[o6#-F`] u6$Fj]n,&F`tF[oF[oF[oC&>FboF[o?(F/F[oF[oF//&Fj]n6$FboFc][mFg`l>FboFdcu@$2F[oFbo >Fj]n-F`cr6%Fj]nFboF[o?(FboFa^nF[oFc`uFdr@$0Fh][mFg`l>Fj]n-Ff_o6&Fj]nF[oFbo,$*& Fh][mF[o&Fj]n6$F[oFc][mFcuFcu@$30&Fj]n6$Fc`uFc`uFg`l0F]_[mF[o>Fj]n-Fhin6%Fj]nFc `u*$F]_[mFcu-Fdit6$Fj]nF^drC(>Fc`uF]]u>F]vFd]u?(FboF[oF[oF^gjlFdrC'?(F]fmFboF[o Fc`uFdr@$FbgjlC$@$FggjlC$>Fj]nFjgjlF^\oFeat>F]crFg`l?(F`tFdcuF[oFc`uFdr@$F`hjl> F]crF[o@$Fh_u@$F`ijl>Fj]nF\`u?(F`tFdcuF[oFc`uFdr@$F`hjl>Fj]nF^[[m?(F`tF[oF[oFg\ [mFdr@$Fi\[m@$F^][mC&>FboF[o?(F/F[oF[oF/Fg][m>FboFdcu@$F\^[m>Fj]nF^^[m?(FboFa^n F[oFc`uFdr@$Fb^[m>Fj]nFd^[m@$F[_[m>Fj]nFa_[m-F`iuFe_[mF/Fe]sF/%(backsubGRF/6FFb dmFd]nF^bnFc]nFcdmFhitFiitF`bnFjitF[jtF\jtF]jtF_]nF`]nF_r%"BGFa]al%&AUG11GF^jtF ^etFe]nF_jtF`jtFajtFabnFbbnF_bnFcjtF]bnFgit%&flag1GF]et%'chksetGFh]jl%&temp2GFd hpFf]nF/C.-Fjcn6#%apThe~purpose~of~this~procedure~is~to~demonstrate~the~steps~i nvolved~in~solving~a~systemG-Fjcn6#%hpof~linear~equations~using~the~back~substi tution~algorithm.~If~you~select~the~interactive~modeG-Fjcn6#%ipthen~you~can~imm ediately~check~if~you~have~learned~the~backsubstitution~process.~The~procedureG -Fjcn6#%ipalso~displays~the~solution~immediately~using~back~substitution~if~you ~select~the~no~step~mode.GF^\oFacelFdcelFgcelFjcel>F^]oFienF^\o@(/F^]oF[oC>@%Fi `elC(-F_en6#%JPlease~enter~the~augmented~matrix~or~the|+G-F_en6#%Jrow~echelon~f orm~of~the~augmented~matrix|+GF^\oF^ael>FezFien?(F/F[oF[oF/4-F:FhcnC&-F_en6#%MM atrix~expected.~eg:~matrix([[1,2],[2,3]]);|+G>FezFien>FhfnFez@$F[gnFbgn@%F^bel> FezF[^nC$-Fjcn6#%0Matrix~expectedGFbgn>Fj]nFgjt>FezFgcn>Fhu-Fj^n6#-F`]u6$FezF[o >F[in,&-Fj^n6#-Fb[oFbe[mF[oFcuF[o>F]v-FchoF`_p>Fi\l-F]\s6#-Fe^u6$-Fh^u6%Fez;F[o FhuFj^u-Ff[n6$F]vFd^n>Fd]lF[f[m>Fccr-Fh^u6%FezFaf[m;FcinFcin-Fjcn6$%FThe~augmen ted~matrix~you~entered~is~:G/Fez-F]\sF]`nFg\o@$FjfnFbgn-Fjcn6#%hoThe~above~augm ented~matrix~is~equivalent~to~the~following~system~of~equationsG?(FboF[oF[oFhuF dr-Fjcn6#/&Fi\lF[hfl&Fez6$FboFcinFg\o@$FjfnFbgn>FjuFedn>FezFchnF^\o@$/FjuF[oC-- Fjcn6#%jnThe~given~augmented~matrix~is~not~in~Row~Echelon~form.~Perform~G-Fjcn6 #%`oGauss~elimination~to~obtain~the~augmented~matrix~in~Row~echelon~form:G-Fjcn 6#/%8Row~echelon~form~of~AUGGF]g[mFg\o@$FjfnFbgn>Fi\lF[f[m-Fjcn6#%fnTherefore~t he~original~system~of~equations~is~equivalent~toG-Fjcn6#%Jthe~following~reduced ~system~of~equationsG?(FboF[oF[oFhuFdrFcg[mFg\o@$FjfnFj\o>Fa_rF_f[m>FadnFie[m?( FboF[oF[oF[inFdr>&FadnFiaoF_\fl>F^al-Fi`u6$Fa_rFadn@%/-FcjnFfdrFbjnC9>FboFhu?(F /F[oF[oF/-F_[o6#-Fb[o6$FezFbo>FboFhau>Fez-Fh^u6%FezFb\v;F[oFcin>FhuF_e[m>F[inFd e[m@$2FboFhuC'-Fjcn6#%TRemove~all~trivial~equations~from~the~system~to~getG>Fi\ lF[f[m?(FboF[oF[oFhuFdrFcg[mFg\o@$FjfnFbgn@$Fjfn-F_o6#%4Execution~SuspendedG@%/ FhuF[inC(-Fjcn6#%eoThe~number~of~independent~equations~=~The~number~of~independ ent~variables,G-Fjcn6#%YTherefore~this~system~of~equations~has~a~unique~solutio nGF^\o-Fjcn6#%jnApply~back~substitution~algorithm~and~find~the~Unique~solution. GFg\o@$FjfnFbgnC)-Fjcn6#%TFrom~the~reduced~row~echelon~form~of~the~matrix,~weG- Fjcn6#%Ycan~deduce~that~the~system~has~infinitely~many~solutionsGF^\o-Fjcn6'%.S ince~we~haveGF[in%3variables~and~onlyGFbjn%6independent~equationsG-Fjcn6%%7we~w ill~have~to~chooseG,&F[inF[oFbjnFcu%1free~variables~.GFg\o@$FjfnFbgn-Fjcn6#%hnR ecall~that~the~original~system~of~equations~is~equivalent~toG?(FboF[oF[oFhuFdrF cg[mFg\o@$FjfnFbgn>FftFie[m?(FboF[oF[oF[inFdr>&FftFiaoF_\fl>Fju-Fcho6$F[inFg`l> FhcrF[in?(F]fmFbjnFcuF[oFdrC->FboF[o?(F/F[oF[oF//&FezFdgjlFg`l>FboFdcu>&FjuFiao F[o?(F`tFdcuF[oF[inFdr@$0&FjuFfuF[oC%>&FftFfu&Fj]n6#,(F[inF[oFhcrFcuF[oF[o>Fa_\ mF[o>Fhcr,&FhcrF[oFcuF[o>Fc`uFg`l?(F`tFdcuF[oF[inFdr>Fc`u,&Fc`uF[o*&&Fez6$F]fmF `tF[oFd_\mF[oF[o>F`^\m*&,&&Fez6$F]fmFcinF[oFc`uFcuF[oFj^\mFcu@%0FboF[inC(@%/F]f mFbjnFeeuFheu?(F`tFdcuF[oF[inFdr-Fjcn6#/FdaglFd_\mF^\o-Fjcn6$Fbfu/&F^alF[_nFe`\ m-Fjcn6$Fgfu/F_\flF`^\mF^\oC$-Fjcn6#%BBy~using~the~last~equation~we~getG-Fjcn6# /&F]vF`_p&FftF`_p-Fjcn6$%Fi[hlFie[m?(F]fmFboF[oF[inFdr>&Fi[hlF[_n&FezFjdu@$34Fic\m0 -Fj[s6$Fi[hlFa^nFg`lC(-Fjcn6%%BThe~system~is~inconsistent~unlessGF[d\m%/~is~equ al~to~0GFg\o-Fjcn6%%,If,~howeverG/F[d\mFg`l%K~the~system~is~consistent~and~the~ solutionG-Fjcn6#%,is~given~byG>F[d\mFg`l-Fjcn6#-&Fehn6#Fja[mF]`nF^\o-Fjcn6#%boT his~is~the~end~of~the~demonstration~mode~of~backsub~procedure.~You~mayG-Fjcn6#% aotry~other~examples~to~learn~how~to~solve~a~system~using~this~algorithmG/F^]oF a^nCV>Fj]nFgjt>F]cr-Fcho6#7,FbhelFdhel%#x3G%#x4G%#x5G%#x6G%#x7G%#x8G%#x9G%$x10G @%Fi`elC'-F_en6#%VPlease~enter~an~augmented~matrix~in~row~echelon~form|+GF^\oF^ ael>FezFien?(F/F[oF[oF/F]d[mC&F`d[m>FezFien>FhfnFez@$F[gnFbgn@%F^bel>FezF[^nC$F id[mFbgn>FezFgcn>FhuF_e[m>F[inFde[m>F]cr-Fh^u6%-Ff[n6$F]crFd^nFj^u;F[oF[o>F^drF chn>Fi\l-F]\s6#-Fe^u6$F_f[mFih\m>Fd]l-F]\s6#-Fe^u6$-Fh^u6%F^drFaf[mFj^uFih\m>Fc crFff[m>FjuFedn@$F]h[mC(-Fjcn6#/%HThe~augmented~matrix~you~entered~is~AUGGF]g[m F^\oF_g[m?(FboF[oF[oFhuFdrFcg[mFg\o@$FjfnFbgn?(FboF[oF[o-F_dnFhhuFdrC%>F`tF[o?( F/F[oF[oF/3/&F^drFafmFg`l2F`t-FcdnFhhu>F`tFc][m@$31F`tF\[]m0Fjj\mFg`l>F_fr-Fhin 6%F^drFbo*$Fjj\mFcu-F_en6#%PIs~this~augmented~matrix~in~row~echelon~form~?|+G-F jcn6#F\g[m-F_en6#%5answer~~yes;~or~no;|+GF^\o>F^]oFien?(F/F[oF[oF/45/F^]o%$yesG /F^]oF[awC&-F_en6#%;Please~answer~yes;~or~no;|+G>F^]oFien>FhfnF^]o@$FjfnFbgn>Fj uFedn@)3Fb\]m/FjuFg`l-Fjcn6%%;Good,~the~augmented~matrixGFez%8~is~in~row~echelo n~formG3Fb\]mF]h[m-Fjcn6%%@Incorrect.~The~augmented~matrixGFezF]en3Fd\]mF]h[m-F jcn6%%;Good!~The~augmented~matrixGFezF]en-Fjcn6%Fg]]mFezFc]]mFg\o@$FjfnFbgn@$F] h[mC3-F_en6#%XWhat~would~you~do~next?~Choose~from~the~following~menu|+G-F_en6#% jn~~~~1.~Perform~Gauss~Elimination~to~find~the~row~echelon~form.|+G-F_en6#%Y~~~ ~2.~Write~the~associated~system~of~linear~equations.|+G-F_en6#%6~~~~|+GF^\o>F^]oFien>FhfnF^]o@$FjfnFbgn?(F/F[oF[oF/30F^]oF[o0F^]oFa^nC&>FhfnF^ ]o@$FjfnFbgn-F_en6#%=Please~choose~from~1;~or~2;|+G>F^]oFien@%Fac[mC$-Fjcn6#%IG ood!~One~must~find~the~row~echelon~formG-Fjcn6#%Qbefore~applying~the~back~susbt itution~algorithm.GC$-Fjcn6#%NIncorrect.~One~must~find~the~row~echelon~formGF`` ]m-Fjcn6#%RThe~row~echelon~form~of~the~augmented~matrix~is~:G-Fjcn6#/Fez-F]\s6# F_frFg\o-Fjcn6#%inWe~will,~however,~for~the~rest~of~the~procedure~use~the~matri xG-Fjcn6#-F]\sFhhuFg\o@$FjfnFbgn>FboFhu?(F/F[oF[oF/-F_[o6#-Fb[o6$F^drFbo>FboFha u>Fhu-Fj^n6#-F`]u6$-Fh^u6%F^drFb\vFij[mF[o>F[in,&-Fj^n6#-Fb[o6$F^drF[oF[oFcuF[o @$2FboF_e[m>Fi\lFci\m>FebflFgo-F_en6#%WWrite~down~an~equivalent~system(reduced) ~of~equations|+G-F_en6#%Wbased~on~the~row~echelon~form~of~the~augmented~matrix| +GF^\o-Fjcn6#/%YRecall~that~the~row~echelon~form~of~the~augmented~matrixGFda]m> FhcrF[o?(FboF[oF[o-FcjnFhhuFdrC)@%FdfnC$-F_en6#%doEnter~the~first~equation,~usi ng~row~echelon~form~of~the~augmented~matrix|+G-F_en6#%fnUse~ONLY~the~variables~ x1,x2,..etc.~example:~>~x1~+~x2~=0;|+G-F_en6$%APlease~enter~equation~number~%d| +GFbo>&F`wFiaoFien?(F/F[oF[oF/4-F:6$Fhd]mFa^flC&>FhfnFhd]m@$FjfnFbgn-F_en6#%LAn ~Equation~is~expected.~eg:~>~x1~+~x2~=0;|+G>Fhd]mFien>Fa\hl,&-Fi_n6#Fhd]mF[o-Fb `nFge]mFcu>F]gfl,&F^c\mF[o&F^drFhg[mFcu@&0Fa\hlF]gflC&-Fjcn6$%Fhcr,&FhcrF[oF[oF[o0FboFbjnC$-Fjcn 6#%=This~is~the~correct~equationGF^\o>Febfl-Fjdo6$Febfl<#Fhd]m>Fi\lFci\m>FjbflF go?(FboF[oF[oFhuFdr>Fjbfl-Fjdo6$Fjbfl<#/Ffg[mF[f]m@%/-Feu6#-F_dy6$FjbflFebflFg` lC%-Fjcn6#%MGood~.~You~gave~the~correct~set~of~equationsG-Fjcn6#%RThe~original~ system~of~equations~is~equivalent~toG?(FboF[oF[oFhuFdr-Fjcn6#Fhg]mC'-Fjcn6#%STh e~following~system~of~equations~can~be~extractedG-Fjcn6#%Rfrom~the~row~echelon~ form~of~the~augmented~matrixG?(FboF[oF[oFhuFdrFgh]mF^\o-Fjcn6#%Xwhich~is~equiva lent~to~the~original~system~of~equationsGFg\o@$FjfnFj\o-F_en6#%XBased~on~the~ro w~echelon~form~of~the~augmented~matrix,|+G-F_en6#%hnIs~the~above~reduced~system ~of~equations~always~consistent?~|+G-F_en6#%4Answer~yes;~or~no;|+G>F^]oFien?(F/ F[oF[oF/F`\]mC&Ff\]m>F^]oFien>FhfnF^]o@$Fjfn-F_oFaht>Fa_rF_f[m@%F\j[mCC@%Fb\]m- Fjcn6#%?Yes,~the~system~is~consistent.G-Fjcn6#%TIncorrect.~This~is~a~consistent ~system~of~equationsGF^\o-F_en6#%hnSo~the~system~is~consistent,how~many~solutio ns~does~it~have?|+G-F_en6#%I~~~1.~The~system~has~a~unique~solution.|+G-F_en6#%P ~~~2.~The~system~has~infinitely~many~solutions|+G-F_en6#%6~~~| +GF^\o>F^]oFien>FhfnF^]o@$FjfnFf[\m@%Fj[\mC,@%Fac[mC$>Fa\hlF[o-Fjcn6#%SCorrect. ~The~system~has~a~unique~solution~because,G-Fjcn6#%fnIncorrect~answer.~This~sys tem~has~a~unique~solution~becauseG-Fjcn6#%eothe~number~of~independent~equations ~=~The~number~of~independent~variables,GF^\o-F_en6#%WHow~would~you~start~the~ba ck~substitution~algorithm~?|+G-F_en6$%H~~~1.~Find~x%d~using~the~last~equation|+ GF[in-F_en6#%H~~~2.~Find~x1~using~the~first~equation|+G-F_en6#%2~~~type~1;~or~2 ;|+G>F^]oFien>FhfnF^]o@$FjfnFbgnC5@%Fac[m-Fjcn6#%gnIncorrect.~The~system~has~in finitely~many~solutions~because,G-Fjcn6#%]oCorrect~answer.~This~system~has~infi nitely~many~solutions.~becauseG-Fjcn6#%`pthe~number~of~independent~equations~is ~less~than~the~number~of~independent~variables,GF^\o-F_en6#%QSince~the~system~h as~infinitely~many~solutions,|+G-F_en6#%Ihow~many~free~variables~must~be~chosen ?|+G>F^]oFien>FhfnF^]o@$FjfnFbgn@%/F^]o,&F[inF[oFjc]mFcu-Fjcn6%%7Yes,~we~have~t o~chooseGFa_^m%7indpendent~variable(s)G-Fjcn6%%=Incorrect.~We~have~to~chooseGFa _^m%8independent~varaible(s)GF^\o>F]fmF[o?(F/F[oF[oF//&F^dr6$Fjc]mF]fmFg`l>F]fm Fa^z-F_en6#%;What~would~you~do~next~?~|+G-F_en6$%N~~~1.~Use~the~last~equation~t o~determine~x%d|+GF]fmFb]^mFe]^m>F^]oFien>FhfnF^]o@$FjfnFbgnF^\o@%Fac[m-Fjcn6#% YYes.~This~is~the~idea~of~the~back~substitution~algorithmG-Fjcn6#%`oNo.~The~bac k~substitution~algorithm~determines~x[n],x[n-1]..x[2],x[1]GF^\o-Fjcn6#%ioNow~we ~are~ready~to~apply~back~susbtitution~to~the~reduced~system~of~equationsG>F^alF ci\m?(FboF[oF[oFjc]mFdr-Fjcn6#/&F^alF[hflF[f]mF^\o>FftFie[m?(FboF[oF[oF[inFdr>F `^\m&F]crF[hfl>FjuFb^\m>F_ewFie[m>FhcrF[in>F]gflF[o?(F]fmFjc]mFcuF[oFdrC.>FboF[ o?(F/F[oF[oF//&F^drFdgjlFg`l>FboFdcu>F]_\mF[o>F^ewF[o?(F`tFdcuF[oF[inFdr@$F`_\m C%>Fd_\mFe_\m>FhcrFj_\m>Fa_\mF[o>Fc`uFg`l?(F`tFdcuF[oF[inFdr>Fc`u,&Fc`uF[o*&&F^ drFa`\mF[oFd_\mF[oF[o>F`^\m*&,&&F^drFf`\mF[oFc`uFcuF[oFfb^mFcu@%Fh`\mC+@$0Fa\hl F[oC$?(F`tFdcuF[oF[inFdr@$3/Fa_\mF[o0Fi`glF[oC*-F_en6#%[oAssign~arbitrary~value (s)~from~the~set~|fr_t[1],_t[2],_t[3],....|hr|+G-F_en6#%Xto~the~free~variable(s )~starting~with~_t[1]~and~so~on.|+GF^\o-F_en6$%QAssign~an~arbitrary~value~to~th e~variable~x%d~=|+GF`t>F^]oFien>Fi`glF[o@%/,&F^]oF[o&Fj]n6#,&F`tF[oFboFcuFcuFg` l-F_en6%%HYou~made~a~correct~assignment~x%d~=~%a|+GF`tFd_\m-F_en6%%=A~better~ch oice~is~x%d~=~%a|+GF`tFd_\mF^\oFg\o-F_en6$%FEquation~%d~of~the~reduced~system~i s|+GF]fm-Fjcn6#/&F^al6$F]fmF[oFic^mF^\o-F_en6$%IWhat~is~the~value~of~the~variab le~x%d~?|+GFbo>F^]oFien@$3/-Fffm6$F]gflFa^nF[o/Fa\hlFg`lC%-Fjcn6#%WYou~do~not~n eed~to~simplify~the~answers.~You~may~enterG-Fjcn6#%Wthe~answer~as~a~Maple~expre ssion~eg:~2~-~3*5/2~+~4*a~;G>F]gfl,&F]gflF[oF[oF[o@$-F:6$F^]oFa^fl>F^]o-Fb`n6#F ^]o@%/F^]oF`^\m-Fjcn6#%;This~is~the~correct~answerGC$-Fjcn6#/%Bincorrect.~The~c orrect~answer~is~G-Fbal6#F`^\m>Fa\hlFg`lC)F^\o-F_en6$%F^]oFien@$Fhg^m>F^]oF[h^m>FhfnF^]o@$FjfnFbgn@%/F^]oF_b\m-Fjcn6%%7Corr ect.~We~can~obtainG/&F]cr6$F[inF[oF_b\m%8using~the~last~equationG-Fjcn6$%8Incor rect.~The~value~ofGFhi^m@$0FboFjc]m-Fjcn6$%6The~new~solution~is~:GFftFg\o@$Fjfn Fb`rFcgu?(FboF[oF[oF[inFdrFeb\mFg\o-Fjcn6#%Uand~the~solution~to~the~reduced~sys tem~of~equations:G?(FboF[oF[oFhuFdrF[c\mF^\oFge\m-FjcnFec\mC&@%Fd\]m@%3-F:6$&F^ dr6$Fjc]mFcinFb`al0Fa[_mFg`l-Fjcn6#%Oyes,~the~system~of~equations~is~not~consis tentG@$34F_[_mFc[_mC(-Fjcn6$%Ryes,~the~system~of~equations~is~not~consistent~if GFc[_mFg\o-Fjcn6%%,if,~howeverG/Fa[_mFg`lFfe\mFge\m>Fa[_mFg`l-Fjcn6#-F^f\mFhhu@ %F^[_m-Fjcn6#%Nno,~the~system~of~equations~is~not~consistentG@$Fh[_mC(-Fjcn6$%Q no,~the~system~of~equations~is~not~consistent~ifGFc[_mFg\oF^\_mFge\m>Fa[_mFg`lF c\_mF^\o-Fjcn6#%gnIn~the~above~system~we~observe~that~one~or~more~statement(s)G -Fjcn6#%Tlead~to~an~inconsistent~system~of~linear~equations.GF^\o-Fjcn6#%\pThis ~is~the~end~of~the~interactive~mode~of~backsub~procedure.~Try~other~examples.G/ F^]oFdfqC+@%Fi`elC'-F_en6#%enPlease~enter~the~row~echelon~form~of~the~augmented ~matrix|+GF^\oF^ael>FezFien?(F/F[oF[oF/F]d[mC&F`d[m>FezFien>FhfnFez@$F[gnFbgn@% F^bel>FezF[^nC$Fid[mFbgn>FezFgcn>Fa_rFez>Fhu-Fj^n6#-F`]u6$Fa_rF[o>F[in-Fj^n6#-F b[oF___m>F]v-Fcho6#F^fdl>Fjs-Fh^u6%Fa_rFaf[mF]fdl>Fft-F]\s6#-Fe^u6$FjsFbf[m@%/- FcjnFibjlF]j[mC'-Fjcn6#%>The~solution~to~the~system~:~G?(FboF[oF[oFhuFdr-Fjcn6# /&FftF[hfl&Fa_r6$FboF[inF^\o>Fft-F^f\m6#-FdhnFfdr-Fjcn6$%.is~given~by~:GFftC&>F a_rF`a_m-Fjcn6#%^oThe~echelon~form~of~the~augmented~matrix~that~defines~the~sys tem~isG-FjcnFfdr@%3-F:6$&Fa_rF`inFb`al0F^b_mFg`lF^d\m@$34F\b_mF_b_mC(-Fjcn6%F`e \mF^b_mFae\mFg\o-Fjcn6%Fde\m/F^b_mFg`lFfe\mFge\m>F^b_mFg`l-Fjcn6#-F^f\mFfdrF/Fe ]sF/FcbrRF/6&F]bnF_]nF`bnF`]nF/F/C1F^\o@'Fi`elC'-F_en6#%LPlease~enter~the~matri x~you~want~to~reduce|+G-Fjcn6#%enFor~example~type:~AUG:=matrix([[1,2,3],[0,1,3] ,[0,0,1]]);|+G>F]fmFien?(F/F[oF[oF/4-F:6$F]fmFd^nC&-F_en6#%F]fmFien>FhfnF]fm@$FjfnFbgn-Fjcn6$%F`t-Ff[nF_bel>Fhu-Ff[n6$F_^nFd^n>F]fm-F\cel6$F`tFhu>F]fmFid_m-Fjcn6#%^p The~purpose~of~this~procedure~is~to~demonstrate~the~steps~that~reduce~a~matrix~ intoG-Fjcn6#%_pits~reduced~row~echelon~form.~By~selecting~the~interactive~mode, ~you~can~immediatelyG-Fjcn6#%fpcheck~if~you~have~learned~the~process~of~reducin g~a~matrix~to~its~reduced~row~echelon~form.G-Fjcn6#%gpThe~nostep~mode~yields~th e~reduce~row~echelon~form~without~exhibiting~any~intermediate~stepsGF^\oFacelFd celFgcelFjcel>FboFien>FhfnFbo@$FjfnFb`r@(Fdfn-FjanF[_nFefn-Fa\rF[_nFh^z-Fjcn6#- FabrF[_nF/F_felF/F/, I,ShowExampleRF/F/F/F/C,-Fjcn6#%XHere~is~an~example~to~show~how~to~extract~the~ reduced~~G-Fjcn6#%Ysystem~from~the~row~echelon~form~of~the~augmented~matrixGF^\ o-Fjcn6#%PIf~the~augmented~matrix~in~row~echelon~form~is~G-Fjcn6#/F`bn-F]\s6#-F d^n6#7$7%F[oFa^nFg`l7%Fg`lF[oF[oF^\o-Fjcn6#%fnthen~the~reduced~matrix~is~equiva lent~to~the~reduced~systemG-Fjcn6#%0eq1:=x1+2*x2=0;G-Fjcn6#%+eq2:=x2=1;GFg\oF/F /F/F/, I2type/CM_PIECEWISERF]asF/Fh`nF/-F:6$FK-Fcan6$Fean<$%*PIECEWISEGFb\zF/F/F/6", I1context/jointextR6#'F``dlFdfo6%%2separation_stringG%(NewTextG%)TextListGF-F/C )Fb^p>FboFK>F]fm-Fa_l6#&F\^n6#;Fa^nFcu-F26#-F:6$7#-FjwF[_n-F]dm6#Fh\t@&2FjetF[o -F_oFchy2FjetFa^n-F_o6#-Fjw6$F[oF]fm>F`t-Fio6$RF]asF/F/F/6$FKT#F/F/6$Fci_mFboF] fm-F`go6#-Fjw6$FjnF`tF/F/F/6", 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>F]fm-F`im6$RF]asF/F\eyF/52F[oFb`bl-F:6$Fd[rF^bxF/F/F/F]fm>F]fm-Fio6$RF]as6#Fij `mF/F/C%-F26#F^[r>Fbo-Fdo6$FKFi\t@%/FcfqF[o-Fd[r6#-Fjw6$F[oFboF\pF/F/F/F]fmC&>F ]fm-F]ho6%F:Fbo-Fb^q6#F)>F]fmFi\am>F]fmFa]am>F]fm-F`im6$RF]asF/F\eyF/Fj]amF/F/F /F]fmF/F/F/Fh[am, I9context/cfparse/pushmenuR6$FcqFedp6%FffoF,FipFhdpFidpC*-FSFdio-FS6#.F">FjcxFe ep@%-F56#&F\tF^cx>F^fpFh`am>F^fpF]gp-F26#-F:6$F^fpFb^q>F`ep-F_s6$F[imF^fp@$4-F` fm6$F`ep<$FdjmFbco>FjcxF`ep-F"6$F]gp&FigpF^dxFidpF[hpFidpFidp, I*CM_GlobalFd\tF/, I/help/text/rref-Fg[p6.Fi[pFb\p%`o~linsys[reduced]~:~The~purpose~of~this~functi on~is~to~reduce~a~linearG%gn~~~~~~~~~~~system~or~its~augmented~matrix~to~its~re duced~rowG%^o~~~~~~~~~~~echelon~form.~The~procedure~will~perform~row~operations ~G%fn~~~~~~~~~~~on~the~matrix~A~OR~on~the~augmented~matrix~[A,b]GFb\pFc\p%8~~~~ ~~~~~~~~~~~rref(A);GFe\p%:~~~~~~~~~~~~~~~rref(A,b);GFb\pF/, I.type/CF_CLASSRFOF/F-F/3333-F:6$FK-Fcan6$Fean.Fcp-Fdfs6$FKF_do-Fdfs6$FKFjhm-Fd fs6$FKFbv-Fdfs6$FKF`[[lF/F/F/Fcv