Transcription of 1.5 Consistent and Inconsistent Systems
1 And Inconsistent followingsystem:3x+2y 5z=4x+y 2z=15x+3y 8z=6To ndsolutions,obtaina row-echelonformfromtheaugmentedmatrix:0 BBB@32 5411 2153 861 CCCAR1$R2 !0 BBB@11 2132 5453 861 CCCAR2!R2 3R1 !R3!R3 5R10 BBB@11 210 1110 2211 CCCAR2 ( 1) !0 BBB@11 2101 1 10 2211 CCCAR3!R3 + 2R2 !0 BBB@11 2101 1 1000 11 CCCAR3 ( 1) !0 BBB@11 2101 1 100011 CCCA(Row-EchelonForm)Thesystemof equationscorrespondingto thisREFhasas itsthirdequation0x+ 0y+ 0z= 1i:e:0 = 1 Thisequationclearlyhasnosolutions- noassignment of numericalvaluestox; yandzwillmake thevalueof theexpression0x+ 0y+ 0zequalto systemof linear equationsis calledinconsistentif it has no a solutionis a systemis Inconsistent , a REFobtainedfromitsaugmentedmatrixwillinc ludea row oftheform0 0 0: : :0 1, a leading1 in its a row correspondsto anequationof theform0x1+ 0x2+ + 0xn= 1, which (MA203 Summer2005,Q1)(a)Findtheuniquevalueoftfo rwhich thefollowingsystemhasa solution.
2 X1+x3 x4=32x1+2x2 x3 7x4=14x1 x2 9x3 5x4=t3x1 x2 8x3 6x4=1 Solution: @ 101 1322 1 714 1 9 5t3 1 8 611 CCCCCCAR1 ( 1) !0 BBBBBB@10 11 322 1 714 1 9 5t3 1 8 611 CCCCCCAR2!R2 2R1R3!R3 4R1 !R4!R4 3R10 BBBBBB@10 11 3021 970 1 5 9t+ 120 1 5 9101 CCCCCCAR3!R3 R4 !0 BBBBBB@10 11 3021 970000t+ 20 1 5 9101 CCCCCCAF romthethirdrow of thismatrixwe canseethatthesystemcanbe Consistent onlyift+ 2 = 2.(b)Findthegeneralsolutionof : Sett= 2 omitthethirdrow,which consistsfullyof zeroes @10 11 3021 970 1 5 9101 CCCAR4 ( 1) !R3$R40 BBB@10 11 30159 10021 971 CCCAR3!R3 2R2 !0 BBB@10 11 30159 1000 9 27271 CCCAR3 ( 19) !0 BBB@10 11 30159 100013 31 CCCAR1!R1 +R3 !R2!R2 + 5R30 BBB@1004 6010 650013 31 CCCAH avingreacheda reducedrow-echelonform,we canseethatthevariablesx1; x2andx3areleadingvariables,andthevariabl ex4is have fromtheRREFx1= 6 4x4; x2= 5 + 6x4; x3= 3 3x4:If we assigntheparameternamesto thevalueof thefreevariablex4in a solutionof thesystem,we canwritethegeneralsolutionas(x1; x2; x3; x4) = ( 6 4s;5 + 6s; 3 3s; s); s2R:Summaryof PossibleOutcomeswhenSolvinga Systemof LinearEquations:1.
3 Thesystemmay be Inconsistent . Thishappensif a REFobtainedfromtheaugmentedmatrixhasa leading1 in Thesystemmay be Consistent . In thiscaseoneof thefollowingoccurs:(a)Theremay be a if all variablesareleadingvariables, a REFobtainedfromtheaugmentedmatrixhasa leading1. In thecasethereducedrow-echelonformobtained fromthe20augmentedmatrixwillhave thefollowingform:0 BBBBBBBBB@100: : :0 010: : :0 001: : :0 ..000: : :1 1 CCCCCCCCCA withpossiblysomeadditionalrowsfullof zeroes at : If a systemof equationshasa uniquesolution,thenumber of equationsmustbe at leastequalto thenumber of variables(sincetheaugmentedmatrixmusthav eenoughrowsto accommodatea leading1 foreveryvariable).(b)Theremay be in nitelymany thesystemis Consistent butat leastoneof thevariablesis thiscasetherankof theaugmentedmatrixwillbe lessthanthenumber of variablesin
