Transcription of 1.5 Consistent and Inconsistent Systems
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1.5 Consistent and Inconsistent Systems
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.
Example 1.5.3 (MA203 Summer 2005, Q1) (a) Find the unique value of t for which the following system has a solution. x1 + x3 x4 = 3 2x1 + 2x2 x3 7x4 = 1 4x1 x2 9x3 5x4 = t 3x1 x2 8x3 6x4 = 1 Solution: First write down the augmented matrix and begin Gauss-Jordan elimination.
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