Solve the following linear system: x + 2 y − 3 z = 4 3 x − y + 5 z = 2 4 x + y + ( s 2 − 14 ) z = s + 2

Question

Solve the following linear system:  x  +  2  y  −  3  z  =  4  3  x  −  y  +  5  z  =  2  4  x  +  y  +  (      s    2    −  14  )  z  =  s  +  2

Emmy Sparks · Accepted Answer

First we eliminate the first terms taking 3 of the first row from the second and 4 of the first row from the third, to get  x  +  2  y  −  3  z  =  4  −  7  y  +  14  z  =  −  10  −  7  y  +  (      s      2    −  2  )  z  =  s  −  14Then we simply take 1 of the new second row from the new third row, and our transformed system is just  x  +  2  y  −  3  z  =  4  −  7  y  +  14  z  =  −  10  (      s      2    −  16  )  z  =  s  −  4If   s  =  +  4, the last equation becomes   0  =  0 and your set is underdetermined, with solution  x  =  z  +      20    7    ,    y  =  2  z  +      10    7  If   s  =  −  4, the last equation becomes   0  =  −  8, and your set is overdetermined, with no solution.If   s is any other value, then simply   z  =      1          s      +      4       and the other variables can be found by back-substitution. This solution is unique.

Solve the following linear system: x + 2 y − 3 z = 4

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