find all the values of a and b so that the system has a) no solution b) 1 solution c) exactly 3 solutions d) infinitely many solutions { x − y + 2 z = 4 3 x − 2 y + 9 z = 14 2 x − 4 y + a z = b I know that a and b has to either equal to something or not in order to satisfy the 4 conditions stated above.my matrices looked like [ 1 − 1 2 0 1 3 0 0 a + 12 ] [ x y z ] = [ 6 2 b + 8 ]

Question

find all the values of a and b so that the system has a) no solution b) 1 solution c) exactly 3 solutions d) infinitely many solutions      {                                    x                          −                          y                          +                          2          z                          =          4                                              3          x                          −                          2          y                          +                          9          z                          =          14                                              2          x                          −                          4          y                          +                          a          z                          =          b                        I know that   a and   b has to either equal to something or not in order to satisfy the   4 conditions stated above.my matrices looked like      [                            1                          −          1                          2                                      0                          1                          3                                      0                          0                          a          +          12                      ]        [                            x                                              y                                              z                              ]    =      [                            6                                              2                                              b          +          8                              ]

Leah Conley · Accepted Answer

Apply Gaussian Elimination to the reduced matrix.      (                            1                          −          1                          2                          4                                      3                          −          2                          9                          14                                      2                          −          4                          a                          b                      )    →      (                            1                          −          1                          2                          4                                      0                          1                          3                          2                                      0                          −          2                          a          −          4                          b          −          8                      )    →      (                            1                          −          1                          2                          4                                      0                          1                          3                          2                                      0                          0                          a          +          2                          b          +          4                      )  So- if you want a unique solution,             b      +      4              a      +      2       must be defined, i.e.   a  ≠  −  2.- if you want infinite number of solutions,   a  =  −  2 and   b  =  −  4 removes the last equation.- if you want no solutions,   a  =  −  2 and   b  ≠  −  4.

find all the values of a and b so that the system has a) no solution b) 1 solution c) exactly

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