Determine the value of b for which the system x 1 + 4 x 2 − 3 x 3 + 2 x 4 = 2 2 x 1 + 7 x 2 − 4 x 3 + 4 x 4 = 3 − x 1 − 5 x 2 + 5 x 3 − 2 x 4 = b 3 x 1 + 10 x 2 − 5 x 3 + 6 x 4 = 4 is soluble, and determine the solution set.

Question

Determine the value of   b for which the system                              x          1                +        4                  x          2                −        3                  x          3                +        2                  x          4                                    =        2                            2                  x          1                +        7                  x          2                −        4                  x          3                +        4                  x          4                                    =        3                            −                  x          1                −        5                  x          2                +        5                  x          3                −        2                  x          4                                    =        b                            3                  x          1                +        10                  x          2                −        5                  x          3                +        6                  x          4                                    =        4            is soluble, and determine the solution set.

Harley Fitzpatrick · Accepted Answer

We have this system:                              x          1                +        4                  x          2                −        3                  x          3                +        2                  x          4                                    =        2                            2                  x          1                +        7                  x          2                −        4                  x          3                +        4                  x          4                                    =        3                            −                  x          1                −        5                  x          2                +        5                  x          3                −        2                  x          4                                    =        b                            3                  x          1                +        10                  x          2                −        5                  x          3                +        6                  x          4                                    =        4            If you notice that by multiplying the first equation by   (  −  1  ) and the second one by   2 and adding these two things together you get precisely the fourth equation   3      x    1    +  10      x    2    −  5      x    3    +  6      x    4    =  4, you see, that you can ignore the fourth equation and the original system is equivalent to:                              x          1                +        4                  x          2                −        3                  x          3                +        2                  x          4                                    =        2                            2                  x          1                +        7                  x          2                −        4                  x          3                +        4                  x          4                                    =        3                            −                  x          1                −        5                  x          2                +        5                  x          3                −        2                  x          4                                    =        b            Now if you notice that by multiplying the first row by   (  −  3  ) and adding the second row you get the equation   −      x    1    −  5      x    2    +  5      x    3    −  2      x    4    =  −  3 you can see that the system cannot be solvable unless   b  =  −  3.Anyway, (one possibility of) the standard solution without guesswork would be using elementary row operation to get      (                            1                          4                          −          3                          2                          2                                      2                          7                          −          4                          4                          3                                      −          1                          −          5                          5                          −          2                          b                                      3                          10                          −          5                          6                          4                      )    ∼      (                            1                          0                          5                          2                          −          2                                      0                          1                          −          2                          0                          1                                      0                          0                          0                          0                          b          +          3                                      0                          0                          0                          0                          0                      )  What can you say from the last matrix whether the system is solvable or not (depending on   b)?

Determine the value of b for which the system <mtable columnalign="right left right left

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