How find this linear equationlet a ∈ R,then solve this follow equation: { x 1 − x 2 − x 3 − ⋯ − x n = 2 a − x 1 + 3 x 2 − x 3 − ⋯ − x n = 4 a − x 1 − x 2 + 7 x 3 − ⋯ − x n = 8 a ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ − x 1 − x 2 − x 3 − ⋯ + ( 2 n − 1 ) x n = 2 n a

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How find this linear equationlet   a  ∈  R,then solve this follow equation:      {                                        x                      1                    −                      x                      2                    −                      x                      3                    −          ⋯          −                      x                      n                    =          2          a                                      −                      x                      1                    +          3                      x                      2                    −                      x                      3                    −          ⋯          −                      x                      n                    =          4          a                                      −                      x                      1                    −                      x                      2                    +          7                      x                      3                    −          ⋯          −                      x                      n                    =          8          a                                      ⋯          ⋯          ⋯          ⋯          ⋯          ⋯                                      −                      x                      1                    −                      x                      2                    −                      x                      3                    −          ⋯          +          (                      2            n                    −          1          )                      x                      n                    =                      2            n                    a

lofoptiformfp · Accepted Answer

Write   s  =      x    1    +      x    2    +  …  +      x    n  . Then the system becomes      2    k        x    k    −  s  =      2    k    a  ;    1  ⩽  k  ⩽  n  .Rearranging that, we obtain      x    k    −  a  =      2          −      k        s  .Summing these equations yields      ∑          k      =      1        n    (      x    k    −  a  )  =  s  −  n  a  =      (          ∑              k        =        1            n              2              −        k              )    s  =      (    1    −          2              −        n              )    s  .From that we obtain  s  =  n      2    n    aand      x    k    =  a  (  1  +  n      2          n      −      k        )for   1  ⩽  k  ⩽  n.

How find this linear equation let a ∈ R ,then solve this follow equatio

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