Given the linear system of equations: { x 1 + x 2 + x 3 = n x 1 + x 2 + x 3 + x 4 + x 5 = 3 n 2 x 1 + 2 x 2 + 2 x 3 + x 4 + x 5 + 3 x 6 + 3 x 7 + 3 x 8 = 10 n how many solutions are in N ∪ { 0 }?The solution must not be using sum notation like ∑ y.I know how to find the number of solutions to the regular equations like x 1 + x 2 + x 3 + ⋯ = n but I'm not sure how to do this for a system of equations. I thought of substituting some x's: x 1 + x 2 + x 3 = 3 n − ( x 4 + x 5 ) x 4 + x 5 = 10 n − 2 ( x 1 + x 2 + x 3 ) − 3 ( x 6 + x 7 + x 8 ) ⟹ x 1 + x 2 + x 3 = 3 n − ( 10 n − 2 ( x 1 + x 2 + x 3 ) − 3 ( x 6 + x 7 + x 8 ) ) ⟹ x 1 + x 2 + x 3 + 3 ( x 6 + x 7 + x 8 ) = 7 n ∗As far as I understand finding the number of solutions for the system is equivalent to finding the number of solutions to the equation *.The only next step from here I can think of is using generating functions: ( 1 + x + x 2 + … ) 3 ( 1 + x 3 + x 6 + x 9 + … ) 3 and we need to find the coefficient of x 7 n .From the closed form identities we have: ∑ k = 0 ∞ ( 3 − 1 + k k ) x k ⋅ ∑ i = 0 ∞ ( 3 − 1 + i i ) x 3 i But I have no idea now how to find the coefficient of 7n from here and certainly not without using some kind of sum notation.

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Given the linear system of equations:      {                                        x            1                    +                      x            2                    +                      x            3                    =          n                                                  x            1                    +                      x            2                    +                      x            3                    +                      x            4                    +                      x            5                    =          3          n                                      2                      x            1                    +          2                      x            2                    +          2                      x            3                    +                      x            4                    +                      x            5                    +          3                      x            6                    +          3                      x            7                    +          3                      x            8                    =          10          n                        how many solutions are in       N    ∪  {  0  }?The solution must not be using sum notation like   ∑  y.I know how to find the number of solutions to the regular equations like       x    1    +      x    2    +      x    3    +  ⋯  =  n but I&#039;m not sure how to do this for a system of equations. I thought of substituting some   x&#039;s:      x    1    +      x    2    +      x    3    =  3  n  −  (      x    4    +      x    5    )        x    4    +      x    5    =  10  n  −  2  (      x    1    +      x    2    +      x    3    )  −  3  (      x    6    +      x    7    +      x    8    )      ⟹        x    1    +      x    2    +      x    3    =  3  n  −  (  10  n  −  2  (      x    1    +      x    2    +      x    3    )  −  3  (      x    6    +      x    7    +      x    8    )  )      ⟹        x    1    +      x    2    +      x    3    +  3  (      x    6    +      x    7    +      x    8    )  =  7  n    ∗As far as I understand finding the number of solutions for the system is equivalent to finding the number of solutions to the equation *.The only next step from here I can think of is using generating functions:  (  1  +  x  +      x    2    +  …      )    3    (  1  +      x    3    +      x    6    +      x    9    +  …      )    3  and we need to find the coefficient of       x          7      n      .From the closed form identities we have:      ∑          k      =      0              ∞                                    (                                      3          −          1          +          k                k                              )                          x    k    ⋅      ∑          i      =      0              ∞                                    (                                      3          −          1          +          i                i                              )                          x          3      i      But I have no idea now how to find the coefficient of 7n from here and certainly not without using some kind of sum notation.

sleuteleni7 · Accepted Answer

This is more of a hint rather than a solution but still:You have      x    1    +      x    2    +      x    3    =  n  .Substituting this into your second equation gives      x    4    +      x    5    =  2  n  .Substituting both in the third gives      x    6    +      x    7    +      x    8    =  2  nIf I&#039;ve understood correctly you can solve this new system of equations since each       x    i   appears in exactly 1 equation.

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