Hi. You asked to solve one such system for the values you gave and I tried to think of the most compact way of doing so in Mathematica:

which returns six sets of solutions.Code:coeff = {2, 3, 4, 6, 12, 32}; mycoeff = Table[Subscript[a, n] -> coeff[[n]], {n, 1, 6}]; myvars = Join[Table[Subscript[x, n], {n, 1, 3}], Table[Subscript[y, n], {n, 1, 3}]]; myequations = Table[ Sum[Subscript[x, j]*Subscript[y, j]^ (n - 1), {j, 1, 3}] == Subscript[a, n], {n, 1, 6}]; Solve[myequations /. mycoeff, myvars]