Sets and sums of different elements

s = {w, x, y, z};
sum = {-1, 3, 5, 8};

add = Plus @@@ Subsets[s, {3}]
(* {w + x + y, w + x + z, w + y + z, x + y + z} *)

Solve[add == sum, s]
(* {{w -> -3, x -> 0, y -> 2, z -> 6}} *)

mat = ConstantArray[1, {4, 4}] - IdentityMatrix[4];
LinearSolve[mat, {-1, 3, 5, 8}]

{6, 2, 0, -3}