This linear system of equations is inconsistent. One way to see this is to recognize that the first 6 equations imply that the variables a to f all have value zero. But the last equation dictates that their sum is equal to 1. Clearly, if they are all zero then they cannot add up to 1.
eqs:=[a*(1-x)-x*f-x*e = 0,
b*(1-x)-x*c-x*d = 0,
c*(1-z)-z*b-z*a = 0,
d*(1-z)-z*e-z*f = 0,
e*(1-y)-y*d-y*c = 0,
f*(1-y)-y*a-y*b = 0,
a+b+c+d+e+f-1 = 0]:
vars:=[a, b, c, d, e, f]:
with(LinearAlgebra):
Now compare results from
linsys:=GenerateMatrix(eqs[1..6],vars,augmented);
LinearSolve(linsys);
LUDecomposition(GenerateMatrix(eqs[1..6],vars,augmented),output=R);
%[1..-1,1..6].Vector(vars)=%[1..-1,7];
with that from,
linsys:=GenerateMatrix(eqs,vars,augmented);
LinearSolve(linsys);
LUDecomposition(GenerateMatrix(eqs,vars,augmented),output=R);
%[1..-1,1..6].Vector(vars)=%[1..-1,7];