The set of equations is inconsistent.
here are your equations:
In[15]:= eqns = {xa == .10,
1 - x == .89 - xb, xc == .01, xa + (1 - x) d == .1,
xb + (1 - x) e == 0, xc + (1 - x) f == .9, (1 - x) d == 0,
x + (1 - x) e, (1 - x) f == .89}
Now substitute the explicitly set values that are already in them
In[16]:= eqns /. {xa -> .1, xc -> .01}
Out[16]= {True, 1 - x == 0.89 - xb, True, 0.1 + d (1 - x) == 0.1,
e (1 - x) + xb == 0, 0.01 + f (1 - x) == 0.9, d (1 - x) == 0,
e (1 - x) + x, f (1 - x) == 0.89}
Ok, from d (1 - x) == 0, one sees that x==0 so let's use that too:
In[17]:= eqns /. {xa -> .1, xc -> .01, x -> 1}
Out[17]= {True, 0 == 0.89 - xb, True, True,
xb == 0, False, True, 1, False}
So the set of equations is inconsistent and there are therefore no solutions.