Solve wants equations using explicit variables. That can be done along the lines below.
mat = {{1, 1, -2}, {3, 4, -3}, {1, 2, 1}};
b = {1, 1, -1};
vars = Array[x, 3];
In[30]:= mat.vars == b
(* Out[30]= {x[1] + x[2] - 2 x[3], 3 x[1] + 4 x[2] - 3 x[3],
x[1] + 2 x[2] + x[3]} == {1, 1, -1} *)
