I don't follow you with regard to "system assignment."

There may be a more abbreviated way to append a column, but the code below constructs the traditional augmented matrix form the coefficients.

In[5]:= (* an augmented matrix *)
am = Transpose[Append[Transpose[s[[2]]], -s[[1]]]]

Out[5]= {{a, b, k1}, {c, d, k2}}

Are you trying to do something like this?

In[1]:= (* some equations *)
eqs = {a x + b y == k1, c x + d y == k2};

In[2]:= (* this contains same information as augmented array *)
s = Normal[CoefficientArrays[eqs, {x, y}]]

Out[2]= {{-k1, -k2}, {{a, b}, {c, d}}}

In[3]:= (* LinearSolve can use it *)
LinearSolve[s[[2]], -s[[1]]]

Out[3]= {(d k1 - b k2)/(-b c + a d), (c k1 - a k2)/(b c - a d)}

In[4]:= (* Solve gives the same answer *)
Solve[eqs, {x, y}]

Out[4]= {{x -> -((d k1 - b k2)/(b c - a d)), 
  y -> -((-c k1 + a k2)/(b c - a d))}}