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))}}

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}}

The [[ ]] is a form of Part. (look up Part in the help system.)

In 5 above, we transpose the 2nd part of am, which is the coefficient matrix, so columns become rows. Then append a row which is the constants, then transpose it all back so columns are columns again.

Using your equations:

In[14]:= eqs = {-19 u - 19 w + 6 x - 18 y + 10 z == -2, -15 u - 
     17 w - 16 x - 11 y + 11 z == -15, 14 u - 8 w + 13 x - 11 y == 16,
    13 u - 17 w - 10 x + 18 y + 12 z == 17, 
   4 u + 9 w + 20 x - 16 y - 9 z == 13};

In[15]:= (* this contains same information as augmented array *)
s = Normal[CoefficientArrays[eqs, {u, w, x, y, z}]];

In[16]:= (* an augmented matrix *)
am = Transpose[Append[Transpose[s[[2]]], -s[[1]]]]

Out[16]= {{-19, -19, 6, -18, 10, -2}, {-15, -17, -16, -11, 
  11, -15}, {14, -8, 13, -11, 0, 16}, {13, -17, -10, 18, 12, 17}, {4, 
  9, 20, -16, -9, 13}}