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}}

Can you please explain the syntax? specifically the s[] and -s[] Thank you

SoQ

(-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)

i'm simply trying to construct the augmented matrix. Looking at the instructions, it's actually asking to use the system assignment. Thank you for your help.