From choice of x, y, and z we have:
  x11 = x + y
  x12 = x - z
  x22 = x

We already showed that the sum of every column, row and diagonal is 3x

Finish the top row:
  3x = S
     = x11 + x12 + x13
     = (x + y) + (x - z) + x13
     = 2x + y - z + x13
  So
    x13 = x - y + z

Now let's do the bottom row:
  3x = S
     = x13 + x22 + x31
     = (x - y + z) + (x) + x31
  So
    x31 = x + y - z

  3x = S
     = x12 + x22 + x32
     = (x - z) + (x) + x32
     = 2x - z + x32
  So
    x32 = x + z

  3x = S
     = x11 + x22 + x33
     = (x + y) + (x) + x33
     =  2x + y
  So
    x33 = x - y

Now do the 2 sides

  3x = S
     = x11 + x21 + x31
     = (x + y) + x21 + (x + y - z)
     = 2x + 2y - z + x21
  So
     x21 = x - 2y + z

  3x = S
     = x13 + x23 + x33
     = (x - y + z) + x23 + (x - y)
     = 2x - 2y + z + x23
  So
    x23 = x + 2y - z