On your TI-83 or 84 calculator,

Press 2ND x-1 to get the MATRIX menu
Scroll right to highlight EDIT
Press ENTER
Press 4
Press ENTER
Press 5
Press ENTER

Press 3 ENTER -2 ENTER 2 ENTER -1 ENTER 2 ENTER
for the first row and ENTER after each of these 
on the other three rows:
   4 1  1 6 8 
  -3 2 -2 1 5 
   5 0 3 -2 1  (pressing a 0 for the missing term in y
                on the 4th row.

Press 2ND MODE (for QUIT to go to the main screen)
Press 2ND x-1 to get the MATRIX menu again
highlight MATH
Scroll to B:rref
Press ENTER
Press 2ND x-1 to get the MATRIX menu again
Press ENTER
Press )
Press ENTER

To make the decimals into fractions press MATH ENTER ENTER

Scroll right and see that the matrix is

1 0 0 41/13  0
0 1 0 -9/13  0
0 0 1 -77/13 0
0 0 0 0      1

The bottom row 

0 0 0 0      1

means this equation:

0x + 0y + 0z + 0w = 1 

0 = 1, so the system is inconsistent. 

After all that, no solution exists! Phooey!

Good thing we did it by calculator instead of
by hand.  That would have really been a let-down! :)

Edwin