
Solution:
2y  3z = 2 2x + z = 3 x  y + 3z = 1.

write the system as an augmented matrix  

interchange first and third row (to make top left entry nonzero) 


Add 2 times first row to second row (to get 0 in first column of row 2) 


Divide second row by 2 (to get a leading 1 in row 2) 


Add second row to first Add 2 times second row to third (to get 0's in the second column) 


Divide third row by 2 (to get a leading 1) 


Add 1/2 times third row to first row Add 5/2 times third row to second row (to get 0's in third column) 

Check that the answer satisfies the initial equations (in case we made arithmatic errors):
All of these check out, so our solution is correct.2y  3z = 2(7/4)  3(1/2) = 7/2  3/2 = 4/2 = 2
2x + z = 2(5/4) + 1/2 = 5/2 + 1/2 = 6/2 = 3
x  y + 3z = (5/4)  (7/4) + (1/2) = (2/4) + 3/2 = (1/2) + 3/2 = 2/2 = 1

