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


Add second row to first row 

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

