SOLUTION: Use row of operations to solve x+y+z=-5, x-y+3z=-1, 4x+y+z=-2

Swap row 1 and 2 to get a 1 in the upper left corner



Multiply Row 1 temporarily by -2, getting 


and add that to row 2:
  
Swap rows 2 and 3:

 

Multiply Row 2 temporarily by -3, getting 


and adding that to row 3:

 

Mutiply row 3 by 

 

Convert back into a system of equations:



or just



Substitute  from the bottom equation
into the middle equation 





Substitute  and 
into the top equation 





So the solution is 

Edwin