Lining the letters up vertically as below, makes it easier
to see what's what:

 3x + 3y      =   3
-3x      - 2z = -11
       y +  z =  -1

The idea is to reduce this 3x3 system to a 2x2 system.
This is done by using the three equations to obtain
only two equations in only two unknowns.  

1.  We observe that x is already eliminated from the 3rd
equation 

and 

2. We observe that x is easily eliminated from the 1st
and 2nd equations just by adding them term by term
just as they are, since the coefficients of x in them
are equal in absolute value but opposite in sign.
So we add them term by term:

 3x + 3y      =   3
-3x      - 2z = -11
-------------------
      3y - 2z =  -8
 
3.  Next we solve the system

3y - 2z = -8
 y +  z = -1

We can eliminate z if we multiply the 2nd equation (which
was the 3rd original equation) by 2 and adding them term by term:

3y - 2z = -8
2y + 2z = -2
------------
5y      = -10
      y = -2

4. Then we substitute -2 for y in

  y + z = -1
 -2 + z = -1
      z = 1

5. Then finally we substitute -2 for y in
the first original equation
 
   3x + 3y = 3
3x + 3(-2) = 3
    3x - 6 = 3
        3x = 9
         x = 3

Solution (x,y,z) = (3,-2,1)

Edwin