Your problem

3x  +  2y  +   z  =  -5  --- eqn (1)
x   -  3y   + 2z  =  28/3 ---- eqn (2)
-6x +   y   + 2z  =  -5  ----- eqn (3)

Solution.

Step 1) We first try to eliminate 1 variable (say z) from the 3 equations, to get 2 equations in x and y, which we can solve.

Step 2) Subtract eqn 3 from eqn 2 to get

   (note that the two 2z terms cancel out)

  ---- eqn (4)

Step 3) Multiply eqn (1) by 2 to get another 2z term that can be cancelled.

  ---- eqn (5)

Step 4) Subtract eqn 5 from eqn 3 (both have 2z terms which will cancel out)

 ----- eqn (6)

Step 5) Now we have 2 eqns (4) and (6) in x and y. We will solve it by eliminating y.

Multiply eqn (4) by 3 to get

   ---- eqn (7)

Multiply eqn (6) by 4 to get

   ---- eqn (8)

Step 6)
Subtract eqn 8 from eqn 7 to get

 or 

Step 7) Substitute for x in eqn (7)


 or 

Step 8)
Now substitute for x and y in eqn(1)

 or 

So the final solution is 




Check: Substitute in original 3 equations.

Eqn 1: 3*1/3 + 2*(-3) + 0 = 1 - 6 = -5 - correct
Eqn 2: 1/3 -3*(-3) + 2*0 = 1/3 + 9 = 28/3 - correct
Eqn 3: -6*1/3 -3 + 2*0 = -2 - 3 = -5 --- correct

:)