(1)    x  +  y -  z =  3
(2)    x  - 3y + 2z = -2
(3)   3x  +  y - 3z = -1

1. Pick 2 equations and a letter to eliminate from them

I will pick (1) and (2) and the letter y to eliminate from them.

(1)    x  +  y -  z =  3
(2)    x  - 3y + 2z = -2

I will multiply (1) by 3 to make the term +y into +3y so it will
cancel with the -3y in (2) whe we add them term by term vertically:

      3x + 3y - 3z =  9
(2)    x - 3y + 2z = -2
(4)   4x      -  z =  7

2. Pick a different pair of equations and eliminate the same letter
   from them that you elimenated before.

I will pick (1) and (3) this time and eliminate y from them:

(1)    x  +  y -  z =  3
(3)   3x  +  y - 3z = -1

I will multiply (1) by -1 to make the term +y into -y so it will
cancel with the -y in (3) whe we add them term by term vertically
   
      -x  -  y +  z = -3
(3)   3x  +  y - 3z = -1
(5)   2x       - 2z = -4 

Now take (4) and (5) together:

(4)   4x -  z =  7
(5)   2x - 2z = -4

I will multiply (5) by -2 to make the term 2x into -4x so it will
cancel with the 4x in (4) when we add them term by term vertically

(4)   4x -  z =  7
     -4x + 4z =  8
           3z = 15
            z = 5

Substitute 5 for z in (4)

      4x - (5) =  7
      4x       = 12
       x       =  3

Substitute 3 for x and 5 for z in (1):

(1)    x  +  y - z = 3
       3  +  y - 5 = 3
             y - 2 = 3
                 y = 5

Solution (x,y,z) = (3,5,5)

Edwin