Note: It would be faster to substitute the y=2 back into, say, 4x + 4y = 4 to get x, but the instructions say to use the addition method, so I used only the addition method.
x + 3y + z = 6
3x + y - z = -2
2x + 2y - z = 1
add the first equation to the 2nd and third equations
4x + 4y = 4
3x + 5y = 7
multiply the top equation by 3 and the bottom equation by -4
12x + 12y = 12
-12x - 20y = -28
add the equations
-8y = -16
y = 2
back to the pair of equations
4x + 4y = 4
3x + 5y = 7
multiply the top equation by 5 and the bottom equation by -4
20x + 20y = 20
-12x - 20y = -28
add the equations
8x = -8
x = -1
back to the original equations
x + 3y + z = 6
3x + y - z = -2
2x + 2y - z = 1
multiply the top equation by -6, the 2nd equation by 2, and the 3rd equation by 3
-6x - 18y - 6z = -36
6x + 2y - 2z = -4
6x + 6y - 3z = 3
add the top equation to the 2nd and 3rd equations
-16y - 8z = -40
-12y - 9z = -33
multiply the top equation by -3 and the bottom equation by 4
48y + 24z = 120
-48y - 36z = -132
add the equations
-12z = -12
z = 1
z = 0