first take the first equation
3x + 2y + z = -2
then subtract z
3x + 2y + z - z = -2 - z
simplify
3x + 2y = -2 - z
add 2
3x + 2y + 2 = -2 - z + 2
simplify
3x + 2y + 2 = -z
divide by -1
-3x - 2y - 2 = z
plug z into last equation
2y + ( -3x - 2y - 2 ) = -1
simplify
-3x - 2 = -1
add 2
-3x = 1
divide by -3
x = -1/3
plug x and z into second equation
-( -1/3 ) - y - 2 ( -3 ( -1/3 ) - 2y - 2 ) = 2
simplify
1/3 - y - 2 ( 1 - 2y - 2 ) = 2
distribute
1/3 - y - 2 + 4y + 4 = 2
combine like terms
2 1/3 + 3y = 2
subtract 2 1/3
3y = -1/3
divide by 3
y= -1/9
plug in x and y into second equation
-( -1/3 ) - ( -1/9 ) - 2z = 2
simplify
1/3 + 1/9 - 2z = 2
add
4/9 - 2z = 2
subtract
-2z = 14/9
divide
z = -7/9
answer:
x = -1/3
y = -1/9
z = -7/9