Since you can find the value of x in the last equation,
x+2y=0
x=-2y
You can substitute this in for a different equation -
x+6y-2z=-2
(-2y)+6y-2z=-2
Then combine like terms -
4y -2z=-2
and since -2z = -4y - 2
You can substitute this one in to the top equation as well as the value for x -
3x -2y +2z = -2
3x - 2y = -2z - 2
3x - 2y + 2 = -2z
3(-2y) - 2y + 2 = (-4y - 2)
Then combine like terms -
-6y - 2y + 2 = -4y - 2
-8y + 2 = -4y - 2
4 = 4y
or
y = 1
With this information, you can find x
x = -2y
x = -2(1)
x = -2
Then you can substitute these values into another equation to find z -
3x -2y +2z = -2
3(-2) - 2(1) + 2z = -2
-6 -2 +2z = -2
-8 +2z = -2
2z=6
z=3
So,
x=-2
y=1
z=3