There's only one system of equations here.
Let's eliminate x first,
x - y + z = -1
x = -1 + y - z
Substituting into x + y + 3z = -3 gives
(-1 + y - z) + y + 3z = -3
2y + 2z = -2
y + z = -1
Substituting for x into 2x-y+2z = 0 gives
2(-1 + y - z) - y + 2z = 0
-2 + 2y - 2z - y + 2z = 0
y = 2
Got lucky on that one.
z = -1 - y = -1 - 2 = -3
x = -1 + y - z = -1 + 2 - -3 = 4
Answer: x=4, y=2, z= -3
Check:
x-y+z = 4 - 2 + -3 = -1, good
x+y+3x = 4+2+3(-3) = -3, good
2x-y+2x = 2(4)-2+2(-3) = 0, good
