x-y+2z = -1 (1)
2x+z = 0 → z = -2x (2)
x+5y+z = 5 (3)
Putting (2) into (1) and (3), we have
x - y + 2(-2x) = -1
x + 5y - 2x = 5 simplify both of these
-3x - y = -1 → y = -3x + 1 (4)
-x + 5y = 5 (5)
Putting (4) into (5), we have
-x + 5(-3x + 1) = 5
-16x + 5 = 5 subtract 5 from both sides
-16x = 0 → x = 0
And using (4)
y = -3(0) + 1 = 1
And using (2)
z = -2(0) = 0
So {x, y, z} = {0, 1, 0}
