solve the system of equations

x-y+2z=-1

y=x+2z+1

2x+z=0

z=-2x 

x+5y+z=5

x=5-5y-z 

z=-2(5-5y-z)

z=-10+10y+2z 

z+10=10y+2z

10=10y+z

z=10-10y

x=5-5y-(10-10y)

x=5-5y-10+10y

x=5y-5

y=5y-5+2(10-10y)+1

y=5y-5+20-20y+1

x=0

y=1

z=0

y=-15y+16

16y=16

y=1

z=10-10y

z=10-10

z=0

x=5y-5

x=5-5

x=0

 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}