Q:

How to solve this kind of linear system?

IF there is a linear system such as :
                             y=-2x-2z+1
                             x=-2y-z+2
                             z=x-y

I want a way of solving this problem different from Gaussian-Jordan method
I tried the elimination method but it didn't work with me (even though I rearranged the equations and tried solving this problem by this way several times but the answer was wrong ) I don't know if I made mistakes or this way can't go further with this kind of system
:(

A:

Substitute the third equation into the 1st and 2nd:
$$
y=-2x+1-2x+2y  \quad \quad  x=-2y-x+y+2
$$
$$
y=4x-1 \quad  \quad y=2-2x
$$
Equating the 2 equations you get:
$$
4x-1=2-2x
$$
so $x=1/2$. Then $y=1$ and $z=-1/2$

