How do I solve the following system?
$$ \left\{ \begin{array}{} x_o = 4 - x_r \\ x_r = -2 - x_s \\ x_s = 2 - x_r \end{array} \right. $$
All the techniques i've found for solving 3-equation systems rely on all equations containing all the variables, whereas I don't see it in this case. When I try to isolate one of the factors and substitute it, it carries another factor with it:
$$ x_s = 2 - (-2 -x_s) \\ x_s = 2 + 2 + x_s \\ -2 = 2 $$
Your system does not have any solutions. The second equation says $x_s+x_r=-2$, whereas the last equation says $x_s+x_r=2$. This is not possible, hence inconsistent system.