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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:

$$ x_s = 2 - (-2 -x_s) \\ x_s = 2 + 2 + x_s \\ -2 = 2 $$

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This does contain all the variables - write $x_o=0*x_o + -1*x_r + 0*x_s + 4$, etc. – mathematician Jun 1 '14 at 23:58
up vote 5 down vote accepted

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.

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