# Find rational solutions to a system of equations

I have the following system of equations: $$(Q-E).x=C.z_{1}-z_{2}+C.q_{1}-q_{2}$$ where $C,Q\in \text{GL}(n,\mathbb{Q})$, $q_{1},q_{2}\in \mathbb{Q}^{n}$ and $E$ the identity matrix. How can I find all solutions $x\in \mathbb{Q}^{n}$ between $0$ and $1$ for all possible integers $z_{1},z_{2}\in \mathbb{Z}^{n}$?

Edit: the system of equations can be transformed to have integers coefficients. If we define $A=(Q-E)$ and $w=C.q_{1}-q_{2}$ then $$A.x=C.z_{1}-z_{2}+w$$ $$A'.x=C\ '.z_{1}-mz_{2}+w'$$ where $m=\text{LCD}(A,C,w)$ (least common denominator) and $A'=mA$, $C\ '=mC$ and $w'=mw$. Note that $A'$, $C\ '$ and $w'$ all have (known) entries from $\mathbb{Z}$. I'm not sure whether this helps, but it's a start.

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I have changed your $X$ to $x$ to avoid confusion. If you think that's inappropriate, please feel free to roll back. –  user1551 Dec 29 '12 at 16:03
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