Just having some difficulties with this system of inequalities...

We know *E* is a system of **m** linear inequalities of the form:

a_{1},_{1}x_{1}+ ··· +a_{1},_{n}x_{n} ≤ b_{1}

...

a_{m},_{1}x_{1}+ ··· +a_{m},_{n}x_{n} ≤ b_{m}

And *E'* an equivalent system, derived from *E*:

a'_{1},_{2}x_{2}+ ··· +a'_{1},_{n}x_{n} ≤ b'_{1}

...

a'_{m},_{2}x_{2}+ ··· +a'_{m},_{n}x_{n} ≤ b'_{m}

I have proven that given that if *E* has a solution, then so does *E'* and vice-versa, on the set of *R* (real numbers). However, now we suppose that the coefficients *a _{i}_{j}* and

*b*in the original system

_{i}*E*are in fact integers with a maximum absolute value M.

Since the coefficients in the system *E* are integers, then the various coefficients in the system *E'* will be rational numbers, and so can be written in the form *a/b*, where *a* is the numerator and *b* the denominator.

How can I obtain a bound on the absolute values of these numerators and denominators as a function of M? I'm not quite sure which would be the right approach in order to correctly prove it..

Thanks for taking your time!