Just having some difficulties with this system of inequalities...
We know E is a system of m linear inequalities of the form:
a1,1x1+ ··· +a1,nxn ≤ b1
am,1x1+ ··· +am,nxn ≤ bm
And E' an equivalent system, derived from E:
a'1,2x2+ ··· +a'1,nxn ≤ b'1
a'm,2x2+ ··· +a'm,nxn ≤ 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 aij and bi in the original system 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!