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Assume that $(F_i)_i$ is a system of linear inequalities in $n$ variables, of the form $F_i(x_1,\ldots, x_n) > 0$, where $F_i(x_1, \ldots, x_n) = a_{i,1}x_1 + a_{i,2} x_2 +\ldots + a_{i,n} x_n + c_i$. One way to prove that this system is contradictory (assuming it is) would be to show that there exists a linear combination $\sum_i \lambda _i F_i = -K$, with $K\geq0$ an $\lambda_i \geq 0 $ for all $i$.

I am wondering if the converse is not true: "if the system is contradictory, there must exist a linear positive combination of the $F_i$ that is equal to $-K$ for some $K\geq 0$".

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The answer is Yes. See Fourier-Motzkin Elimination and Its Dual, GEORGE B. DANTZIG AND B. CURTIS EAVES (Feasibility theorem).

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