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I have a question regarding feasibility checking. I need to check whether the system $\{ x: Ax=b , x \geq 0 \}$ has a feasible solution.

1- What is the best worst case running time for this decision problem if it is considered as an LP?

2- Is it possible to decide the feasibility using Gauss-Elimination method or some other methods like this?

Note, however, that the entries of the matrix $A$ can be rational.

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