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Linear Programming : Proof using Duality and the Farkas Lemma

This question is from linear programming.

I want to use duality (it's so obvious), farkas lemma (alternative solution) and all.

(See attached file for full problem description with equations)

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(a) Let . Prove that one of the following systems has a solution but not both:

(b) Prove or disprove the following claim:

Assume that both the linear program
min
s.t.

and its dual
max
s.t.

are feasible. Then at least one of them has an unbounded feasible region.
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Duality and the Farkas Lemma are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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