The Fourier-Motzkin elimination method has been recently extended to linear inequality systems that have infinitely many inequalities. It has been used in the study of linear semi-infinite programming by Basu, Martin, and Ryan. Following the idea of the conjecture for semi-infinite programming in a paper by Kortanek and Zhang recently published in Optimization, which states ``all the duality results proved by applying FM (the Fourier-Motzkin elimination method) first can also be obtained by working with the problem directly", in this paper without using the Fourier-Motzkin elimination, we reproduce all the results presented in a recent paper by Basu, Martin, and Ryan on the strong duality and dual pricing properties in semi-infinite programming in which the main mechanism is the Fourier-Motzkin elimination. We also present some new results regarding the strong duality and dual pricing properties, which are the main topics in Basu-Martin-Ryan's paper.