Farkas lemma and duality

Question

In his blog, Terence Tao proves the following this version of Farkas lemma.

Propsition.   For $i=1,\dots,m$ let $P_i:{\mathbb R}^n\to{\mathbb R}$ be affine linear functions. Then TFAE

1. there is an $x$ such that $P_i(x)\ge 0$ for all $i=1,\dots,m$;
2. there are no $y_i\ge0$ such that $\displaystyle\sum^m_{i=1}y_iP_i=-1$

This is an unusual phrasing Farkas' lemma. My field is quite distant fro linear programming, so I love this phrasing because it mimics that of Hilbert's Nullstellensatz.

In the same blog TerenceTao assigns the following exercise.

Exercise.   Use Farkas’ lemma to derive the duality theorem in linear programming.

Can someone help me to understand how the two propositions are connected?