= Solution
Write the primal in the form $\min c^Tx$ with $Ax\geq b$ and $x\geq0$. Its <dual linear program> is $\max b^T\lambda$ with $A^T\lambda\leq c$ and $\lambda\geq0$. Indeed, nonnegative combinations of the primal inequalities give $c^Tx\geq\lambda^TAx\geq\lambda^Tb$, which is <weak duality>. The specific dual is
$$
\boxed{\begin{aligned}
\text{maximize }&3\lambda_1+4\lambda_2+\lambda_3,\\
\text{subject to }&\lambda_1-2\lambda_3\leq2,\\
&2\lambda_1+2\lambda_2\leq4,\\
&-\lambda_1+\lambda_2+3\lambda_3\leq3,\\
&-3\lambda_2+\lambda_3\leq1,\\
&\lambda_1,\lambda_2,\lambda_3\geq0.
\end{aligned}}
$$
The four dual inequalities correspond, in order, to the four primal variables.
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