I have been given the problem of solving the dual of the linear program
$$\max_{x\in\mathbb{R}^3}\{2x_1+6x_2+x_3\}$$
subject to
$$x_1-2x_2-x_3 \le -3 \\ 2x_1+3x_2-x_3 \ge -3\\x_1 \ge 0, x_2\ge 0, x_3\ge 0$$
I went ahead and reformulated the problem to the dual one and ended up with
$$\max\{3y_1-3y_2\}$$
where
$$-y_1+2y_2 \le -2\\ 2y_1+3y_2 \le -6\\ y_1 - y_2 \le -1\\ y_1\ge 0, y_2 \ge 0$$
The constraints of the dual problem however, leave me with an empty set. This confuses me, since the problem was formulated in a way that you expect the dual to have a solution. That's why I wanted to ask, whether or not I did make any mistakes in formulating the dual problem. Any help would be greatly appreciated!
You can't expect your dual problem to have solution, since the primal(original) problem has an unbounded feasible region: you can take $x_1=x_3 = 0$ and $x_2>2$, so your maximum is $+ \infty$. This means your dual problem must be unfeasible.