Find all the optimal solutions to the dual problem of this maximization problem

This was posted a year ago, but no answer was posted. Here's my crack at it and hopefully I can get some critique because I want to get better at this.

Maximize $$z=-3x_1-x_2-x_3$$
$$x_4=5-x_1-2x_2$$
$$x_5=0-x_1-x_2+x_3$$
$$x_6=20-7x_1-3x_2+5x_3$$
$$x_1,x_2,x_3\ge0$$

The dual problem is:
$$w=5y_1+0y_2+20y_3$$
$$y_1+y_2+7y_3\ge-3$$
$$2y_1+y_2+3y_3\ge-1$$
$$-y_2-5y_3\ge-1$$
$$y_1,y_2,y_3\ge0$$

So obviously the primal problem is already at it's optimal dictionary with an optimal solution of (0,0,0,5,0,20). Then by complementary slackness theorem we know that because $$x_4$$ and $$x_6$$ are greater than 0 then $$y_1$$ and $$y_3$$ are 0 in the optimal solution for the dual. Then we can see in the dual problem that the optimal value is 0, but we can see this will be satisfied for any y, because our constraints in the dual are all negative so the solution to the dual is (0,$$y_2\ge0$$,0)