# Does optimal solution from primal problem follow from optimal solution to dual?

In a linear programming context, does the primal optimal solution yield an explicit way to find the primal dual solution? I vaguely remember something like this from an optimization class but can't recall the precise statement.

The closest I've found is that if $x, y$ are primal and dual feasible respectively, and the value of the primal objective at $x$ is equal to the value of the dual objective at $y$, then $x$ and $y$ are primal and dual optimal.

Thanks!