I am interested in the following problem for its own sake. This is not a homework problem.

Given a bipartite graph $G = (V = X cup Y, E)$ and a weight function $w: E rightarrow mathbb{R}_{+}$, how does one reduce the problem of finding a minimum weight edge cover to that of finding an optimal weight perfect matching?

The following link

https://cstheory.stackexchange.com/questions/14690/reducing-a-minimum-cost-edge-cover-problem-to-minimum-cost-weighted-bipartie-per

apparently provides an answer which clearly I am not following. When discussing an edge cover, please use the definition — that is, a subset $C subseteq E$ such that each $v in V$ is incident to at least one $e in C$.

I would like to use the Hungarian algorithm as part of a solution to this problem. Any insight, whether it is a clarification of the linked answer or otherwise, is greatly appreciated.