I have this weed/plant that I am trying rid my yard of but nothing is working!! Please help identify this plant and through some ideas at my on ways to get rid of it. It will get about 3-4 feet tall, has long leaves, and purple flowers. The root system is rather large and very hard to pull.

My first encounter with trying to remove them was in an old vegetable garden that was overgrown with them. I started with a weed wacker and cut them all down then tilled and planted the garden only to have taken over with these plants. when I started pulling the smaller plants it became evident that every root structure that I tilled resulted in little chunks of root that started sprouting. Round two had me pulling these out the following year only to find that as I pulled, I left off-shoots of the main root behind that would sprout. I have tried weed killers, burning, pulling, and nothing works...AARRRGGGGGGG! the more I try to get rid of these, the more I end up with! PLEASE HELP!! I have several locations around my yard where these are growing...literally thousand of plants!!! Here are some pics of it...

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