Getting rid of bromegrass

The family farm has a valley that used to be a cattle pasture, the cattle usually hung out on top of one particular hill, which killed all the native grasses, my grandfather planted bromegrass there so it would be covered by something. Now the bromegrass is very well established and spreading into the garden and around the valley, I would like to get rid of it sometime in the next couple years, Any ideas?

I have killed it with the rototiller and with pigs. If you rototill be sure not to water and then till again every few days as the roots are very good at restarting if they have time. So it takes multiple disturbances and drying out to kill them. I have also pushed it out with lawn grass by intelligent mowing. This is a slow process maybe getting a foot a decade. Might work for native grasses too. Indentify the edge between the grass you want to keep and the brome and mow that zone really short on a regular basis being sure you get as little of the good grass as possible while getting basically all the brome. Let the good grass make viable seed before you mow it and then mow it so you are blowing the residue out over the area you want to seed. Because the other grass is kept strong and the brome reduced in strength it will gradually be pushed out. The area I have recovered is about 40 feet wide and 100 feet long and started from just a few square feet of good grass. But it has taken a decade to do it. Occasionally if a small patch pops up in the good grass it is pulled but mostly this has been done by simple adjusting mower height and watching where I mow when.

Thanks C, I will try mowing it as you suggested, A big part of the problem is that that it has been spreading so if I can just stop the spread and get it out of the garden that would be a huge help. I won't till because that would cause some serious erosion as it is on a slope. Thank you for your suggestion.