Mowing will help with weed control but I suggest using one of the selectivegrass killers like Poast or Arrest. You only have about 1/4 acre and Poastis only offered in large quantities. I would Google Arrest herbicide and buy the pint. It will treat the area at least once maybe twice. I think it will cost about $20.

NOTE: If you grass is mature add in 2 oz of a seed oil to make it more effective on the bigger grasses. It will make a defference.

Like I mentioned, mowing is fine but with the price of gas a one time herbicide application is more effective and probably less expensive.

Wait until you get some rain. The plants need to be actively growing for any herbicide to work properly.

Good Luck

POP