First off, NEVER trust a hardware store to tell you how much fertilizer a pasture needs. It is so easy to bring your nitrogen levels well above toxic levels you wouldn't believe it. You can go out one morning to find everyone of your horses dead with their feet sticking straight up in the air. It's even more dangerous to ruminant animals when nitrate levels are high.
You need to find a co-op or agricultural supply center near you. The first one I used here was recommended by the ag extension agent. Since then I've done a lot of shopping around and found a cheaper one that lets us borrow their big spreader for as long as we need it. It holds about 4 tons of nitrogen and they fill it and set it to the rate I need for me so all I have to do is pull it home with the truck then hook it to the tractor and go.
Your fertilizer, if done properly, will get the grass growing fast and tall and really go a long way towards helping it choke out the weeds on it's own. Soil samples are extremely important! Plant analysis should be done also before turning out animals on it unless you set up a rotational grazing system which you should probably do anyway. Horses are real bad about eating to the dirt before moving on and that's not good for any type of grass.
Plus if you have a rotational system, you can graze one section of it and then fertilize another till the grass is tall and nitrate levels are at an acceptable level for them.
We've had our giant Bermuda pasture in operation for three years now. Every month from April untill October we bale most of it for hay. It's 7 acres just like yours. I always temp fence off a small part of it to graze the horses on after it's reached full height and is at least a month after I fertilized it. We cut on a 28 day cycle here. According to our soil samples we require 60 pounds of nitrogen per acre after each cutting and that whole 7 acres only costs me around 300 bucks even at that high level when buying it from a co-op or elevator that caters to farmers, NOT the happy home owner.
60 pounds of Nitrogen is not like one sixty pound bag of some store mix. It means 100 pounds of 60-0-0 per acre. You will have to learn to read your soil analysis. The co-ops here are great about doing it for me. The stuff I usually buy is 46-0-0 and the man at the co-op does all the math for me to figure the proper amount to bring it to a 60 pound per acre amount. I still have a hard time doing that one in my head.
Just be very careful, I'm serious. Call the ag department at your local college, veterinarians, ag extension agent and whoever else you need to verify what I say about the nitrogen being dangerous. It is needed on any grass crop but if you aren't careful you will not have a need for it for long. I highly recommend doing a lot of samples untill you get a good feel for your land especially with the horses.
We have always had horses here. And I board them too. Before I started growing my own hay we had two mysterious horse deaths that my vet blamed on the hay I was buying. One very old horse we had suffered kidney failure and a yearling we had the same thing plus she also had chronic choke. Both were blamed on way too high nitrate levels in the hay we were buying and also dirty mouldy hay. Some hay growers get real carried away with the nitrogen trying to maximize their yield and have no idea what damage they can do with it. I get 80 to 100 bales per acre off my bermuda sticking to the maximum recommended (according to soil analysis) every month and I'm happy with that. I know some morons going all the way up to 100 or 200 pounds of nitrogen though. That's the kind of junk that killed my two horses.
Our soil analysis reports used to have two rates listed on them. I think it was only 20-0-0 with a 7 day withdrawal time for grazing and 60-0-0 for haying with a 28 day withdrawal time for animals and 28 days between cuttings after spreading. We just always wait those times now and have only the haying recommendation listed.