The only thing stopping you is the power supply. That PC includes a 280W unit that probably would not be strong enough to power the 7790 comfortably. I would suggest upgrading to at least a 400W unit with the proper 6 pin connector.
Something like this would work:http://www.newegg.com/Product/Product.aspx?Item=N82E16817139026
There is plenty of room in the case.