Is it possible to power an ultralight airplane with a single-cylinder weed-whacker engine? Plane:
Weighs 200 pounds.
16 FT wingspan,
36in. by 5 in. airfoil

Engine: 3,000 RPM, 2 HP

Propeller: 4ft with 4 blades and weighs 4 lbs. 

Is there perhaps an equation that could determine this?
There are methods of calculating the power required to fly a particular plane.  In your case, assuming that you'll also need to carry ~18 kg of fuel and a pilot (roundly 75 kg), the answer is NO.  A 2 hp (1.5 kw) power plant will be insufficient to give safe takeoff and climb performance -- with the possible exception of a high performance sailplane airframe (some self-launching sailplanes use ridiculously small engines).