A dual cycle system for generating shaft power using a supercritical fluid and a fossil fuel. The first cycle is an open, air breathing Brayton cycle. The second cycle is a closed, supercritical fluid Brayton cycle. After compression of air in the first cycle, the compressed air flows through a first cross cycle heat exchanger through which the supercritical fluid from the second cycle flows after it has been compressed and then expanded in a turbine. In the first cross cycle heat exchanger, the compressed air is heated and the expanded supercritical fluid is cooled. Prior to expansion in a turbine, the compressed supercritical fluid flows through a second cross cycle heat exchanger through which also flows combustion gas, produced by burning a fossil fuel in the compressed air in the first cycle. In the second cross cycle heat exchanger, the combustion gas is cooled and the compressed supercritical fluid is heated.