In order to solve for this equation for heat capacity, do I need to treat it as a system of equations?
In the first equation, what does C designate.


Since the final value is energy in joules, then it should be the the unit for temperature. She should have used °C to be clear.


  If it designates the heat capacity of the system, do I solve for q by solving a system of equations?


No, because you don't have the correct equations. You need the amount of heat released from the hot water minus the amount of heat absorbed by the cold water to give you the heat absorbed by the calorimeter.