If you do 1,000 J of work on a system and remove 500 J of heat from it, what is the change in its internal energy

Here I advice doing thought experiment and not remembering equations. By doing thought experiment you will write down equation by yourself.

Doing some work on a system means that system will "absorb" that energy. Here, system absorbs 1000J of energy. Now you are cooling system down by removing 500J of energy. That means that system will lose 500J of energy

All of that can be written as 1000 - 500 = 500 which means that after cooling your system has 500J left.