You cannot solve for x and y in the equation:
x + y = 5
because you do not have enough information. If you knew what x was, you could find y, and you can find what x and y are in relation to each other (e.g.
x=5-y), but you cannot find both values without more info.
Likewise, once two signals are mixed, you cannot find what either sounds like by itself from the mix alone because you have the sum of those signals.
Now... If you have the original drum signal you want to remove from the mix, you could add an inverted version of it--i.e., subtract the signal from the mix. You will want to tweak the volume and tone to be identical to what is mixed prior to inverting it. This is a basic noise reduction technique. I understand noise reduction works this way. If you do not get the volume (and any other effects) pretty close and the timing perfect, it will not work. Besides, if you have access to the original sounds, you can still edit the mix, right?