Is it possible to add sound to a system to make it quieter?

Sure, but you have to be really fast.

First, you use a microphone to pick up the sound that's already there.  Then you generate a new wave that exactly copies the existing sound BUT it's a half wavelength (180 degrees) AFTER the original sound.

We know that if you have two waves with equal amplitudes, and they overlap, and the crest of one wave is on top of the trough of the other one, they cancel out and the result is NO WAVE.  So we overlap the new wave with the existing sound, and if we got the amplitude of the new one right, it can cancel out the original sound.

That's exactly what's going on in "noise canceling" headphones.  

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