At first I could wiggle the cables where they plugged into the inputs and the audio would come through, but eventually wiggling and putting pressure on them had no effect.

How can you fix this problem without buying a new system?

http://3.bp.blogspot.com/-oD8XxVy75RY/T86eQCZ0KgI/AAAAAAAAAF4/fp-1m81mnwA/s1600/altec+Lansing+board+burn.JPG

http://2.bp.blogspot.com/-h7xSkIjQ9RU/T86eRgclgCI/AAAAAAAAAGA/cSyVkoKMwJY/s1600/alteck+Lansing+rca+jacks.JPG

Please provide the makes and models of all your equipment involved, anything you've tried to fix the problem, and how everything is connected.

Also, have you tried replacing the cables that you wiggled? Did you experience any change?

Lastly, if you have any photos of the connections/connectors in question, please provide them.

http://3.bp.blogspot.com/-oD8XxVy75RY/T86eQCZ0KgI/AAAAAAAAAF4/fp-1m81mnwA/s1600/altec+Lansing+board+burn.JPG

http://3.bp.blogspot.com/-oD8XxVy75RY/T86eQCZ0KgI/AAAAAAAAAF4/fp-1m81mnwA/s1600/altec+Lansing+board+burn.JPG

It is an Altec Lansing 5 speaker surround audio system for computers.

I did not try to replace the cables because they are permanently connected to the speakers.

I removed the board and noticed the blackened mark circled in red. Wondering if that would cause the RCA inputs to short.

http://2.bp.blogspot.com/-h7xSkIjQ9RU/T86eRgclgCI/AAAAAAAAAGA/cSyVkoKMwJY/s1600/alteck+Lansing+rca+jacks.JPG

Looks like the black mark is just glue/adhesive; without a clearer photo, it is difficult to tell.

Are you wiggling the wire, the connector, or the input connection?

If it's the wire or connector, it's possible to just splice in a new wire/connector. If it's the input connection itself that seems to be the issue, I would touch up the solder connections on the input board (although just looking at your photo they seem fine).