Nope, the mixer's main outputs won't let you have individual track outputs unless it's an 8-bus mixer, and I've never seen an 8-bus mixer with only 7 inputs; usually more like 24, 32 or more.
If your mixer has "insert" points next to each XLR input, it's a fairly safe bet that you can patch out of those points and into a multi-input sound card (1 insert point per sound card input) and get separate tracks into your 'puter that way - one way to do this is get some adapters from Radio Shack - they offer one that looks like a 1/4" stereo male but ends up in a single RCA female on the other end - plugging this into your insert points will "tap off" the signal from each channel and let you run cables from there into your sound card... Steve