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How to feed a list of xy pairs into Solve to output xyz triplets


Hi:

Let's say that I have a list of xy coordinate pairs:

list= {{1,2}},{3,1},{5,4},{6,3},{2,5}}.

Now I'd like to find xyz triplets using Solve for the equation 6*x + 4*y + z == 1.

xyz results:  {{1,2,-13}},{3,1,-21},{5,4,-45},{6,3,-47},{2,5,-31}}. (I used Solve 5 times.)


Question: How do I code that into Mathematica for a "one shot output?" (Use Solve once.)



Thanks,

Bill


Mathematica 6.0.1 using Win XP on a pc.


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