I haven't taken a close look at your first problem; I'm assuming that you read the chemical equation carefully, copied the numbers right, and got all the info out of it that you can. If so, you did it right. To have at least one free variable is something you'd expect. A chemical reaction could happen on a small scale or a large scale, right? You might produce one resultant molecule, or a large number, right?
As for the second problem ... I wonder what book you got it from, what comments or instructions the text might've provided? The idea seems to be that something is flowing into and out of each intersection, but I don't know what, vehicle traffic perhaps, measured in number of vehicles, or blood cells, or dollars. In that case, the number of whatever going in must equal the number going out, so at the upper left intersection we'd have
300 + 500 = x1 + x3
x1 + 0.x2 + x3 + 0.x4 + 0*x5 + 0*x6 + 0*x7 =800
You'll wind up with six equations in 7 unknowns, which you'll represent as a 6 x 8 matrix (the eighth column is the sums, it's an augmented matrix). Every entry, except in the 8th column, will be either 0 or 1. To row-reduce it should be straightforward, though tedious.
Mathematica is a great program, but entering matrices is tedious, at least in the older version I know. Better to use a TI-89 if you have one. I use SWP (Scientific Workplace) where entering the matrices is very easy.
Oops, I just noticed you have another png posted. So it's about traffic, yes, number of vehiclesle. With only six equations, yes, you should have a free variable, you'll need more info to produce a unique solution.