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Matrix question
Hey guys new here to the forum, I am just learning about linear algebra in college and can do the basics. My only problem is when we apply this knowledge to more abstract ideas, for example these two questions on chemistry and road maps.
I think I am on the right track with the first one, but I am completely stumped by the second, I have no idea how to attempt it.
For the first one (I must note I am no chemistry expert so forgive me if I'm completely wrong) but is it basically asking you to find a value for a,b,c,d and e such that:
aNaHC(O3)+b(H3)(C6)(H5)(O7)=c(Na3)(C6)(H5)(O7)+d(H 2)O+eC(O2)
???
from there i equated the amount of Na, H, C and O on each side such that:
Na: a=3c
H:a+8b=5c+2d
C:a+6b=6c+e
O:3a+7b=7c+d+2e
Then I moved everything onto one side of the 4 equations to we have:
(1)...a3c=0
(2)...a+8b5c2d=0
(3)...a+6b6ce=0
(4)...3a+7v7cd2e=0
With these 4 simultaneous equations, I constructed a matrix which I then solved in mathematica. It gave me b,c,d and e in terms of a. I assume this is because we have 4 equations but need 5 because we have 5 variables. so I guess the solution isn't unique and maybe I've answered my own question by myself?
Anyway I have no idea with the second. Thanks for your time everyone
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Re: Matrix question
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 rowreduce 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 TI89 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.

Re: Matrix question
Thanks a lot, I was just unsure about the first one because I wasn't sure how chemical equations are meant to work exactly. I now understand the second one, just had trouble getting that first step about intersections.