Solving this linear system based on the combustion of methane, has no constants I recently discovered that I could solve a chemical reaction using a linear system. So I thought I would try something simple like the combustion of methane.
where x y z and w are the moles of each molecule
x $CH_4$ + y $O_2$ = z $H_2$O + w C$O_2$
the linear system for this would be:
x = w
2y = z + 2w
4x = 2z

I got as far as y  = z and y = 2w but without any constants, I am stumped. Can anyone help me? I was assuming that elimination and substitution would suffice, but I must be wrong.

Each of your equations can be rearranged to give
$$x-w=0$$
$$2y-z-2w=0$$
$$4x-2z=0$$
Your idea of using Gaussian elimination is then a very good one. Or you can solve directly. From the first equation, you get $$x=w.$$ From the third equation, you get $$z=\frac{x}{2}=\frac{w}{2}.$$ Then from the second equation you get $$y=\frac{z}{2}+w=\frac{5w}{4}.$$
In each subsequent equation, we have used previous information and any value for $w$ will suffice ($w$ is called a free variable).
Note that this makes sense as $w$ is a particular number of type atoms and changing it will change how the equation is to be balanced.
