(O Level Chem) Using matrices to balance chemical equations

Balance chemical equations by counting atoms on the LHS and on the RHS… 

Let's balance this equation for the reduction of iron(III) oxide by carbon monoxide.

Fe₂O₃ + CO --> Fe + CO₂

You see there're 2 iron atoms on the left, and so you put a coefficient of 2 for iron on the RHS.

Fe₂O₃ + CO --> 2Fe + CO₂

So far so good. You then count there are 4 oxygen atoms on the LHS and so you put a coefficient of 2 for carbon dioxide on the RHS to balance the oxygen atoms.

Fe₂O₃ + CO --> 2Fe + 2CO₂

But by putting that 2, you can't balance the carbon atom. So you got to start over again. After some tries, if you happen to put a 3 as the coefficient of carbon monoxide, you're in the right path. And you'll eventually get:

Fe₂O₃ + 3CO --> 2Fe + 3CO₂

There's another interesting way of balancing equations. I just found it on the Internet. It's by using matrices if you're familiar with your secondary 3 or 4 elementary math.

Using matrices to balance chemical equations involves solving simultaneous equations… 

The steps are:

1. Write the coefficients a, b, c, d etc in the equation.
   
2. Form four vectors for the four reactants and products. Make sure you account for all the elements in all the vectors.
   
3. Form simultaneous equations and solve for a, b, c, d etc.
   

But when I apply the matrices method to the equation for the decomposition of iron(III) oxide by carbon monoxide, I found there are four variables a to d and only three equations. You can't solve it unless you make a guess that b = d = 3, then you'll arrive at a = 1 and c = 2.

You can even skip the construction of the matrices and form the simultaneous equations. Look at the combustion of methane.

CH₄ + O₂ --> CO₂ + H₂O

Put the coefficients a-d.

aCH₄ + bO₂ --> cCO₂ + dH₂O

Carbon atoms: a + 0 = c + 0 ==> a = c
Hydrogen atoms: 4a + 0 = 0 + 2d ==> 2a = d
==> d = 2c
Oxygen atoms: 0 + 2b = 2c + d ==> 2b = 2c + d
==> 2b = d + d ==> 2b = 2d ==> b = d

Let a = c = 1

Therefore, d = 2 x 1 = 2, b = d = 2

CH₄ + 2O₂ --> CO₂ + 2H₂O 

Now, have fun using this interesting method to other equations in your Chemistry books.