I want to take a detour to talk about linear algebra. It's featured quite a bit so far in this book, although it was scarcely mentioned by name. In fact linear algebra underlies every chapter we've done so far.
Imagine you have two equations:
Let's say 
and 
is 
and 
, respectively. We can now write the following equations as such:
And we can solve it using basic algebra (please do work it out on your own): 
and 
.
What if you have three, four, or five simultaneous equations? It starts to get cumbersome to calculate these values. Instead, we invented a new notation: the matrix notation, which will allow us to solve simultaneous equations faster.
It had been used for about 100 years without a name (it was first termed "matrix" by James Sylvester) and formal rules were being used until Arthur Cayley formalized the rules in 1858. Nonetheless...