1. Suppose we want to solve this system of equations:
First notice that we don’t need the x’s.
, or the plus signs.
We do need to separate the left and right sides of the equations, but we don’t need equals signs.
This “augmentation line” works just fine.
Throw in parentheses just cuz.
, and fill any missing spots with zeros.

2. We now have this (augmented) matrix:
To solve the original system of equations we proceed as follows.
Swap the first and third rows. -3
-12 +9 -6
-15 +9

3. We now have: -3 +6 -6 -3 +9
This process is called Gaussian Elimination. It results in a matrix in Row-Echelon Form.
These numbers are called “pivots.”