Learning about matrices on KhanAcademy, I came to a great video from Sal Khan explaining how to use matrices to solve equations. Thanks!
What occured to me while watching the demonstration was that it seems inefficient to reuse the same rows to drop results, because we lose potentially valuable previous rows, which could help us in further steps. My initial thought was : "we could probably make this more easy/efficient by create new rows for the results", so I decided to play with that idea.
Here is the equation system Sal started from: