You can put this solution on YOUR website! I have not heard of an echelon method, but I have heard translations of the word echelon from the French into other languages to refer to a strategy to solve systems of linear equations.
You want to keep an equation involving both variables and replace the other equation with a combination of the two where one of the variables has been eliminated.
I would keep because it's nice and simple and replace the other equation, using the sum of the two -->
The system is now
Even better, we can further simplify, dividing both sides by 7. -->
So we could say that we now have the system
Then substituting we get -->
Maybe you were trying to eliminate instead of y to give your modified system the same shape as the one in the book.
We can do that too, if you get points for eliminating variables in alphabetical order.
Just multiply times -6 to get and add it to to get -->
Then you could simplify it by dividing both sides by -7 to get
