Section 16.4 Closing Ideas
This section created a new toolbox of ideas to solve the types of equations that you already knew how to solve. In that sense, this section is extraneous. Why learn a new way to do something you already know how to do? But in another sense, this is a critical section because it teaches you that sometimes the way you've done things in the past isn't necessarily the way you want to do things in the future.
The method of substitution works for a wide range of problems, and it follows a very straightforward pattern: Solve for one of the variables and plug it into the other equation. The method of elimination requires some intuition and intellectual flexibility. You have to actively assess the situation and determine what choice of multiplication values will cause the elimination to happen. But if you choose wisely, the calculations are much simpler. In fact, with a bit of practice you can solve some of these problems mentally (as long as the numbers aren't too big). You would find that to be extremely difficult if you were thinking about using the method of substitution.
So consider this section to be one about simply expanding your toolbox of techniques. Instead of being confined to just one way of looking at these problems, you now have two. You have the option of looking at the equations to decide whether substitution or elimination makes more sense, and you have the freedom to pick the method that suits you the best. As you get further in your mathematical studies, you will find more and more situations where there are multiple approaches to solve a problem, and that you'll be making more active decisions about the methods and techniques that you apply.