Section 15.4 Closing Ideas
This section is primarily an exercise in your core algebra skills. Solve a linear equation for a variable, make a substitution, and then solve for a variable. These are all things that we've already done previously. What's new about it is that those skills are being put together in a brand new way.
It is a common theme in mathematics that we build new ideas on the backs of old ideas. This is also where we can start to identify weaknesses in our understanding, as there may be one step in particular that appears to be more difficult than the others. For example, many students do fine with these problems when all of the numbers are integers, but as soon as fractions start to appear they start to struggle.
This is one of the challenges of the way the field of mathematics is built out. A lot of ideas are built on the foundation of your basic skills. And so even as you move forward into higher levels of math courses, you need to make sure you take the time to go backward to fill in the gaps that you have. And that's one of the main goals of this book.
As you continue to fill in those gaps, you will start to discover that you're able to make connections between ideas more easily, that you're able to use the tools that you have already developed more efficiently, and that you're able to see how to apply your ideas more effectively. In the end, those are the markers of mature mathematical thinking.