Section 12.4 Closing Ideas
This section focused more on developing your intuition for lines than on the algebra. Linear equations are extremely common in practical applications. For example, if a burger costs $3 and you want 5 burgers to bring to your friends, how much will that cost? This can be modeled as a linear equation.
Of course, simple cases such as this can be done with simple methods. As you start to work with "real life" situations, it's usually not going to solved by quick mental arithmetic. You're going to need better tools in your toolbox. And this is where having a background in algebraic reasoning and mathematical thinking will help.
As you worked your way through the worksheets, you might have started to notice that moving from one point to another along a line always required your variables to change in tandem. For example, the might increase by every time the decreased by We will see that this is a reflection of the idea of the "slope" of a line, which we will explore more deeply in the next section.
Without telling you, there were two types of modeling that were introduced in the last couple worksheets. These are known as interpolation and extrapolation. The basic idea of interpolation means to fill in data points between existing data points. (See Section 12.5.) Extrapolation means modeling data beyond the existing data points. Linear equations are often used for models of these types. It's usually more complicated because real life data rarely ever fits exactly on a line, so you have to go through a process of finding the "best" line that matches the data, but that application will have to wait for a science or business course.