2.2 Introduction to Linear Relationships
In this chapter we'll explore linear relationships, relationships which appear as straight lines when graphed.
One of the most basic geometric objects to graph is a line. These mathematical objects appear to us everywhere in the physical world: if you just take a second to look around the room, you'll notice many examples of straight edges, or lines. To get a deeper understanding of these mathematical creatures, let's first study a specific real world example, and then gradually increase our level of abstraction so that we have a bird's-eye view of lines. Just as a scientist's curiosity in one specific animal leads to an understanding of an entire species, a mathematician's curiosity about one particular equation can lead to a deeper understanding of a wide class of equations. The motivating example for this chapter comes from the world of business.
Suppose that you're the owner of a company that develops ipad Apps. While you're in charge of developing the App concepts, you've got to hire a computer programmer ($1,500 per month) and designer ($1,500 per month) to actually create the Apps. In total, you'll pay your two employees $3,000 per month, which will include basic work on the App and monthly revisions to the product. Further, you charge 99 cents for each App and each App costs 30 cents to produce (the fee taken out by the company through which you sell your App).
Being a good business person, you want to be on top of your finances: you'd like to quickly be able to figure out how much money you make for a given number of Apps sold. For each specific number of Apps that you sell, you can compute your total profit by determining the amount of money that you take in, known as revenue, less the amount of money that goes out, also known as your costs. In other words, you can use:
Total Profit = Total Revenue - Total cost
As an example, if you sell 100 Apps in a given month, your total revenue is .99 × 100 = $99 while your total cost is $3,000 + .3 × 100 = $3,300. Then, your total profit is:
Total Profit = $99 - $3,300 = -$3,201.
In other words, if you sell 100 Apps, then you've lost $3,201.
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For a different number of Apps, you'd have to do the computation once again, figuring out your revenue and your costs, and then subtracting the two quantities.
For example, if you sell 400 Apps, your profit would be: $ .
This approach of "thinking through" our profit each time works just fine. But, as mathematicians, we ask ourselves, "Is there a way that I can generalize this process? Is there some underlying pattern?" By generalizing, we often find ways to both streamline a specific process AND develop a deeper understanding of the thing that we're studying.
Finding an abstract pattern by just looking at one example is next to impossible. So, in the next section, let's first develop a table, with numerous specific examples to see if we notice a relationship between them.
