1.4 Variables and Equations
Numbers are abstract creatures that we use to represent how many "things" we have. Variables, in turn, can abstractly represent numbers. When there's a relationship between different variables in the form of a pattern, we can develop an equation to represent this relationship.
We started talking about counting numbers, mathematical creatures that you've worked with since you were just a little kid, and then wound talking about irrational numbers, which are numbers that you probably won't run into in the day-to-day physical world. Together, the counting numbers, integers, rational numbers, and irrational numbers, make up all of the real numbers. These are the types of numbers that we'll be using for this course.
Now that we have a sense of our abstract numbers, can we move up a level of abstraction? This is the type of question that the mathematician always asks. By representing objects from the real world as numbers, such as having 2 tables or having 3 chairs, we can use the language of mathematics to answer questions about quantity. If we combine 2 tables with 3 tables, then instead of thinking about the tables, we can first just think about adding the 2 and the 3 to get 5 and then translate back to the specific object, namely 5 tables. Perhaps we'll receive a similar benefit if we can think of numbers as specific objects and develop an abstract representation of the numbers themselves.
To see one real world benefit, let's say you're selling Apps for the iphone and you make 30 cents per App. If you sell a total of 1000 Apps, you'd make $ dollars. If you're a business person, and have to figure out how much you make on a monthly basis, you'd probably want to have some sort of spreadsheet or computer program that will compute the profit for you.
You can let x, known as a variable, represent the number that you sell without specifying a particular number –that's why "x" is more abstract than a number – and then in your spreadsheet program, use .3x to represent your profit; you then tell the program the number that you sell and you let it do the computation for you.
Using variables, we can then create equations that can then be used to model a real world situation. We can think of an equation as a way to describe a process using the language of mathematics. As an example, if you wanted your profit to be $300 and wanted to know how many units you'd need to sell in order to achieve this profit, then you could set up the equation:
In other words, "Find the unknown quantity x, so that .3x is equal to our desired profit of $300". We've translated a real world problem into the mathematical language: now using the rules of algebra, one branch of mathematics, we can solve for the unknown variable. In this case, we can divide both sides by .3 to get that x = 1000.
In this book we'll learn many different techniques that we can use to solve equations. But, before solving an equation, it's super important to understand how to translate the problem from English to mathematics. While there are no hard and fast rules as to how to do this, some key tips are:
- Know what it is that you're actually looking for
- Start with a "rough" equation – one that contains both English and math
- Draw a picture if possible, labeling unknown quantities
You're trying to construct a rectangular pen for your dear pig Wilbur. You'd like the length of the pen to be 4 more than the width so that Wilbur has ample room to run around. If x represents the width of the pen and you have 100 feet of wire available to construct the pen, then which equation represents the relationship between the unknown width and the other information given?
Try to work it out before seeing the full solution below.
From our picture, we see that the two widths are each x while each length is x+4. Adding up the total number of x's (four of them) plus the non-x values (8) gives us the total perimeter, which is 100. This tells us that the equation:
can be used to describe the relationship. Once we've translated the problem into the language of mathematics, we'll be able to use tools from algebra to solve for the unknown.
Truth be told, the above example probably doesn't have much of a real world application. In fact, to be completely honest, most problems that you'll see in any introductory algebra course don't have much "practical" value. However, these basic concepts are the building blocks of more advanced problem solving techniques that are used in the real world to solve some very, very complicated problems.
As an example, consider an airline looking to maximize their profit. They need to figure out how many planes they should have, the number of routes, what to charge customers, among a host of other variables. Do you think that they're just going to "guess" what number to assign each of the unknown variables in order to maximize their profit? Of course not! They use some very advanced techniques from algebra to figure out all of their unknowns (potentially hundreds), in order to maximize profit.
After working through some problems in this section, which will give you a chance to practice translating some English statements into mathematical ones, you'll begin to notice that one helpful tool in developing equations is being able to visualize what's going on. In the next section, we're going to study the number line, which will connect the concepts of numbers, variables, and representing them both visually.