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Teachers & SchoolsStudy Guide

A **variable** is a letter, like *x* or *y*, that represents an unknown number. Another way of thinking about it is that a variable is "code" for a number. Mathematicians can spend hours at a party talking to one another in variables and no one will have the slightest clue what they're talking about.

The most common variable used is *x*. When we have multiple variables, we might use *x*, *y*, or even *z*. Really, any letter will work, as long as we're consistent with it. We tend to use *x*, *y*, and *z* because we feel bad for them having to hang out at the back of the alphabet all the time. This is their time to shine.

### Variables as Unknown Quantities

Sometimes we're given some tantalizing information about a number, and we need to figure out what the number is.

### Sample Problem

Fill in the blank: 2 + ? = 10.

The answer is 8, since 2 + 8 = 10. This solution may be particularly helpful for someone with 2 fingers on one hand and 8 fingers on the other.

### Sample Problem

What number doubled equals 24?

In other words, if you were to sleep half the day away, how long would that be?

In symbols, we know 2 · ☐ = 24, and we need to fill in the box with the appropriate number. In this case, the answer would be 12, since 2 · 12 = 24. You must've eaten a ton of turkey before you went to bed last night. Or a double dose of NyQuil.

The examples above use the placeholders "?" and "☐" to represent unknown numbers. While these placeholders work fine for basic examples like these, they wouldn't work as well for problems that have more than one unknown quantity. Also, most keyboards don't have a "box" key. If you're truly upset about this fact, you'll need to ask the Geek Squad at Best Buy. For these reasons, we use letters as placeholders for unknown numbers, and we call these letters variables. Because they are "very able" to represent numbers. Oh, great. We

*are*the Geek Squad.A variable expressing an unknown quantity is like a box waiting to be filled in. To rewrite the previous examples with variables, we write letters instead of question marks or boxes. Don't throw the boxes away though. You'll probably need to move soon.

Instead of writing "Fill in the blank: 2 + ? = 10," we write "Find the value of

*x*for which 2 +*x*= 10."And instead of "What number doubled equals 24?" we could write "For what value of

*x*does 2 ·*x*= 24?"Sometimes the information we've been given isn't enough to force a variable to be one particular number. The info may restrict the possible values the variable can have. In that case, we need to determine all possible values, or the range of values, that'll solve the given problem. This step can sometimes be slightly more work, but don't worry: we'll pay you overtime. It'll look great at your annual review.

### Variable Notations

Remember that mathematicians love to abbreviate things (RTMLTAT, for short). To write "3 multiplied by 4" in symbols, we could write 3 · 4, 3 × 4, or (3)(4). To write "3 multiplied by

*x*" we could also write 3 ·*x*, 3 ×*x*, or (3)(*x*).However, there's a much shorter way: write 3

*x*. When multiplying a number by a variable, we can write the number and the variable side by side. They get along swimmingly, so there's no need to separate them with a symbol. We can't do the same when multiplying numbers together, because if we write 2 next to 4, for example, we get 24. If you think that 2 times 4 is 24, then you may have taken a 2 × 4 to the back of the head.When multiplying two (or more) variables, we also write the variables next to each other to show that they're being multiplied. For example,

*xy*means "*x*times*y*." This is another reason that we go with such rarely-used letters as our variables. If we used*a*and*b*most of the time, you might see*ab*and think we're talking about somebody's six-pack.The mathematical convention (the usual way of doing things) is to write the number before the variable when multiplying numbers by variables. In other words, we write 3

*x*, not*x*3. If you do write*x*3 people will probably know what you mean, but you probably won't be invited back to the convention.Also, you should know that

*xy = yx*since multiplication of real numbers is*commutative.*When multiplying variables together, it can be helpful to write the variables in alphabetical order (*xy*or*xyz*), so we have a standard order in which to write them. Writing*yx*instead of*xy*isn't nearly as bad as writing*x*17 in place of 17*x*, but it's still frowned upon in certain circles. Generally the circles frequented by us math nerds. You scoff, but our frowns can be intimidating.When we multiply a variable by itself several times—almost like cloning, but much less controversial—we can use

*exponent*notation. For example,*x*·*x*·*x*=*x*^{3}. We can read*x*^{3}as "three copies of*x*," since*x*^{3}is an abbreviation for three copies of*x*multiplied together. Too bad we don't need 100 copies, 'cause then we'd get a price break.When dividing a variable by a number, there are a couple of different ways to write the division in symbols. Since , then and both mean "

*x*divided by 4." In this expression, the*x*could not possibly stand for the United States of America, because our nation is indivisible. Pledge of allegiance, represent.**Be careful**: It's safer to write division using fraction notation than it is to write division using the slash. Not that you'll be in any real physical danger if you do the latter, but it isn't advisable and here's why. The expression 1/4*x*is ambiguous, since it could mean either or*.*Avoid the problem by simply not writing 1/4*x*. No, your solution of avoiding the problem by skipping algebra altogether isn't a valid one. Nice try.### Constants

**Constants**are quantities that do not change. They're like crotchety old curmudgeons, forever stuck in their ways.- 3, 57, 2
^{99}, and are constants.

- π and
*e*are also constants.

We can also use letters to represent unknown constant values.

### Sample Problem

Joey gets

*d*dollars for each odd job he does for his grandpa. His grandpa literally told him he would pay him*d*dollars for each job, probably so that old coot could stiff his grandson when it came time to tell him how much*d*is. No matter what the job is, Joey is paid the same. If Joey does*t*odd jobs for his Grandpa in a day, how much money does Joey make that day?We don't have any amounts, and therefore, we can only answer this question in terms of the variables we're given. Joey makes

*d*·*t*dollars in a day, since that's his pay per job times the number of jobs he does. Because Joey gets paid the same amount per job,*d*will never change; therefore, we call it a constant. The varying quantity in this problem is*t*, since the number of jobs Joey does per day will vary depending on what his grandpa forces—pardon us—*asks*him to do.- 3, 57, 2