At a Glance - Variables as Unknown Quantities
Sometimes we're given some tantalizing information about a number, and we need to figure out what the number is.
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.
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.
For what values of y does y2 = 9?
For what values of y is |y| ≤ 1?