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# Defining Variables

Often students are confused with variables: those x's, y's, z's and other letters that begin to replace numbers beginning in algebra and continuing on into geometry, algebra II, Trigonometry, Precalculus, and Calculus. While there are several aspects to variables, one of the best ways to start is to understand a variable as a placeholder for a number. Take the equation: 7 + 3 = 10. Inserting x into this equation for the number 3 produces the "same" equation 7 + x = 10. Yet this equation also shows how a variable is "defined" in that x must equal 3 for the equation to work. More often variables are defined, however, not by a specific number, but with an idea that covers either a to-be-determined range of numbers or an as yet unknown number.

Try this example. Suppose I go to the store planning to put a cheese and cracker tray together for an upcoming football game. I have 24oz. of cheese and 28oz. of crackers and I expect each person to eat 4 oz. of cheese and they need 2oz of crackers for every 1oz. of cheese, so how many people can I serve cheese to, and how many more crackers do I need for all my cheese? Before scrolling down, think about how you would solve this in real life and then think about how you would work this problem for class.

I would begin to define y as the number of people and z as the additional amount of crackers in ounces that I need to buy, i.e. the two questions being asked that I do not know. Then I would link those variables to what I do know. I know that for every person y, they will eat 4oz of cheese, and that should equal the total amount of cheese (24oz), i.e. 4y = 24. I also know that for each of the 24oz's of cheese, I need 2oz of crackers. I also know that the total amount of crackers for that cheese is the 28oz I have plus the z crackers I need to buy. Thus, 24 x 2 = 28 + z. Solving these equations, I find I have enough cheese for y = 6 people, and that I need z = 20 oz. of crackers.

While your teachers may try to complicate things with unneeded information, additional variables or other mathematical operators, most problems involve identifying what you know (the amounts of cheese and crackers you have and the rates at which those items are needed), defining as variables what you need to find out, and preparing equations that relate what you know to the variables you don't know.

John, nice explanation, especially the placeholder analogy. If you don't mind, I'll use it with my math tutee's.

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John M.

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