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Algebra is the study of properties of operations carried out on sets of numbers. That may sound like mumbo-jumbo, but understand that algebra is really a generalization of arithmetic in which symbols, usually letters, stand for numbers. You study algebra to solve equations and to find the value of a variable. For example, how often have you heard the command, "Solve the equation for x"? The SAT II Math test has a liberal dose of algebra problems in both Level IC and Level IIC.
Variables are symbols that stand for numbers. Generally, when letters are used as variables, they represent a certain value for a number. Just like their names imply, the valuable of variables can change, depending on the equation they are used in. And now, to introduce the most famous variables of all time, let us present: x, y, and z.
Constants, on the other hand, and as their name implies, are numbers that don't change their values in a specific problem. In algebra, while some letters may also refer to constants, they don't change their values in an equation as a variable may. To distinguish constants from variables, and because we have lots of letters in the alphabet to play with, algebraic problems generally designate constants with the letters a, b, or c.
Terms are any set of numbers and/or constants you can multiply or divide together to form a single unit in an equation. You can combine these single parts of an equation by addition or subtraction to become a sum or difference. As an example, the following algebraic expression has three terms.
ax2 + bx + c
An expression in algebra is a collection of terms that are combined by addition or subtraction, and the terms are often grouped by parentheses, such as (x + 2), (x – c3), (2x – 3y), and the like. While an expression can contain as few as one term, it's more common to think of an expression combining two or more terms.
Think of variables as abbreviations for discrete things. For example, if you buy two apples and four oranges, the clerk would not simply ring them up together by adding 2 + 4 to get 6. After all, that would be like mixing apples and oranges, wouldn't it!
So in algebra, you could actually make an expression for combining 2 apples and 4 oranges, and it may look something like
2a + 4o
In this expression, you know that the variables are a and o. The numbers 2 and 4 are called coefficients. So the coefficient of the variable a is 2, and the coefficient of the variable o is 4. Likewise, in the expression ax2 + bx + c, you can say that the coefficient of the variable x2 in the term, ax2, is a, while b is the coefficient of the variable x in the term bx. Finally, c is not a coefficient because it doesn't multiply another number. Instead, c is just a constant with a fixed numerical value.
In an algebraic expression, terms involving the same variable, even if they have different coefficients, are called like terms. For example, in the expression
3x + 4y – 2x + y
3x and 2x are like terms for obvious reasons. Likewise, 4y and y are also like terms. Like terms can be combined with each other, while other terms may not.
In the preceding expression, you can subtract the like terms, or 3x – 2x = x; and you can also add the like terms 4y + y = 5y. The end result of the expression, by performing these operations of adding and subtracting like terms, will become x + 5y, a much simpler expression to work with.
3x – 2x + 4y + y = x + 5y
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