Introductory Algebra Tutorials

Variables

• The chief challenge in introductory algebra is mastering the concept of variables. It may sound like a complicated word, but the concept is much easier than it seems. A variable is simply a symbol--usually a letter--that represents a number.

  By now, the following problem should be a piece of cake for you:
  5 + 7 =

  Imagine if it were written:
  5 + 7 = ?

  In this problem, the question mark is our variable. Since answer is 12, the question mark represents or replaces the 12.
  ? = 12
  Simple, right?

  Now imagine if the problem were written this way:
  5 + 7 = x

  Now x is our variable, and x = 12.

  Consider this problem:
  5 + x = 12

  In this case, the x is replacing the 7, so x = 7.

  It's not hard to solve this problem:
  x - 3 = 6
  You know that 9 - 3 = 6, so the x replaces the 9, so x = 9.

  The previous problem can also be written:
  a - 3 = 6
  z - 3 = 6
  n - 3 = 6
  # - 3 = 6

  Don't let different letters or symbols trick you. No matter what symbol is used, the problem remains the same.

Solving an Equation

• But what about a problem like:
  4x - 24 = 68

  Problems that are too difficult to simply look at and answer must be solved through a series of steps. The goal of these steps is always to get the variable by itself on one side of the equal sign and the rest of the problem on the other side.

  Remember the order of operations (PEMDAS, or Please Excuse My Dear Aunt Sally: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? When solving equations, we reverse this order.

  So we begin with addition and subtraction. Our first step to get the x by itself is to add 24 to both sides of the equal sign. (Whatever you do to one side of the equal sign, you must do to the other.)

  4x - 24 + 24 = 68 + 24

  Which leaves us with:
  4x = 92

  Now we move on to multiplication and division. Since the problem multiplies x by 4, we must divide by 4 to get x by itself.

  4x / 4 = 92 / 4

  x = 23

  In this case, with just two steps, our problem was solved. In introductory algebra, most problems you encounter will only require addition, subtraction, multiplication, and division. As you progress to more advanced problems where the x is still not by itself after performing those operations, you will continue to exponents and then to parentheses.

Working With Variables

• We can add and subtract variables just like numbers.

  x + x = 2x

  7x - 2x = 5x

  We can also multiply and divide variables just like numbers. Remember that a number multiplied by itself is the number squared, and that a number divided by itself is 1. In the example below, a carat (^) is used indicate the exponent, so that "x^2" means x squared.

  4x * 3x = 12x^2
  9x / 3x = 3