Variables and Algebra

Any symbol can represent a variable, but most often a letter is used. Let's say that a is a variable that describes the weight of an apple. For a particular apple, you can find out what a is by weighing the apple.

Now, maybe you want to find out the relationship between the weight of an apple and its height. The variable h can be used to describe the height of the apple.

Small Apple Large Apple

a = 130 grams a = 265 grams

h = 6 cm h = 7.5 cm

You can use graphing to begin to show the relationship between weight and height. If each height gives a different weight of apple, you might be able to write the relationship as a function.

Graphing in two dimensions, shows us part of the relationship between the height and weight of an apple. With only two points of data, we don't know what the graph will look like. We need more data.

The horizonal line labeled "h" is called the horizontal axis and each square to the left represents 1 cm of height.

The vertical line labeled "a" is called the vertical axis and each square is labeled to represent 50 grams.

We will explore more about linear equations later, but first we need to know more about what you can do with variables. Variables can be added, subtracted, multiplied, divided and squared, just like any number.

You can practice plotting points on a graph at webmath.com or graphing functions at graphsketch.com

Variables can also be raised to an exponent. It is useful to review what an exponent means and in what ways you can combine exponents. Manipulating exponents means moving them around according to set rules. There are 6 rules for manipulating exponents using variables. All of them rely on the fact that to combine exponents, the base number (the number you are raising to an exponent) must remain the same. The only exception is when you have two multiplied (or divided) numbers raised to the same power (then you can distribute the power over the two different numbers).

You can review the rules and some practice problems here.

You can also review many other math concepts at math-aids.com or basic-mathematics.com.

An expression describes something. An algebraic expression uses variables, numbers and operators (+, -, *, /, exponents and roots) to describe a value. When you set this expression equal to another expression, it creates and algebraic equation.

All sorts of problems can be described by algebraic equations. You may even solve a problem without realizing that you are using algebra. If an ice cream cone costs $2 and you have $8 to spend, how many ice cream cones can you buy?

2 times the number of ice cream cones = $8 or

2 * I = 8

So, dividing $8 by the cost of an ice cream cone ($2) gives you the number of ice cream cones you can buy. I = 4 ice cream cones.

Paying close attention to the language of the problem helps you to set up the right equation. Then, to solve the equation (find the value of the variable), you can move around parts of the equation using the rules of algebra. Basically, the rules are like a balance, you can add anything to one side as long as you add the same to the other side. You can also subtract anything from both sides of the equation.

You can also multiply both sides by a constant. You can divide both sides by a constant. You can also multiply or divide by a variable. You can even square both sides of the equation without changing the relationship, because both sides are equal expressions.

In the above balance there are 2 yellow 20 gram weights on one side. How many 5 gram weights are on the other side?

If our variable is n, the number of weights with the value 5 grams, then

5 * n = 2 * 20

This is the same as: 5 * n = 40

So, to find n, we can divide both sides by 5.

(5 * n) / 5 = 40 / 5

so, n = 8

If we add 2 green weights to each side, will it change the balance between the sides?

No, because (4 * 8) + 20 = 40 + 20

or (4 * 8) + 100 = 40 + 100

If we take away 20 grams from each side, will it change the balance between the sides?

No, because 4 * 8 - 20 = 40 - 20

We could also multiply the weight 4 times on each side. It is still true that

4 * (4 * 8) = 4 * 40