Hidden Values in Variables

There are expressions and then there are equations. Expressions are math statements with no equal sign. 2+2 is an expression. 2+2=4 is an equation. Remember it this way. An equation shows an equality.

4 + 8
5 - (-12)
4 * 8
16 ÷ 4

4 + 8 = 12
5 - (-12) = -7
4 * 8 = 32
16 ÷ 4 = 4

Variables are Symbols and Letters

Sometimes an expression doesn't have two numbers. Sometimes you will see a letter or symbol. Those letters or symbols can represent any number in the Universe and they are called variables. Usually you will be asked to solve the value of the variable when you are given math problems.

4 + a
5 - n
4 * #
16 ÷ ρ

Start with What You Know

You are already used to solving problems where you don't know the answer. When variables are around, the answer has just been jumbled into the equation. An easy equation would go like this. 3 + 8 = n
11 = n (You can do this.)

We're just switching things around when we think of equations with variables. 3 + n = 11
n = 8

When you are working with variables in expressions they can mean anything. When you place them in an equation they usually have specific values. If we give the expression 2+x it really doesn't mean anything. 'x' could be equal to 5 or 5 million. But if we set up the equation 2+x=6 then there is only one value for 'x'. We can solve it this way.

2 + x = 4
(-2) + 2 + x = (-2) + 4 (you can add a value to both sides in an equation)
0 + x = 2 (any number added to it's opposite is equal to zero)
x = 2 (a number added to zero is equal to the original number)

The above example shows how you can 'zero out' one side of the equation to figure out the value of the variable ('x'). One more.

5 * n = 25
n * 5 = 25 (some math rules let you move values around in an equation)
(n * 5) ÷ 5 = 25 ÷ 5 (you can divide both sides of an equation by the same value)
n * (5 ÷ 5) = 25 ÷ 5 (when using multiplication and division, you can regroup values)
n * 1 = 5 (a number divided by itself is equal to one)
n = 5 (any value multiplied by one is equal to that value)

You could have probably done the last one in your head. "Five multiplied by what number is equal to twenty-five?"

We'll do more work with variables but the thing you need to remember now is that a variable represents an unknown value. Most of your math problems will ask you to solve for those unknown values.


We are starting to use some rules in our examples. As you learn more algebra, you will learn about these special rules that control equations. If we show you how they are used now, you won't be scared when you start learning about proofs. You probably know all of the rules, no one has told you their names yet.


1 + 0 = 1 -> When you add zero to a number, the number doesn't change.

5 * 1 = 5 -> When you multiply a number by one, the number doesn't change.

See? You know these.


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