Algebra developed from a need to solve equations. When we talk about algebra today, we no longer think exclusively about equations, but working with expressions with variables in them. Here’s an example of an algebraic expression:

$$4x$$ |

But what does $4x$ really mean?

The first thing you need to know is that when a number and a variable stand right next to each other, like above, there is an invisible multiplication sign between them. Look at this:

$$4x=4\cdot x$$ |

But what is $4x$?

$4x$ tells you that you have four $x$’s. Mathematically, you can think of the expression as

$$4x=4\cdot x=x+x+x+x$$ |

Rule

$$a\cdot x=ax=\underset{\text{}a\text{times}}{\underbrace{x+x+\cdots +x}}$$ |

When you see $ax$, $a$ is telling you how many $x$’s you have. That means the letter $a$ is a number. You use a letter—a variable—to describe the number, because you don’t know what the number is. If you had written a number in place of $a$, $4$ for example, you would just be describing one single case, instead of all the possibilities.

You use a variable to show that the formula applies no matter how many $x$’s you have! That means that $a$ can be any number.

Pay close attention to the examples and make sure you understand what’s going on.

Example 1

Write $2a$ as a sum

$$2a=a+a$$ |

Example 2

Write $3b$ as a sum

$$3b=b+b+b$$ |

Example 3

Write $6x$ as a sum

$$6x=x+x+x+x+x+x$$ |

Example 4

Write $4ab$ as a sum

$$4ab=ab+ab+ab+ab$$ |

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