In arithmetic, we work with numbers we know. For example, $3 + 5 = 8$. It’s straightforward. But what if we don't know one of the numbers? What if you know you have some money in your wallet, and a friend gives you $5, but you don't count the new total yet?

This is where algebra begins. We use a symbol to hold the place of an unknown amount. This symbol is called a variable. Most of the time, we use a letter like $x$ or $y$ as a variable.

So, the money in your wallet could be represented by the variable $m$. The total amount after your friend's gift is $m + 5$. This combination of variables, numbers, and operations (like addition, subtraction, multiplication, and division) is called an algebraic expression.

Here are a few more examples of expressions:

• $x - 10$ (10 less than some number)
• $4y$ (Some number multiplied by 4)
• $\frac{z}{2} + 1$ (Some number divided by 2, plus 1)

We can translate everyday situations into algebraic expressions. The key is to identify the unknown quantity and the operations involved.

Let's say you buy a ticket to a movie for $12 and some snacks. If you don't know the cost of the snacks, you can use a variable, like $s$, to represent it. The total cost for your trip would be:

In algebra, we often omit the multiplication sign between a number and a variable. If a box of pens costs $3 and you buy several boxes, you could write the total cost as $3 \times b$, where $b$ is the number of boxes. But it's more common to just write $3b$.

Sometimes, expressions can look long and complicated. We can often make them simpler by combining "like terms." Like terms are parts of the expression that have the same variable raised to the same power. Numbers without variables are also like terms.

For example, in the expression $5x + 2 + 3x$, the terms $5x$ and $3x$ are like terms because they both have the variable $x$. We can combine them.

Imagine you have 5 apples and your friend gives you 3 more apples. You now have 8 apples. It's the same idea. If $x$ represents an apple, then $5x + 3x$ is just $8x$.

The full expression $5x + 2 + 3x$ simplifies to $8x + 2$. We can't combine the $8x$ and the $2$ because they are not like terms. One has a variable, and the other doesn't.

Notice that when a variable stands alone, like $a$, its coefficient is 1. So, $4a + a$ is the same as $4a + 1a$, which equals $5a$. Simplifying expressions by combining like terms makes them easier to work with later on.

Ready to check your understanding? Let's try a few questions.

Mastering variables and expressions is your first big step into the world of algebra. It's the language we use to solve all kinds of interesting problems.