Think of a variable as a placeholder for a number. It’s a symbol, usually a letter like $x$ or $y$, that represents a value we don't know yet, or one that can change. Instead of writing "a mystery number plus 3," we can simply write $x + 3$. This shorthand is the foundation of algebra.

Using variables allows us to describe relationships in a general way. For instance, the formula for the perimeter of a square is $4s$. Here, $s$ is a variable representing the side length of any square. If you know the side length, you can find the perimeter. If the side length changes, so does the perimeter.

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators (like +, -, ×, ÷). An expression doesn't have an equals sign. It's like a recipe for a calculation. For example, $y - 7$ is an expression. Its value depends entirely on the value of $y$.

We can evaluate an expression by substituting a specific number for the variable. Let's look at the expression $2a + 5$. What is its value if $a = 3$?

We just replace the variable $a$ with the number 3 and perform the arithmetic. When $a=3$, the expression evaluates to 11.

An equation is different from an expression. An equation is a statement that two expressions are equal. It's like a complete sentence, while an expression is just a phrase. The key feature of an equation is the equals sign (=).

For example, $x + 5 = 12$ is an equation. It states that the expression on the left ($x + 5$) is equal to the expression on the right (12). Our job is usually to find the specific value of $x$ that makes this statement true.

Think of an equation as a perfectly balanced scale. If you add or remove something from one side, you must do the exact same thing to the other side to keep it balanced. This is the central rule for solving equations.

To solve an equation, our goal is to isolate the variable, which means getting it by itself on one side of the equals sign. We do this by using inverse operations. Addition and subtraction are inverses of each other, and multiplication and division are inverses of each other.

Let's solve our equation: $x + 5 = 12$.

To get $x$ by itself, we need to undo the "+ 5". The inverse of adding 5 is subtracting 5. To keep the scale balanced, we must subtract 5 from both sides.

On the left side, $5 - 5$ is 0, leaving just $x$. On the right side, $12 - 5$ is 7.

We found the solution! The value that makes the equation true is 7. We can check our work by plugging 7 back into the original equation: $7 + 5 = 12$. It's a true statement.

Let's try one with multiplication, like $4a = 20$. Remember, $4a$ is just a shorthand for $4 \times a$. The inverse of multiplying by 4 is dividing by 4. So, we divide both sides by 4.

The solution is $a = 5$. By understanding how to manipulate variables using inverse operations, you can solve for the unknown in many different situations.