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Let's start with the idea of variables in algebra. A variable is a symbol used to represent a number. It's like a placeholder for a value we don't know yet or that can change. Variables are usually represented by letters like \( d \), \( f \), or \( x \).

For example, in our exercise, the variables are \( d \) and \( f \). These could represent any numbers, but their exact values aren't specified. Understanding variables is essential because they allow us to write general mathematical expressions that can apply to multiple situations.

Think of a variable like a box that can hold any number inside it. This flexibility makes it easier to solve many different problems using the same methods and rules.

Now, let's talk about addition in algebraic expressions. Addition is a basic mathematical operation where you combine the values of two or more numbers to get a total. In algebra, you can also add variables together.

For instance, when you see the word 'sum' in a problem, it tells you to add. In our exercise, the phrase 'the sum of \( d \) and \( f \)' means we need to add these two variables together.

We use the '+' symbol to indicate addition. So, adding the variables \( d \) and \( f \) is written as \( d + f \). This expression means 'whatever value \( d \) has, add it to the value of \( f \)'.

Addition helps us combine values and find totals, a fundamental operation in algebra and all math.

Finally, let's look at what a sum is. A sum is the result you get after adding two or more numbers or variables together. Think of it as the answer to an addition problem.

In algebra, the sum doesn't change whether you're dealing with numbers or variables. For example, the sum of 3 and 4 is 7. Similarly, the sum of \( d \) and \( f \) is written as \( d + f \).

This concept is crucial because it allows us to combine unknown values in a logical way. By understanding sums, we can solve more complex algebraic problems where multiple terms are involved.

So, in our exercise, we translated 'the sum of \( d \) and \( f \)' into the algebraic expression \( d + f \). This simple expression shows the total when you add whatever values \( d \) and \( f \) represent.