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When solving equations, one of the most important steps is isolating the variable you're solving for. In our example, we need to find the value of W in the equation \( P = 2L + 2W \). To do this, we need to get W by itself on one side of the equation. Think of it like peeling an onion layer by layer until you get to the core.

First, identify the term that contains the variable you want to isolate. In this case, it's the term \( 2W \). To isolate this term, we need to remove the other terms from the right side of the equation.

We start by subtracting \( 2L \) from both sides to get:
\[ P - 2L = 2W \]

This step effectively moves the \( 2L \) to the other side, leaving us with only the term containing W on the right side. This is the key step to isolating the variable. Always remember: whatever you do to one side of the equation, you must do to the other.

Algebraic manipulation involves performing operations on equations to simplify or solve them. In our example, we used two main types of operations: subtraction and division.

After isolating the term with W, the next step is to solve for W by performing division. Our equation at this point is:
\[ P - 2L = 2W \]

Now we need to divide both sides by 2 to get W by itself. When dividing, each term on the left side of the equation must be divided by 2:

\[ W = \frac{P - 2L}{2} \]

This operation ensures W is completely isolated. The intuitive way to understand this is by thinking about balancing scales: if you take away or add weights (operations) from one side, you have to do the same to the other side to keep it balanced.

Pre-algebra exercises often involve understanding basic principles of algebra, like isolating variables and performing algebraic manipulations. These exercises build a strong foundation for more complex algebra topics.

Let's revisit our example for clarity. We started with:
\[ P = 2L + 2W \]

Using pre-algebra skills, we subtracted 2L:
\[ P - 2L = 2W \]

Then, we divided by 2 to isolate W:
\[ W = \frac{P - 2L}{2} \]

Such exercises help students get used to the logical and balanced nature of algebra. Practicing these can make solving more complicated equations easier in the future. Always approach each step methodically, understanding the why behind each operation, and soon enough, these principles will become second nature.