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The perimeter of a rectangle is the total distance around the edge of the rectangle. This is a fundamental concept in geometry. Understanding it helps when you need to solve related algebraic equations.

In a rectangle, the opposite sides are equal in length. The formula to calculate the perimeter (denoted as \(P\)) is:

• \(P = 2l + 2w\)

Here, \(l\) represents the length and \(w\) represents the width of the rectangle. The equation shows that the perimeter is the sum of all the sides of the rectangle.

Breaking it down, the formula adds the lengths of two lengths \( (2l) \) and two widths \( (2w) \). If you know the perimeter and one of these measurements (either length or width), you can find the other using algebraic manipulation.

Algebraic manipulation involves rearranging an equation to isolate a variable. It's a key skill in solving equations. Let's explore how this works using our perimeter equation.

Given the perimeter equation \(P = 2l + 2w\), if you want to solve for \(w\), you need to rearrange the equation. Here are the steps:

• Start with the original equation: \(P = 2l + 2w\).
• Subtract \(2l\) from both sides to eliminate the term involving length on the right: \(P - 2l = 2w\).
• Now, you have \(2w\) by itself on one side of the equation.

This process of separating the terms is essential in algebra. It allows you to focus on the variable of interest, which in this case is \(w\).

Once you've rearranged an equation using algebraic manipulation, the next step is solving for a variable. Solving for a variable means isolating it on one side of the equation so you can find its specific value.

In our example, after rearranging, we get \(P - 2l = 2w\). To solve for \(w\), divide both sides of the equation by 2:

• \(w = \frac{P - 2l}{2}\)

This operation leaves \(w\) alone on one side and shows \(w\) in terms of \(P\) and \(l\).

Solving for a variable is helpful when you need to express one quantity in terms of others, especially when making predictions or evaluating scenarios in real-life problems.