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The perimeter of a rectangle is a fundamental concept in geometry. It's the total distance around the outside of the rectangle. Imagine walking around the border of a rectangle; the perimeter is the distance you would cover. To calculate the perimeter, add up the lengths of all four sides. However, in a rectangle, opposite sides are equal, so a simpler formula is used: - The perimeter, denoted as \( P \), is given by the formula \( P = 2l + 2w \), where \( l \) is the length of the rectangle and \( w \) is the width.This formula effectively doubles the sum of the length and the width, as it considers both pairs of opposite sides. Understanding this formula is crucial, especially when needing to manipulate it to solve for other variables, like width or length.

Variable isolation is a technique used to solve equations and involves manipulating the equation so that the unknown variable stands alone on one side of the equation. In our case, we want to solve the equation for \( w \) in terms of the other known quantities. Here are the steps to isolate a variable:- Identify the variable you need to solve for. In this exercise, it's \( w \).- Perform operations that will leave this variable by itself of one side. This may involve addition, subtraction, multiplication, or division.In solving \( P = 2l + 2w \) for \( w \), the first step is subtracting \( 2l \) from both sides of the equation. This gets rid of the term that doesn't contain \( w \) on the \( w \)-side:- Move: \( P - 2l = 2w \)This result shows that only the terms involving \( w \) remain on one side, paving the way to solving the equation by further simplifying.

Algebraic manipulation refers to the systematic process of rearranging equations by applying mathematical operations, such as addition, subtraction, multiplication, division, and factoring.To solve for \( w \) in the equation \( P - 2l = 2w \), we need to continue manipulating terms:- Our goal is to have \( w \) by itself. We see it's being multiplied by 2.- Use division to isolate \( w \): divide both sides of the equation by 2.This gives:- \( w = \frac{P - 2l}{2} \)Through division, the expression \( 2w \) simplifies to \( w \), leaving the equation solved for \( w \). Each algebraic step simplifies the equation, making it easier to extract the value of the variable of interest. This skill is critical when dealing with various algebraic problems, allowing one to pinpoint specific values efficiently.