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When we talk about the perimeter of a rectangle, we're dealing with the total distance around the rectangle. A rectangle has two pairs of parallel sides: length (\( l \)) and width (\( w \)). The formula to calculate the perimeter (\( P \)) is:
\[ P = 2l + 2w \]This equation shows that we need to add up the lengths of all sides to find the perimeter. Multiplying the length and width by 2 accounts for both pairs of sides.
Understanding this formula is crucial for any problem involving the perimeter as it forms the foundation for solving many geometry-related problems. It's a simple yet powerful tool in math to determine one dimension when the perimeter and the other dimension are known.

Isolating a variable means rearranging an equation to get the variable of interest alone on one side of the equation. This process helps solve for that variable. For instance, using the perimeter formula \( P = 2l + 2w \) to find the width (\( w \)) involves isolating \( w \).
First, move all other terms to the opposite side of the equation. Subtract \( 2l \) from both sides to separate terms containing \( w \):

• \( P - 2l = 2w \)

Now, \( w \) is almost isolated, except for the multiplication by 2. The final step is to divide every term by 2:

• \( w = \frac{P - 2l}{2} \)

This process of isolating is fundamental in algebra to solve equations for unknowns. It helps us express one variable in terms of others, providing a clear path to finding numerical values or expressing relationships between variables.

Algebraic manipulation involves using mathematical operations to rearrange equations. In our example, the goal was to solve for the width (\( w \)) of the rectangle using the perimeter formula. Let's break down the key steps:

• Subtracting Terms: To isolate a specific term, like \( 2w \), you start by removing other terms from that side. Here, subtracting \( 2l \) from both sides left us with \( P - 2l = 2w \).


• Dividing Coefficients: Once we've isolated \( 2w \), divide all terms by the coefficient of the variable. This gave us \( w = \frac{P - 2l}{2} \).

Through applying these basic algebraic principles, we solved the formula to express one unknown (\( w \)) in terms of known values. Mastery of these techniques allows for flexible problem-solving in algebra and beyond, letting you manipulate various equations to find the solution you need.