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Chapter 9: Problem 467

In the following exercises, solve. Solve the formula P = 2L + 2W for W.

Short Answer

Expert verified
W = \frac{P - 2L}{2}

Step by step solution

01

Understand the formula

The formula given is: \[ P = 2L + 2W \] Here, P represents the perimeter, L represents the length, and W represents the width. The goal is to solve this formula for W.
02

Isolate the term containing W

To isolate the term containing W, subtract 2L from both sides of the equation: \[ P - 2L = 2W \]
03

Solve for W

To solve for W, divide both sides of the equation by 2: \[ W = \frac{P - 2L}{2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perimeter Formula
The concept of perimeter is fundamental in geometry. The perimeter of a shape is the total distance around it. For a rectangle, this is calculated as:
\[ P = 2L + 2W \]
Here,
  • \( P \) is the perimeter.
  • \( L \) is the length.
  • \( W \) is the width.

Remember, the formula tells us that the perimeter is twice the length plus twice the width. This is handy for finding the boundary of a space, like a garden or a room.
Isolating Variables
Isolating a variable means getting the variable by itself on one side of the equation. Imagine you have the equation for the perimeter of a rectangle, but you need to find the width (\( W \)).
Start with the formula: \[ P = 2L + 2W \]
To isolate \( W \), we need to remove the term involving \( L \). Subtracting \( 2L \) from both sides gives us: \[ P - 2L = 2W \]
Now, \( W \) is not completely alone yet. To finish isolating \( W \), divide both sides by 2: \[ W = \frac{P - 2L}{2} \]
This final equation shows the width in terms of the perimeter and length.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging and simplifying equations using algebraic rules. When working with the equation: \[ P = 2L + 2W \],
we use algebraic manipulation to isolate \( W \). Here are the steps:
  • Subtract \( 2L \) from both sides: \[ P - 2L = 2W \].
  • Divide both sides by 2 to solve for \( W \): \[ W = \frac{P - 2L}{2} \].

These steps show how we can transform an equation to find the value of one variable in terms of others. This is a crucial skill in algebra that helps in solving more complex problems.

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