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Chapter 2: Problem 83

Solve each formula for the specified variable. $$ P=2 L+2 W \text { for } W $$

Short Answer

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

Step by step solution

01

Identify the given formula

The given formula is: \[ P = 2L + 2W \] We need to solve this formula for the variable \( W \).
02

Isolate the term containing W

Subtract \( 2L \) from both sides of the equation to isolate the term containing \( W \): \[ P - 2L = 2W \]
03

Solve for W

Divide both sides of the equation by \( 2 \) to solve for \( W \): \[ W = \frac{P - 2L}{2} \]

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Key Concepts

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

Algebraic Formulas
Algebraic formulas are equations involving variables and constants that represent relationships between different quantities. In our example, the formula is \( P = 2L + 2W \) It represents the perimeter \( P \) of a rectangle in terms of its length \( L \) and width \( W \). When you know an algebraic formula, you can derive unknown values by plugging in known values. This is a powerful tool in algebra. Understanding how to manipulate these formulas is crucial for solving problems.
To effectively work with algebraic formulas:
  • Identify the variables and constants.
  • Understand the relationships between different parts of the equation.
Knowing how the variables interact helps in solving for any unknowns.
Isolation of Variables
Isolation of variables is a critical step in solving equations. To isolate a variable, you need to get it alone on one side of the equation. This involves rearranging the equation using algebraic operations.
In our task, we need to solve the formula for \( W \). The formula is: \( P = 2L + 2W \).
The goal is to isolate \( W \). Here鈥檚 how to do that step-by-step:
  • Start with the given equation: \( P = 2L + 2W \).
  • Subtract \( 2L \) from both sides: \( P - 2L = 2W \).
Removing \( 2L \) from the right side leaves us with \( 2W \) alone, which makes the next step a lot easier.
Algebraic Manipulation
Algebraic manipulation is the process of using algebraic operations to simplify or rearrange equations. This is essential for solving equations and involves operations such as addition, subtraction, multiplication, and division of terms.
In the final step of our example, we perform algebraic manipulation to solve for \( W \):
  • We have the equation \( P - 2L = 2W \).
  • To solve for \( W \), divide both sides by 2: \( W = \frac{P - 2L}{2} \).
This division step isolates \( W \), giving us the solution in a simple form.
Mastering algebraic manipulation lets you solve complex formulas and equations methodically, reducing errors and ensuring accurate solutions.

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