Problem 31 - Precalculus Mathematics for Calculus

Question

Solve the equation for the indicated variable. $$P=2 l+2 w ; \quad \text { for } w$$

Step-by-Step Solution

Answer

The solution is \( w = \frac{P - 2l}{2} \).

Step 1: Understand the Equation

The given equation is the formula for the perimeter \( P \) of a rectangle, where \( l \) is the length and \( w \) is the width. The formula is \( P = 2l + 2w \). We need to solve this equation for the variable \( w \).

Step 2: Isolate Terms with \( w \)

In order to solve for \( w \), we need to isolate terms containing \( w \) on one side of the equation. Start by subtracting \( 2l \) from both sides of the equation: \( P - 2l = 2w \).

Step 3: Solve for \( w \)

Now that we have \( 2w = P - 2l \), we can solve for \( w \) by dividing both sides of the equation by 2: \( w = \frac{P - 2l}{2} \).

Key Concepts

Perimeter of Rectangle

The perimeter of a rectangle is a fundamental geometric concept that expresses the total distance around the rectangle's boundary. It is determined by the lengths of the rectangle's two pairs of opposite sides.
To calculate the perimeter, we use the formula:\[P = 2l + 2w\]where:

• \( P \): Perimeter of the rectangle
• \( l \): Length of the rectangle
• \( w \): Width of the rectangle

This formula originates from the sum of the lengths of all four sides of the rectangle, where each pair of opposite sides is equal. Understanding this formula is crucial for solving geometric problems that involve the perimeter, as it provides a direct relationship between the shape's dimensions.

Isolation of Variables

Isolation of variables is a key technique used to solve equations, especially those with multiple terms. The goal is to rearrange the equation to express a particular variable by itself on one side.
In mathematical problems, this often involves:

• Adding or subtracting terms on both sides
• Multiplying or dividing by constants

In our example of solving for the width \( w \) in the perimeter formula:\[P = 2l + 2w\]we start by eliminating other terms to isolate \( w \). This involves subtracting \( 2l \) from both sides, resulting in:\[P - 2l = 2w\]This step makes it easier to solve for \( w \) by focusing only on those terms that include the variable. By isolating \( w \), we are setting the stage for a straightforward algebraic manipulation to find its value.

Algebraic Manipulation

Algebraic manipulation consists of operations that transform equations to solve for unknown variables. It helps to simplify equations and effectively find solutions.
This technique often involves:

• Basic arithmetic operations such as addition, subtraction, multiplication, and division
• Rearranging terms logically to maintain the equation's equality

In our task with the rectangle's perimeter, after isolating the variable \( w \), the equation becomes:\[2w = P - 2l\]To solve for \( w \), apply division to both sides of the equation:\[w = \frac{P - 2l}{2}\]This final step of algebraic manipulation concludes the process by clearly expressing \( w \) in terms of the other known values, \( P \) and \( l \). Mastering this skill allows for efficient problem-solving across a variety of mathematical contexts.