Problem 63 - A Graphical Approach to College Algebra

Question

Solve each formula for the specified variable.} \(P=2 L+2 W\) for \(W\) (Perimeter of a rectangle)

Step-by-Step Solution

Answer

The solution is \( W = \frac{P - 2L}{2} \).

Step 1: Understand the formula

The given formula is for the perimeter of a rectangle, where \(P\) represents the perimeter, \(L\) is the length, and \(W\) is the width. The formula is expressed as \(P = 2L + 2W\). Our task is to solve for \(W\).

Step 2: Rearrange the formula

We want to isolate \(W\) on one side of the equation. Begin by subtracting \(2L\) from both sides of the equation to get:\[ P - 2L = 2W \]

Step 3: Solve for the variable W

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

Key Concepts

Solving for a Variable

When we talk about solving for a variable, we mean isolating one variable on one side of an equation. This process allows us to find the value of that variable in terms of other variables or constants. Consider the equation of the perimeter of a rectangle:

• Given: \[ P = 2L + 2W \]

Our goal in this problem was to solve for the width, \( W \). The steps generally involve:

• Identifying the term with the variable you wish to isolate.
• Rearranging the equation so that this term is by itself on one side of the equation.

For example, to solve for \( W \), we first subtracted \( 2L \) from both sides and then divided by 2, effectively isolating \( W \) to find its expression: \[ W = \frac{P - 2L}{2} \] These steps can be applied to any algebraic equation where you're required to solve for a specific variable.The key is to perform the same operation on both sides of the equation to maintain equality.

Formulas in Algebra

In algebra, formulas are essentially equations that express relationships between different quantities. The formula for the perimeter of a rectangle, in this case, shows the relationship between the perimeter (\( P \)), the length (\( L \)), and the width (\( W \)) of a rectangle. When working with formulas in algebra, it's essential to understand:

• What each variable represents.
• How these variables are mathematically related.
• How to manipulate the formula to solve for different variables depending on the requirements of the problem.

For instance, in the formula \( P = 2L + 2W \):

• \( P \) is the total distance around the rectangle (perimeter).
• \( L \) and \( W \) are the length and width, respectively.

To solve algebraic formulas for a specific variable, rearranging the formula is key. This involves using basic arithmetic operations like addition, subtraction, multiplication, and division. The familiarity with different algebraic formulas is crucial to understanding how to manipulate these formulas effectively.

Geometry in Algebra

Geometry often makes use of algebraic concepts to solve problems related to shapes and space. In this exercise, we worked with a geometric formula describing the perimeter of a rectangle. The integration of geometry in algebra allows us to solve for dimensions and understand spatial relationships through math.Let's take the perimeter of a rectangle:\[ P = 2L + 2W \] Here:

• Perimeter is a measure of the distance around a two-dimensional shape.
• The relationship in the formula indicates that you calculate perimeter by adding together twice the length and twice the width.

This example shows how geometric concepts like shapes and their properties can be expressed using algebraic equations, enabling us to leverage algebra techniques in solving geometry-based problems. Understanding this link is particularly useful in real-world applications where measurement and calculation are essential.