Problem 73 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the given equation

The equation given is \( P = 2l + 2w \). This equation describes the perimeter \( P \) of a rectangle with length \( l \) and width \( w \). Our goal is to solve for the width \( w \).

Step 2: Isolate the terms with the variable w

We need to isolate \( w \) on one side of the equation. Start by subtracting \( 2l \) from both sides:\[ P - 2l = 2w \]

Step 3: Solve for w

Now, divide both sides of the equation by 2 to solve for \( w \):\[ w = \frac{P - 2l}{2} \]

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate a particular variable of interest. This process requires understanding and applying basic arithmetic operations such as addition, subtraction, multiplication, and division.
By using algebraic manipulation, you can rewrite an equation to make a specific variable the subject, which means that variable is isolated on one side of the equation. This is essential for solving equations since it helps you find the numerical value of the variable given the known quantities in the equation.

• Start with performing opposite operations: If a variable is being added, subtract to remove it; if it’s being multiplied, divide to isolate it.
• Keep operations balanced: Whatever you do to one side, do to the other.
• Aim to get the variable by itself on one side of the equation.

In the context of the problem, we needed to solve for the variable \( w \) in the equation \( P = 2l + 2w \). This involved subtracting \( 2l \) from both sides and then dividing by 2 to isolate \( w \). Thus, through algebraic manipulation, we expressed \( w \) as \( \frac{P - 2l}{2} \).

Rectangular Perimeter

The perimeter of a rectangle is a key concept in geometry. It represents the total length around the rectangle, which is the sum of all its sides.
A rectangle has two pairs of equal sides: the lengths \( l \) and widths \( w \). Therefore, the formula for perimeter \( P \) is \( P = 2l + 2w \). This formula helps in understanding how changes in length or width impact the total perimeter.

• The perimeter gives an idea of the boundary length of a rectangle, which can be critical in landscaping, architecture, and design.
• Knowing one part of the perimeter equation allows you to solve for others, such as finding the width if you know the perimeter and length.

By using the given formula of perimeter, you can diplomatically manipulate it to solve for other variables, like isolating \( w \) when given \( P \) and \( l \). Understanding the perimeter equation aids in tackling practical problems involving rectangular spaces.

Variables in Equations

Variables in equations are symbols used to represent unknown values. They are crucial for formulating mathematical relationships and expressing general laws or patterns.
In our exercise, \( w \) represents the width of a rectangle, \( l \) the length, and \( P \) the perimeter. These variables help create a formula that can adapt to any particular instance of a rectangle, offering flexibility and generality in problem-solving.

• Understanding variables: Recognize each variable's role and what it represents in a formula or equation.
• Substitution: Once you've solved for a variable, you can substitute numerical values to find specific measurements.
• Multiple variables: Learn how they interact through the equation to produce meaningful results or new insights.

In the problem, solving for \( w \) involved understanding what each variable represented and how rearranging them would allow us to find the needed measure. With algebraic steps, we expressed \( w \) in terms of the other variables, showing how equations model dynamic, real-world relationships through symbolic means.