Problem 69 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

\( W = \frac{P - 2L}{2} \)

Step 1: Understand the Formula

The given formula for the perimeter of a rectangle is \( P = 2L + 2W \). Our task is to solve this formula for \( W \).

Step 2: Isolate Terms with W

Subtract \( 2L \) from both sides of the equation to isolate the terms containing \( W \). This gives us:\[ P - 2L = 2W \]

Step 3: Solve for W

Now, divide both sides of the equation by 2 to solve for \( W \):\[ W = \frac{P - 2L}{2} \]. This isolates \( W \) as the subject of the formula.

Key Concepts

Isolation of Variables

When solving equations, our goal is often to make one variable the center of attention, or in technical terms, the "subject" of the equation. This is known as isolating variables. Imagine you're peeling away layers of an onion to get to the core. Likewise, in equations, we remove other terms to focus on the variable we're interested in.
The strategy involves several steps:

• Choose the variable you need to solve for, let's say it's \( W \) in our case.
• Move other terms to the opposite side of the equation. Usually, you can do this by adding, subtracting, multiplying, or dividing both sides by certain numbers or expressions.
• Continue until the variable stands alone on one side of the equation.

In our example, we start with the equation \( P = 2L + 2W \). To isolate \( W \), we subtract \( 2L \) from both sides, leaving \( 2W \) isolated on one side: \( P - 2L = 2W \). By dividing both sides by 2, we create a nice and tidy expression for \( W \). Remember, isolation helps in understanding how \( W \) relates to \( P \) and \( L \).

Algebraic Manipulation

Algebraic manipulation is like solving a puzzle where each piece must fit perfectly into place. The operations we apply to the equation are akin to adjusting puzzle pieces so that they form a coherent picture.
The key operations in algebraic manipulation include:

• Addition or Subtraction: Used to simplify or eliminate terms from one side of the equation.
• Multiplication or Division: Employed to combine terms or simplify expressions.
• Factoring and Expanding: Useful for expressing formulas in simpler or more useful forms.

In the case of our perimeter formula \( P = 2L + 2W \), we pay particular attention to how multiplication and division are used. After subtracting \( 2L \), we divide both sides by 2 to keep the integrity of the equation. This action ensures \( W \) is effectively extracted, giving us the expression \( W = \frac{P - 2L}{2} \). By mastering algebraic manipulation, you not only solve equations but also reinforce the relationship between different mathematical elements.

Geometry Formulas

Geometry is all about shapes, sizes, and the properties of space. Formulas in geometry provide shortcuts to calculate various attributes, such as area, volume, and perimeter, without needing to measure directly. For rectangles, the perimeter formula \( P = 2L + 2W \) gives us a way to calculate the total boundary length easily.
To effectively use geometry formulas, follow these guidelines:

• Understand the shape and what each variable represents. For rectangles, \( L \) is the length, and \( W \) is the width.
• Identify which formula applies to your need, whether it's perimeter, area, or another attribute.
• Be ready to manipulate the formula as needed, often requiring isolation of variables or algebraic manipulation.

In our example, rearranging the perimeter equation reveals insights into how each dimension affects \( P \). Understanding these relationships is critical in both academic exercises and real-world applications, such as construction and design scenarios.