Problem 68 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\( l = \frac{P - 2w}{2} \)

Step 1: Write down the equation

Start with the given formula for perimeter, which is expressed as \( P = 2l + 2w \). Our goal is to solve this equation for \( l \).

Step 2: Isolate the term with the variable

Subtract \( 2w \) from both sides of the equation to isolate the terms involving \( l \). Thus, the equation becomes \( P - 2w = 2l \).

Step 3: Solve for the variable

Divide both sides of the equation by 2 to solve for \( l \). This gives us \( l = \frac{P - 2w}{2} \).

Key Concepts

Isolating Variables

Whenever you need to solve an equation for a specific variable, isolating that variable is essential. This means getting the variable you are solving for, such as \( l \) in this exercise, alone on one side of the equation.
To isolate a variable, you'll often need to perform the same operation on both sides of the equation. In our original problem, we start with the equation for perimeter: \( P = 2l + 2w \). Here, we want \( l \) by itself on one side.
Here's what you should do:

• Identify the terms: Look at the equation and find where your variable is involved. In this case, terms involving \( l \) are \( 2l \).
• Use opposite operations: If \( 2w \) is added to \( 2l \), subtract \( 2w \) to cancel it out, moving it to the other side: \( P - 2w = 2l \).
• Complete the isolation: Finally, to get \( l \) by itself, divide by 2, yielding \( l = \frac{P - 2w}{2} \).

These steps will help you isolate variables in any equation, letting you solve for the unknown easily.

Perimeter Formula

The perimeter formula is crucial when talking about the boundaries of geometric shapes. It’s the total distance around an object or shape. For rectangles, the formula is standard: \( P = 2l + 2w \), where \( P \) represents the perimeter, \( l \) the length, and \( w \) the width.
Understanding this formula:

• Calculation of the perimeter: The formula adds up all sides of a rectangle. Since opposite sides of rectangles are equal, you multiply the width and the length each by 2 and then add the results.
• Use of the formula: You can rearrange it to express any variable, for example, find \( l \) when \( P \) and \( w \) are known, which means using algebraic techniques to manipulate it.

Knowing and manipulating perimeter formulas is beneficial in everyday tasks, like constructing frames or evaluating property size.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations or expressions to find unknown variables. It requires understanding the properties of equality and performing operations correctly on both sides of the equation to maintain balance.
In our specific task of solving \( P = 2l + 2w \) for \( l \):

• Subtraction: Subtract \( 2w \) from both sides to get \( P - 2w = 2l \). This step utilized the inverse operation of addition (subtracting \( 2w \)).
• Division: Then, divide both sides by 2 to isolate \( l \), giving \( l = \frac{P - 2w}{2} \). This step uses division, the inverse of multiplication.

Proper algebraic manipulation ensures you can rearrange formulas, simplify expressions, and solve equations accurately and efficiently.