Problem 85 - College Algebra with Modeling and Visualization

Question

Solve the equation for the specified variable. $$ P=2 L+2 W \text { for } L $$

Step-by-Step Solution

Answer

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

Step 1: Analyze the Equation

The given equation is \( P = 2L + 2W \). Our goal is to solve this equation for \( L \). This means we need to isolate \( L \) on one side of the equation.

Step 2: Subtract 2W from Both Sides

To isolate \( L \), we can first remove the term \( 2W \) from the right side by subtracting \( 2W \) from both sides of the equation. This gives us:\[P - 2W = 2L\]

Step 3: Divide by 2

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

Step 4: Write the Final Solution

The variable \( L \) has been successfully isolated. Therefore, the expression for \( L \) in terms of \( P \) and \( W \) is:\[L = \frac{P - 2W}{2}\]

Key Concepts

Solving for a Variable

Solving for a variable in an equation means isolating that variable on one side to determine its value in terms of other known quantities. Think of variables as unknowns that you need to find. To solve an equation for a specific variable, follow these simple steps:

• Identify the variable you need to solve for.
• Use operations such as addition, subtraction, multiplication, or division to isolate the variable.
• Perform the same operation on both sides of the equation to keep it balanced.

In our example, the equation is \( P = 2L + 2W \). The target variable is \( L \), which we solve for by rearranging the equation so that \( L \) stands alone.

Equation Manipulation

When dealing with equations, manipulation refers to the process of rearranging or altering the equation to isolate the variable of interest. This is a key skill in solving linear equations efficiently.
To manipulate an equation appropriately, you can:

• Subtract or add terms from both sides.
• Multiply or divide both sides by the same number to maintain equality.

In the provided equation \( P = 2L + 2W \), we improve the equation by subtracting \( 2W \) from both sides, moving us closer to isolating \( L \). Equation manipulation helps maintain balance in the equation while making it easier to see the relationship between variables.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Unlike equations which have an equality sign, expressions do not equate two sides. Instead, they're used within equations to express relationships between variables and constants.
In \( P = 2L + 2W \), the expression \( 2L + 2W \) represents a relationship between the perimeter \( P \), length \( L \), and width \( W \) of a rectangle.
Understanding algebraic expressions allows us to replace or adjust terms to solve for any variable. Once the equation is rearranged (\( L = \frac{P - 2W}{2} \)), it becomes clear how the expression connects these dimensions. Learning to interpret and manipulate these expressions is fundamental to mastering linear equations.