Problem 32 - College Algebra

Question

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Solve for \(W : P=2 L+2 W\)

Step-by-Step Solution

Answer

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

Step 1: Identify the equation

We are given the equation \( P = 2L + 2W \). Our goal is to solve for \( W \), which means we need to express \( W \) as a function of the other variables present in the equation.

Step 2: Isolate terms involving W

To isolate terms involving \( W \), subtract \( 2L \) from both sides of the equation to get: \( P - 2L = 2W \).

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

Understanding Variables

In the world of mathematics, a variable is a symbol used to represent a number or value that can change or vary. Typically, letters such as \(x\), \(y\), or \(W\) are used as variables. For example, in the equation \(P = 2L + 2W\), \(W\) is our variable of interest. It represents a value we want to figure out based on other known quantities—\(P\) and \(L\) in this case.

• Variables are placeholders for numbers, and their values can change depending on the situation or given conditions.
• Understanding variables is crucial for solving equations, as it allows us to determine unknown values from known information.
• They help in forming equations and inequalities that model real-world situations.

When working with problems like this, knowing which variable to solve for is the initial step in reaching a solution.

Basics of Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions and equations to make them easier to work with. It's a powerful tool that helps in solving equations by moving variables and constants around according to specific rules. In the equation \(P = 2L + 2W\), algebraic manipulation is used to simplify and rearrange the equation to solve for \(W\).

Some basic rules of algebraic manipulation include:

• Adding or subtracting the same value from both sides: This helps in eliminating terms from one side, making it easier to isolate the desired variable.
• Multiplying or dividing both sides by the same non-zero value: Useful when needing to remove coefficients from a variable.
• Using distributive and associative properties: Allows simplification and combination of like terms.

An example from our problem is subtracting \(2L\) from both sides of the initial equation. This is a classic algebraic manipulation step that helps isolate terms involving \(W\).

Isolation of Terms to Solve Equations

Isolating terms in an equation is a crucial step to solve for a particular variable. The aim is to get the variable alone on one side of the equation with nothing else attached. In our equation \(P = 2L + 2W\), we want to isolate \(W\). This involves figuring out how to handle the other terms and constants so that we can clearly solve for \(W\).

The process typically involves:

• Identifying the term that contains the variable we need to solve for (in this case, \(2W\)).
• Using algebraic manipulation to make the targeted variable the subject of the formula (\(W\)).
• Performing operations such as addition, subtraction, multiplication, or division to systematically strip away other terms.

After isolating the term \(2W\) by subtracting \(2L\) from both sides, the next step is to divide each term by \(2\) to further isolate \(W\). This ultimately gives us \(W = \frac{P - 2L}{2}\), highlighting the importance of isolation in deriving the final solution.