Problem 32 - College Algebra with Corequisite Support

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: Isolate the Term with Variable 'W'

The first step in solving for \( W \) is to move all terms not containing \( W \) to the other side of the equation. Our equation is \( P=2 L+2 W \). Start by subtracting \( 2L \) from both sides of the equation:\[ P - 2L = 2W \].

Step 2: Solve for 'W'

Now that we've isolated the term containing \( W \) on the right side of the equation, we divide both sides by 2 to solve for \( W \):\[ \frac{P - 2L}{2} = W \].

Step 3: Result

The formula for \( W \) is now solved. We have:\[ W = \frac{P - 2L}{2} \].

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's like a language used to describe patterns and relationships. The goal is to work with equations and formulas to find unknown values. In algebra, you'll often see variables like \( x \), \( y \), and \( W \) that represent unknown values.
Understanding algebra helps in solving equations systematically.
Here are some key points about algebra:

• **Symbols and Variables:** Symbols represent quantities in algebra, which often include variables like \( L \), \( W \), and constants.
• **Equations and Expressions:** Equations are statements that two expressions are equal, like \( P = 2L + 2W \).
• **Operations:** These include addition, subtraction, multiplication, and division, which are used to manipulate equations and expressions.

Algebra allows the formulation and solving of problems based on certain properties and rules. It's a foundational piece in mathematics that leads to understanding more complex mathematical concepts.

Isolate Variable

Isolating a variable is a critical step when solving equations. The goal is to get the variable you are solving for by itself on one side of the equation. This involves using inverse operations to simplify the equation.
To isolate the variable \( W \) in the equation \( P = 2L + 2W \):

• **Move Other Terms Away:** Perform operations like adding or subtracting terms on both sides. For example, subtract \( 2L \) from both sides to get: \( P - 2L = 2W \).
• **Simplify the Equation:** Once you have the variable term alone, divide by the coefficient of the variable to solve for it. Here, divide both sides by 2 to isolate \( W \).

So, the isolated variable equation becomes \( W = \frac{P - 2L}{2} \). This process helps determine the relationship between the given quantities and the variable you are working with.

Linear Equations

Linear equations are equations of the first degree, meaning each term is either a constant or the product of a constant and a single variable. These equations typically look like \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. They often model real-world situations where there's a constant rate of change.
For example, the equation \( P = 2L + 2W \) is a linear equation because:- It involves addition and multiplication.- The variables \( L \) and \( W \) are both to the power of one.Linear equations can exist in various forms but always graph as straight lines when plotted on a coordinate plane. Understanding linear equations is important because:

• **Predictability:** They allow predictions of one variable when you know the others.
• **Foundation for More Complex Math:** Grasping linear equations sets the groundwork for quadratic equations and other higher-degree polynomials.

Thus, mastering linear equations is a vital skill in mathematics that finds uses in everyday life and advanced scientific computations.