Problem 83 - College Algebra with Modeling and Visualization

Question

Exercises \(83-90:\) Solve the equation for the specified variable. \(A=L W\) for \(W\)

Step-by-Step Solution

Answer

\( W = \frac{A}{L} \)

Step 1: Identify the given equation

We start with the equation given in the problem: \( A = LW \), where \( A \) is the area, \( L \) is the length, and \( W \) is the width. We are asked to solve this equation for \( W \).

Step 2: Isolate the variable

To solve for \( W \), we need to isolate \( W \) on one side of the equation. Currently, \( W \) is being multiplied by \( L \). To undo this multiplication, we can divide both sides of the equation by \( L \).

Step 3: Perform algebraic manipulation

Divide both sides of the equation by \( L \): \[W = \frac{A}{L}\]This step isolates \( W \) and expresses it in terms of \( A \) and \( L \).

Step 4: Verify the rearrangement

Check if the rearranged equation \( W = \frac{A}{L} \) is dimensionally consistent. \( A \) represents an area, and \( L \) a length, so \( \frac{A}{L} \) results in a width, which is consistent with \( W \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to rearrange and simplify equations. This is a fundamental skill in solving equations since it allows us to express equations in different forms, which might be more useful or revealing depending on the problem.

In the example equation \( A = LW \), we use algebraic manipulation to solve for the variable \( W \). Starting with multiplication in the original equation, we apply the rule that says we can perform the same operation to both sides of the equation, as long as it maintains balance. This leads us to divide both sides by \( L \), transforming the equation into \( W = \frac{A}{L} \).

Algebraic manipulation involves these essential steps:

• Identify the operation on the variable of interest.
• Apply the inverse operation to both sides of the equation.
• Continue rearranging until the variable is isolated or the desired expression is achieved.

This process requires understanding various algebraic operations and inverse operations, such as addition and subtraction, or multiplication and division.

Isolation of Variables

Isolation of variables is the technique of rearranging an equation so that one particular variable stands alone on one side of the equation. This is pivotal when solving equations, as it helps in finding the value of the sought variable based on the other known quantities.

For the equation \( A = LW \), we aim to isolate \( W \). Initially, \( W \) is tied up with \( L \) through multiplication. To isolate \( W \), we need to perform an inverse operation, which is division in this case. By dividing both sides by \( L \), we achieve the desired isolation, \( W = \frac{A}{L} \).

This method is of critical importance in algebra because:

• It allows you to express the dependent variable as a function of other variables.
• It clarifies the role of each variable in the equation.
• It simplifies solving for unknowns in more complex equations or systems of equations.

By practicing the isolation of variables, students develop a deeper understanding of how changes to one part of an equation impact others.

Dimensional Analysis

Dimensional analysis is a technique used to verify the consistency of equations, ensuring that the computed result has the appropriate units. This method checks that the dimensions on both sides of an equation match, providing a logical validation of the equation.

In the context of the equation \( W = \frac{A}{L} \), we conduct a dimensional analysis to ensure that this manipulation is correct. Here, \( A \) represents an area with dimensions \( \text{[length]}^2 \), and \( L \) is a length with dimensions \( \text{[length]} \). Dividing them, \( \frac{\text{[length]}^2}{\text{[length]}} \), results in \( \text{[length]} \), aligning with the dimension of width, \( W \).

Dimensional analysis is crucial in:

• Confirming the physical meaning of equations.
• Ensuring that derived equations logically follow from initial assumptions.
• Aiding in the identification of potential errors in computational steps.

Through dimensional analysis, students can cross-verify their results, gaining confidence in their computations and understanding of physical quantities.