Problem 55 - College Algebra

Question

In Exercises \(55-74\), solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A-I w\) for \(w\)

Step-by-Step Solution

Answer

The solution for \(w\) is \(w = \frac{A}{I}\)

Step 1: Identify the Equation

The given equation is \(A = Iw\). The task is to solve this equation for the variable \(w\).

Step 2: Isolate the Variable

In order to isolate \(w\), we must divide each side of the equation by \(I\). This gives us the equation \(w = \frac{A}{I}\).

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In most cases, these symbols represent numbers and help in solving equations. It allows us to understand and express relationships between different quantities.
Algebra is important because:

• It provides a way to describe patterns and relationships numerically.
• It is fundamental for solving almost any mathematical problem that involves unknowns.
• It enhances logical and abstract thinking.

In our original exercise, algebra is used to manipulate an equation to express one variable in terms of another. This skill is crucial for understanding more complex mathematical concepts.

Introduction to Variables

Variables are symbols that represent unknown values in mathematical equations. Commonly used variables are letters like x, y, and z. In algebra, these symbols act as placeholders that can be substituted with different numbers.
Here’s why variables are essential:

• They provide flexibility in problem-solving by allowing the same equation to be used for multiple values.
• They help in formulating mathematical models to represent real-world scenarios.

In the exercise, the variables are A, I, and w. Solving for a specific variable, like w, helps us understand the relationship between the different elements of the equation.

Isolation of a Variable

The process of isolation of a variable involves rewriting an equation to make one of the variables stand alone on one side. It is a crucial step in solving equations as it helps us understand the specific value or relationship we are interested in finding.
To isolate a variable:

• Identify the variable you need to solve for.
• Use arithmetic operations, such as addition, subtraction, multiplication, or division, to separate the variable from others.
• Perform the same operation on both sides to maintain equality.

In the exercise, we isolated w by dividing both sides of the equation by I, resulting in the expression \(w = \frac{A}{I}\). This expression shows that w depends on both A and I.

Formula Manipulation Techniques

Formula manipulation is an essential skill needed to solve equations involving multiple variables. It involves changing the arrangement of variables while keeping the equation balanced and valid. This skill helps in reorganizing formulas to highlight different relationships between variables.
The techniques of formula manipulation include:

• Reversing arithmetic operations: Addition becomes subtraction, multiplication becomes division, and vice versa.
• Using inverse operations to simplify the equation and isolate the variable of interest.
• Maintaining balance by applying operations equally to both sides of an equation.

When applying formula manipulation in the exercise, you learned how to rearrange \(A = Iw\) to solve for w. This process not only isolates w but also gives you insights into how each parameter affects the other.