Problem 20 - Introductory Algebra for College Students

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A=\frac{1}{2}(a+b)\) for \(b\)

Step-by-Step Solution

Answer

After isolating \(b\), we get the equation \(b = 2A - a\)

Step 1: Understand the formula

The formula \(A=\frac{1}{2}(a+b)\) represents the arithmetic mean of two numbers \(a\) and \(b\). The task is to isolate \(b\).

Step 2: Remove the fraction

Multiply the entire equation by 2 to get rid of the fraction to get: \(2A = a + b\)

Step 3: Isolate \(b\)

Rearrange the equation to isolate \(b\). Subtract \(a\) from both sides of the equation to get: \(b = 2A - a\)

Key Concepts

Arithmetic Mean

Understanding the arithmetic mean, or commonly known as the average, is essential in various mathematical and real-world applications. It is defined as the sum of the values in a data set divided by the number of values. In the context of the exercise \( A=\frac{1}{2}(a+b) \) for \( b \), we are looking at the arithmetic mean of two numbers, \( a \) and \( b \). This formula encapsulates the concept by summing \( a \) and \( b \) and then dividing by 2. This formula is beneficial because it gives us a central value that can represent the 'middle ground' between two numbers.

When dealing with the arithmetic mean, it's crucial to understand that it's sensitive to extremely high or low values within the data set. Such values can significantly affect the average, which might not truly represent the data's central tendency. This understanding of the arithmetic mean facilitates better comprehension when manipulating algebraic expressions to solve for a specific variable.

Solving Formulas for a Variable

Algebra often involves solving formulas for a specific variable. This means manipulating an equation in such a way that we isolate the variable of interest on one side of the equation. When solving for a variable, steps are typically required to move terms and coefficients around the desired variable. It's important to perform the same operation on both sides of the equation to maintain equality.

As seen in our exercise, to solve for \( b \) in the formula \( A=\frac{1}{2}(a+b) \), we first removed the fraction by multiplying both sides of the equation by 2. This step is a fundamental application of the principle that what you do to one side of the equation, you must do to the other. It's these algebraic manipulations—along with operations like addition, subtraction, multiplication, and division—that allow us to extract a variable from a more complex expression.

Isolating Variables

Isolating a variable is a key skill in algebra, requiring a strategic approach to manipulate equations. The goal is often to express one variable in terms of others, much like solving a puzzle where each move gets you closer to the solution. This is what we did in Step 3 of our exercise where we isolated \( b \) by subtracting \( a \) from both sides of the equation, resulting in \( b = 2A - a \).

Isolating the variable is the final step in rearranging the equation, but it's preceded by thoughtful selection of arithmetic operations to move other terms across the equality sign. It's like untangling a knot—the sequence of steps matters, and each operation brings clarity. By systematically applying inverse operations, we can effectively 'move' terms from one side to the other, which is crucial when rearranging formulas to make a particular variable the subject.