Problem 19 - 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 \(a\)

Step-by-Step Solution

Answer

Expressed in terms of \(A\) and \(b\), the variable \(a\) is given as \(a = 2A - b\).

Step 1: Identify the variable to isolate

The variable that needs to be isolated is \(a\). Our aim is to express \(a\) in terms of \(A\) and \(b\).

Step 2: Isolate \(a\)

To isolate \(a\), we first multiply the equation by 2 to get rid of the fraction: \(2A=a+b\). Then we subtract \(b\) from both sides: \(2A - b = a\).

Step 3: Simplify the equation

The equation is already simplified, so the final equation is \(a = 2A - b\).

Key Concepts

Isolating Variables

Isolating a variable means solving for that variable in terms of the other variables present within an equation. This becomes particularly useful when you need to determine the value of one variable when the other values are known. Here, our task is to isolate the variable \( a \) in the equation \( A = \frac{1}{2}(a+b) \).
To achieve this:

• Identify the variable you want to isolate; in this case, it is \(a\).
• Consider what mathematical operations will help you separate this variable from the others. 

The primary goal when isolating a variable is to perform opposite operations to those currently in place. For our example, we begin by eliminating the fraction by multiplying through by 2. This is because our formula includes \( \frac{1}{2} \). Removing the fraction gives us a cleaner equation to work with, resulting in \(2A = a + b\). You'll continue to isolate \( a \) by performing algebraic manipulations, like subtracting \( b \) from both sides, ultimately simplifying the expression to \( a = 2A - b \). This process of isolating variables is crucial for solving formulas effectively.

Algebraic Manipulation

Algebraic manipulation involves a series of steps to transform and rearrange equations. It enables us to solve for one variable in terms of others efficiently. In solving the formula \( A = \frac{1}{2}(a+b) \) for \( a \), we perform algebraic operations that revolve around the goal of isolating \( a \).
The operations include:

• Multiplication: Multiply the entire equation by 2 to remove the fraction present in the equation.
• Subtraction: Subtract \( b \) from both sides to continue the process of isolating \( a \).

Algebraic manipulation often requires:

• Using basic arithmetic operations like addition, subtraction, multiplication, and division. Each operation has the goal of simplifying the equation.
• Maintaining balance. Remembering to perform these operations on both sides of the equation is crucial to its integrity.

In our example, after multiplication and subtraction, the equation simplifies to the form \( a = 2A - b \). Simplified equations such as these are easier to interpret and utilize in practical applications.

Recognizing Formulas

Recognizing formulas is about identifying the type of mathematical relationship described by an equation. Understanding these relationships can provide insights and shortcuts for problem-solving. The exercise starts with the formula \( A = \frac{1}{2}(a+b) \), which is actually the formula for the arithmetic mean of two numbers, \( a \) and \( b \). Recognizing the formula quickly clues us into what kind of manipulations might be necessary to solve for a specific variable.
Recognizing formulas involves:

• Having a solid grasp of common mathematical forms and identities.
• Understanding the context of the formula, if provided, to aid in solving the problem confidently.

This type of problem is often found in algebra, where understanding the forms and relationships is just as critical as performing the calculations.
By recognizing the formula, you can often predict what operations will be needed and understand the meaning of the solution in a broader mathematical or real-world context. Knowing that our original equation is an arithmetic mean helps with interpreting the isolated variable and understanding why particular operations were necessary.