Problem 19 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-26,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$A=\frac{1}{2}(a+b) \text { for } a$$

Step-by-Step Solution

Answer

The solution to the equation \(A=\frac{1}{2}(a+b)\) for \(a\) is \(a = 2A - b\).

Step 1: Analysis and Rewriting Equation

Analyse the given equation \(A = \frac{1}{2}(a+b)\). Begin by rewriting the equation to eliminate the fraction. This can be done by multiplying both sides of the equation by 2. This gives: 2A = a + b.

Step 2: Solving for a

The task requires solving for \(a\). To isolate \(a\), subtract \(b\) from both sides of the equation to get: a = 2A - b.

Key Concepts

Rewriting Equations

Rewriting equations is an essential skill in algebra and mathematics as a whole. It involves manipulating an equation into a different form without changing its solution or meaning. One of the most common methods for rewriting equations is to eliminate fractions by multiplying each term by the denominator's least common multiple.

For example, in the given equation \(A = \frac{1}{2}(a+b)\), the fractional coefficient \(\frac{1}{2}\) can make the equation cumbersome to work with. Thus, a useful first step is to multiply every term by 2. This operation removes the fraction, transforming the equation into \(2A = a + b\).

Rewriting the equation lays the groundwork for isolating a specific variable. It simplifies the structure of the equation, making further manipulation easier and more straightforward. Remember, when rewriting an equation, it's crucial to perform equivalent operations on both sides, maintaining equality.

Isolating Variables

Isolating a variable involves rearranging an equation so that a particular variable stands alone on one side of the equation. This is a critical step in solving equations because it allows us to express a variable in terms of other known quantities. The goal is to manipulate the equation until the desired variable is isolated.

In our example, after rewriting \(2A = a + b\), we isolate \(a\) by subtracting \(b\) from both sides of the equation, which gives us \(a = 2A - b\). This subtraction step effectively shifts \(b\) over to the other side, leaving \(a\) isolated on one side of the equation.

• This process requires performing 'inverse operations' - actions that undo each other. For example, addition is undone by subtraction, and vice versa.
• Following each step carefully ensures the equation remains balanced and the isolation correct.

Understanding how to isolate variables is essential for solving a variety of equations, whether in algebra, physics, or other fields.

Arithmetic Operations

Arithmetic operations are the basic processes of addition, subtraction, multiplication, and division that we use to manipulate numbers and expressions. When solving equations, these operations are key to rearranging and simplifying expressions.

In the context of the example equation \(2A = a + b\), arithmetic operations were used to:

• Multiply both sides by 2 to remove the fraction, effectively utilizing multiplication and simplification.
• Subtract \(b\) from both sides to isolate \(a\), demonstrating subtraction.

Each arithmetic operation has specific properties. For instance, addition and multiplication are commutative and associative, meaning the order and grouping of numbers don't change the result. However, subtraction and division are not commutative, which can be crucial when isolating variables.

By mastering these operations, along with their properties, we can solve equations accurately and efficiently. Arithmetic operations are the foundational tools of algebra, and recognizing when and how to apply them is key to success in mathematics.