Problem 16 - 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? $$M=\frac{A}{740} \operatorname{for} A$$

Step-by-Step Solution

Answer

After simplification, the formula becomes \(A = M \times 740 \).

Step 1: Understand the Given Formula

The formula given is \(M=\frac{A}{740}\). In this formula, \(M\) is given as the ratio of \(A\) to 740. The task is to solve for \(A\).

Step 2: Isolate the Variable A

To isolate variable \(A\), multiply both sides of the equation by 740.

Step 3: Simplify the Equation

Simplifying the equation gives \(A = M \times 740 \).

Key Concepts

Formula Rearrangement

Formula rearrangement is the process of modifying a given formula to express one particular variable in terms of the others. This technique is crucial when you need to derive a particular element from an equation. Changing the subject of the formula involves performing valid algebraic operations to both sides of the equation.

To rearrange a formula, consider these essential steps:

• Identify the variable you wish to solve for.
• Decide the operations required to isolate this variable based on the operations performed in the current equation.
• Apply inverse operations to both sides of the equation to maintain equality while moving other terms to the opposite side from the variable of interest.

For example, in the formula from the exercise, we have to rearrange it to solve for the variable \(A\). The initial formula is \(M = \frac{A}{740}\). The aim is to express \(A\) in terms of M and 740.

Variable Isolation

Variable isolation is the technique of maneuvering an equation to place a single variable on one side. Essentially, it is about peeling away any extras that cling to the variable you're trying to solve. In our example, this means rearranging the formula \(M = \frac{A}{740}\) to focus solely on \(A\).

To isolate \(A\), you need to perform operations that systematically eliminate other components affecting it. Here’s how:

• In this context, observe that \(A\) is divided by 740. Thus, to isolate \(A\), simply perform the inverse operation, which is multiplication, by 740 on both sides of the equation.

By executing this, you eliminate the division, making \(A\) stand alone. So, multiply both sides by 740 to achieve \(A = M \times 740\). This formula now clearly reflects \(A\) as the subject, isolated and ready for calculation.

Equation Solving

Equation solving involves finding the value of variables that satisfy the given equation's conditions. Once you've isolated the variable of interest, the task becomes solving this new version of the equation.

In the given problem, after isolating \(A\), you arrive at \(A = M \times 740\). This simple multiplication equation requires you to substitute known values to find \(A\).

To solve such an equation:

• Substitute any known values into the equation, ensuring they correspond to the correct variables.
• Perform straightforward arithmetic operations as indicated.

By performing these steps, you can effectively determine the value of the isolated variable \(A\), and hence complete your solution. This fundamental skill is extensively useful across various mathematical and real-world applications.