Problem 12 - 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? $$P=C+M C \text { for } M$$

Step-by-Step Solution

Answer

The solution to the specified variable M in the equation is \( M = P/C - 1\) .

Step 1: Rearrange the equation

First, all terms containing M should be grouped on one side of the equation and everything else on the other side. This gives \( MC = P - C \).

Step 2: Factor out M

Since we are solving for M, the equation needs to be rearranged to have M isolated on one side. This can be done by dividing the entire equation by C, resulting in \( M = (P - C)/ C\) .

Step 3: Simplifying the equation

The equation can be simplified to the form \( M = P/C - 1\) .

Key Concepts

Solving Equations

When solving equations, the main goal is to find the value of the unknown variable. It's like unwrapping a present to see what's inside. By isolating the variable, we can look at its value and understand the equation better. The exercise we have involves the equation: \[ P = C + MC \] We aim to solve for \( M \), meaning we want \( M \) to stand alone on one side of the equation. To achieve this, we rearrange the terms systematically. Solving equations involves these simple steps:

• Identify all like terms and group them together.
• Apply operations that can help simplify the equation.
• Keep applying these operations until the variable is isolated.

Each step unveils more about the variable's value, leading us to the solution, which is beautifully simple once we get there.

Rearranging Formulas

Rearranging formulas is a skill that involves reshuffling the terms. It’s like organizing a closet so you can find things without hassle. When given a formula, like \[ P = C + MC \] and asked to solve for a specific variable, rearranging is key. We move terms around to achieve this. The key approach involves:

• Bringing all terms with the desired variable on one side of the equation.
• Collecting like terms and simplifying the expression.

In our example, we began by subtracting \( C \) from both sides to group terms with \( M \) together, resulting in:\[ MC = P - C \] By doing so, we set the stage for the final isolation of \( M \), ensuring the formula is ready for the next step.

Isolating Variables

Isolating a variable means making it the subject of the equation. Think of it as putting a single spotlight on our variable of interest in the mathematical "room." In equations, we want to make one variable stand alone, clear of any other terms or numbers. In our equation, the target was to isolate \( M \). Having the equation \( MC = P - C \) from our previous steps, we needed \( M \) to shine on its own:

• Divide both sides of the equation by the coefficient of \( M \), which is \( C \).
• This isolates \( M \), resulting in a clean expression for \( M \).

The final step reveals our solution:\[ M = \frac{P - C}{C} \] This was further simplified to:\[ M = \frac{P}{C} - 1 \]Now, \( M \) is beautifully isolated, offering a clear and direct answer.