Problem 15 - 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? $$M=\frac{n}{5} \text { for } n$$

Step-by-Step Solution

Answer

The formula \(M=\frac{n}{5}\) solved for \(n\) is \(n = 5 * M\).

Step 1: Identify the Variable to be Isolated

In the given formula, \(M=\frac{n}{5}\), it is asked to solve for the variable \(n\). The variable \(n\) is currently in the denominator on the right side of the equation.

Step 2: Isolate Variable n

In order to isolate \(n\) on one side of the formula, we multiply both sides of the equation by \(5\). This gives us \(5 * M = n\). Therefore, the formula \(M=\frac{n}{5}\) solved for \(n\) is \(n = 5 * M\).

Key Concepts

Formula Manipulation

Formula manipulation involves rearranging equations to solve for a particular variable. When dealing with equations, knowing how to manipulate them to isolate a variable of interest is crucial. For instance, consider the equation given in the problem: \( M = \frac{n}{5} \). The goal is to solve for \( n \), which involves formula manipulation.

• Understand the equation: Look at the components and their arrangement within the formula.
• Apply algebraic operations: Use addition, subtraction, multiplication, or division to rearrange the formula as needed.
• Solve for the desired variable: Ensure the solution presents the variable isolated appropriately.

In our example, we needed to multiply both sides by 5, transforming the equation into \( n = 5 \times M \). This step effectively cancels out the 5 in the denominator, making \( n \) the subject of the formula.

Isolating Variables

Isolating a variable in an equation is a foundational skill in algebra. It means modifying the equation so that a specific variable is alone on one side of the equation. This is especially important when you need to find the value of a variable given other values.

• Identify which variable needs isolation: Determine the variable that the problem asks you to solve for, here \( n \).
• Perform inverse operations: Use the opposite operation to move other numbers or variables away from the target variable. In our equation \( M = \frac{n}{5} \), multiplying both sides by 5 helps isolate \( n \).
• Double-check your work: Ensure that the variable is indeed isolated and the solution is simplified correctly.

Once isolated, you have the expression \( n = 5 \times M \), which can be used to compute \( n \) for any given \( M \). This approach not only solves the equation but deepens understanding of algebraic manipulation.

Algebraic Expressions

Understanding algebraic expressions is key to solving equations. These expressions consist of variables, numbers, and operations (like addition and multiplication) combined together.

• Recognize components: Identify numbers, variables, and operations within the equation.
• Follow the order of operations: Apply the rules of arithmetic operations correctly—usually multiplication or division occurs before addition or subtraction.
• Simplify whenever possible: Break down complex expressions into simpler components.

In the exercise \( M = \frac{n}{5} \), the expression involves a fraction and a variable. Simplifying or rearranging such expressions may require multiplying through by a common factor, like we did by multiplying by 5 to eliminate the fraction. This helps in reaching a clear solution like \( n = 5 \times M \). Understanding these components allows for better handling of similar algebraic challenges.