Problem 12 - College Algebra and Calculus: An Applied Approach

Question

Write an equation that represents the statement. The difference of \(n\) and 7 is 4 .

Step-by-Step Solution

Answer

The equation that represents the statement is \(n - 7 = 4\).

Step 1: Identify the Components of the Statement

In the given statement, we are told that 'the difference of \(n\) and 7 is 4'. 'The difference of \(n\) and 7' means that 7 is subtracted from \(n\), and 'is 4' means this 'difference' equals 4.

Step 2: Translate the Statement into an Equation

We can now translate this into an equation. 'The difference of \(n\) and 7' becomes \(n - 7\). The phrase 'is 4' implies equality, so our equation becomes \(n - 7 = 4\).

Key Concepts

Equation Translation

In algebra, translating a statement into an equation is crucial for solving problems. This process involves identifying keywords or phrases that represent mathematical operations. Let's break down the statement, "The difference of \(n\) and 7 is 4."

• Difference: The term "difference" indicates a subtraction operation. This means we are subtracting one number from another.
• Expression: In this context, "the difference of \(n\) and 7" translates to \(n - 7\).
• Equality: The word "is" in math usually represents equivalence, informing us that the expression equals the number following it, which is 4 in this case.

Putting this all together, we translate the statement to an equation: \(n - 7 = 4\). This step of translation is fundamental in algebra to move from words to a solvable mathematical format.

Step-by-Step Solution

To reach a solution, it's essential to work through the problem step by step. This approach helps you clearly see each part of the problem without becoming overwhelmed.

Identify the Keywords

First, identify the keywords or phrases in the statement: "difference" and "is." These alert us to the operations we need to perform.

Construct the Equation

Next, we construct the equation using these keywords:

• Difference of \(n\) and 7: This is expressed as \(n - 7\).
• Equals 4: The statement "is 4" lets us know the result of \(n - 7\) equals 4.

Thus, the complete equation becomes \(n - 7 = 4\).

Check Your Work

Always review your translation to ensure accuracy and confirm that the equation correctly represents the original statement.

Basic Algebra Concepts

Understanding basic algebra concepts lays a solid foundation for tackling more complex mathematical problems. Here are a few they interplay in this exercise.

• Variables: The letter \(n\) represents a variable, which stands in for an unknown number in an equation.
• Operations: Subtraction is the operation in this example, indicated by the word "difference." Recognizing and interpreting operations is crucial in algebra.
• Equations: An equation is a mathematical statement that expresses two expressions are equal. In this case, \(n - 7 = 4\) states the equivalence of the difference and the number 4.

These basics not only help in this exercise but are essential tools for advancing in algebra. Returning to these basic concepts often helps clarify and solve algebraic problems more efficiently.