Problem 61 - Prealgebra

Question

Solve each equation. $$\frac{54}{n}=6$$

Step-by-Step Solution

Answer

The solution is \( n = 9 \).

Step 1: Identify the Equation

The equation given is \( \frac{54}{n} = 6 \). This is a simple equation where \( n \) is the unknown variable we need to solve for.

Step 2: Perform Cross-Multiplication

To isolate \( n \), we can cross-multiply. Multiply both sides of the equation by \( n \) to eliminate the fraction: \[ n \times \frac{54}{n} = n \times 6 \] This simplifies to: \[ 54 = 6n \]

Step 3: Isolate the Variable n

Now, solve for \( n \) by dividing both sides of the equation by 6 to get \( n \) alone: \[ \frac{54}{6} = n \] \[ n = 9 \]

Step 4: Verify the Solution

Substitute \( n = 9 \) back into the original equation to check the solution: \( \frac{54}{9} = 6 \). Since dividing 54 by 9 indeed equals 6, our solution is verified.

Key Concepts

Understanding Prealgebra

Prealgebra is a fundamental branch of mathematics that lays the groundwork for algebra. It deals with basic arithmetic and the introduction of algebraic concepts. In this exercise, we are working with a simple equation where we need to find the unknown variable, often referred to as solving for the variable. Here's a brief guide to understanding the essentials:

• Identify the equation structure: Equations require identifying equalities between two expressions, like \( \frac{54}{n} = 6 \).
• Understanding variables: These are symbols, often letters, representing unknown or varying quantities. In our example, \( n \) is the variable.
• Executing basic arithmetic operations, such as addition, subtraction, multiplication, and division.

These operations help isolate and solve the equation to find the value of the variable. Mastery of prealgebra sets the stage for tackling more complex mathematical problems.

Cross-Multiplication Technique

Cross-multiplication is a useful method when dealing with equations involving fractions. It allows us to eliminate fractions by multiplying across the terms. Let's break this down further:

• Start with an equation that involves a fraction: In our example, \( \frac{54}{n} = 6 \), the aim is to solve for \( n \).
• Multiply both sides by the denominator: By multiplying both sides by \( n \), you remove the fraction: \[ n \times \frac{54}{n} = n \times 6 \]This simplifies to:\[ 54 = 6n \]

This process clears the fraction, making it easier to solve for the unknown variable. Cross-multiplication is a powerful tool that simplifies equations, a crucial skill in algebraic problem solving.

Equation Verification

Equation verification is an essential step in solving equations. It helps ensure that the solution is correct by substituting the found value back into the original equation. Here’s how it works:

• Take the solution from the previous steps: Once we solved \( n = 9 \), we need to verify this value.
• Substitute back into the original equation: Replace \( n \) in \( \frac{54}{n} = 6 \) with 9, and compute:\[ \frac{54}{9} = 6 \]
• Check if both sides of the equation are equal: Confirming that dividing 54 by 9 results in 6 proves our solution is accurate.

Verification acts as a final check against errors. It's an important practice, especially as equations become more complex in advanced mathematics.