Problem 38 - Elementary and Intermediate Algebra

Question

Subtract and simplify the result, if possible. \(\frac{7 x+7}{5 y}-\frac{2 x+7}{5 y}\)

Step-by-Step Solution

Answer

The simplified result is \( \frac{x}{y} \).

Step 1: Understand the problem

We have two fractions that share a common denominator. These fractions are \( \frac{7x + 7}{5y} \) and \( \frac{2x + 7}{5y} \). The task is to subtract the second fraction from the first one.

Step 2: Subtract the numerators

Since both fractions have the same denominator, we subtract their numerators: \[(7x + 7) - (2x + 7)\]. This simplifies to \[7x + 7 - 2x - 7\].

Step 3: Simplify the resulting numerator

Simplify the expression from the subtraction in Step 2: \[7x + 7 - 2x - 7 = 5x\].

Step 4: Construct the new fraction

After simplifying the numerator, our fraction becomes \( \frac{5x}{5y} \).

Step 5: Simplify the entire expression

Since both the numerator and the denominator have a common factor of 5, we can simplify: \[\frac{5x}{5y} = \frac{x}{y}\].

Key Concepts

Understanding Common Denominator

When dealing with fractions, finding a common denominator is crucial, especially for operations like addition and subtraction. In simple terms, a common denominator is a shared multiple of the denominators of two or more fractions. Here, both fractions, \( \frac{7x + 7}{5y} \) and \( \frac{2x + 7}{5y} \), already have the same denominator, which is \( 5y \). This shared base allows us to directly subtract one numerator from another without having to adjust the denominators.
Having a common denominator simplifies the process, making it easy and direct.
Here's why it's important:

• It allows for straightforward subtraction and addition of fractions.
• Prevents errors that can occur when handling different denominators.
• Ensures the resulting fraction is accurate and easy to simplify.

By ensuring the fractions have a common denominator, you're laying down a simple path to solve the problem efficiently.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that helps make complex expressions easier to work with and understand. Once you've subtracted the numerators from the two fractions, you end up with a new expression for the numerator:

• Subtract the numerators: \((7x + 7) - (2x + 7)\)
• Simplify to: \(7x + 7 - 2x - 7 = 5x\)

The simplification step involves combining like terms and eliminating unnecessary numerical values. In the subtraction process, you are focusing on the coefficients of \(x\) and the constant terms. Notice how by rearranging and simplifying the terms, the expression reduces to \(5x\), which is much easier to handle.
Simplifying expressions is important because it:

• Reduces the complexity of mathematical problems.
• Helps make more precise calculations.
• Makes it easier to identify and apply further simplification or factorization.

By practicing simplification, you strengthen your algebraic skills and enhance your ability to tackle more challenging problems.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. After subtracting the fractions and simplifying the numerator, you arrive at the fraction \( \frac{5x}{5y} \).
Both the numerator and denominator have a common factor of 5. By dividing both by this factor, we simplify the fraction to:

• Divide the numerator and the denominator by 5: \( \frac{5x}{5y} = \frac{x}{y} \).

This is the simplest form of the fraction, where no further common factors exist between the numerator and denominator.
Simplifying fractions is beneficial because:

• It brings clarity by reducing fractions to the most understandable form.
• Facilitates easier comparison and computation with other fractions.
• Promotes a better understanding of the underlying mathematical relationship.

Mastering the art of fraction simplification not only makes math easier but also helps in many practical applications, from cooking to engineering.