Problem 1 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{6 x}{14 y}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{3x}{7y} \).

Step 1: Identify common factors

The first step is to look for any common factors in the numerator and the denominator of the fraction. For the numerator, which is 6, the factors are 1, 2, 3, and 6. For the denominator, which is 14, the factors are 1, 2, 7, and 14. Notice that both 6 and 14 have a common factor of 2.

Step 2: Divide by the greatest common factor

Now that we know both 6 and 14 have a common factor of 2, we divide the numerator and the denominator by 2. This will help simplify the fraction:\[ \frac{6 x}{14 y} = \frac{6 \div 2}{14 \div 2} \times \frac{x}{y} = \frac{3x}{7y} \]

Step 3: Verify simplification

Lastly, we verify that the simplified fraction \( \frac{3x}{7y} \) has no further common factors in the numerator and denominator other than 1. The factors of 3 are 1 and 3; the factors of 7 are 1 and 7, confirming there are no further simplifications possible.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is an important concept in mathematics, especially when dealing with fractions. The GCF of two numbers is the largest number that can divide both numbers without leaving a remainder. Finding the GCF is a crucial step in simplifying algebraic fractions.
To determine the GCF:

• List all factors of each number in the numerator and denominator.
• Identify any common factors shared by both numbers.
• The greatest of these common factors is the GCF.

In our example, with the numbers 6 and 14, the GCF is 2, because 2 is the largest number that divides both 6 and 14 without a remainder. Identifying the GCF allows you to divide both the numerator and the denominator by this factor, which simplifies the fraction.

Numerator and Denominator

Understanding the terms 'numerator' and 'denominator' is essential for working with fractions. The numerator is the top part of the fraction, representing the part of the whole, while the denominator is the bottom, showing the total number of equal parts.
In the algebraic fraction \( \frac{6x}{14y} \), 6x is the numerator, and 14y is the denominator. To simplify, we perform operations on both parts of the fraction.

• Identify common factors in both the numerator and the denominator.
• Divide each by these factors, usually starting with the GCF.

For our fraction, dividing both 6x and 14y by their GCF (2) results in the simplified form \( \frac{3x}{7y} \). This reduction reflects the simplest form of the fraction, making it easier to work with in further calculations.

Algebraic Fractions

Algebraic fractions are similar to numerical fractions but they include variables, such as \( x \) and \( y \). Simplifying algebraic fractions follows the same basic rules as simplifying numerical fractions. The process usually involves reducing both the numbers and variables.

• Look for terms with common factors in both numbers and any variables.
• Cancel out the common factors from the numerator and the denominator.

In our exercise, the algebraic fraction \( \frac{6x}{14y} \) is simplified to \( \frac{3x}{7y} \) because the common numerical factor 2 is divided out. Since there are no common variables to cancel between \( x \) and \( y \), the variables remain in their original terms. Understanding and simplifying algebraic fractions is vital in algebra as it forms the basis for solving more complex equations.