Problem 2 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{8 y}{18 x}$$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{4y}{9x}\).

Step 1: Identify Common Factors

Look for common factors in the numerator and the denominator. For \(8y\), the factors are \(8 \) and \(y\). For \(18x\), the factors are \(18\) and \(x\). The numbers 8 and 18 share the common factor 2.

Step 2: Divide by the Greatest Common Factor (GCF)

Divide both the numerator and the denominator by their greatest common factor. The GCF of 8 and 18 is 2. So, we divide: \(\frac{8y}{18x} = \frac{8 \div 2 \times y}{18 \div 2 \times x} = \frac{4y}{9x}\).

Step 3: Simplified Expression

After dividing by the GCF, the algebraic fraction is now in its simplest form: \(\frac{4y}{9x}\). There are no more common factors that can be divided out.

Key Concepts

Greatest Common Factor

The greatest common factor, or GCF, is the largest number that divides two or more numbers without leaving a remainder. Finding the GCF is a crucial step when simplifying algebraic fractions. For example, to simplify fractions like \( \frac{8y}{18x} \), we first determine the GCF of the numerical parts of the numerator and the denominator.

Here's how to identify the GCF:

• List the factors of each number. For 8, the factors are 1, 2, 4, and 8.
• For 18, the factors are 1, 2, 3, 6, 9, and 18.
• Identify the largest common factor. Here, the largest shared factor is 2.

Once the GCF is found, you can divide the numerator and the denominator by this factor to simplify the fraction.

Numerator and Denominator

In any fraction, the numerator is the top part, and the denominator is the bottom part. Algebraic expressions often appear as fractions in mathematics, where understanding these terms is essential in simplifying fractions.

Take the fraction \( \frac{8y}{18x} \):

• "8y" is the numerator.
• "18x" is the denominator.

By identifying and simplifying the numerator and denominator, we can reduce the fraction to its simplest form. In this case, dividing both by their GCF of 2 gives the simplified expression \( \frac{4y}{9x} \). This operation reduces the complexity of algebraic expressions without changing their value.

Algebraic Expressions

Algebraic expressions contain numbers, variables, and sometimes operators like addition and multiplication. Simplifying these expressions often requires factoring and reducing fractions. Understanding how to manipulate these expressions is a key skill.

Let's break it down with \( \frac{8y}{18x} \):

• The "8y" and "18x" are algebraic expressions involving variables "y" and "x".
• You simplify them by factoring out common numerical factors, such as 2 in this example.

After factoring, you're left with a simpler algebraic expression \( \frac{4y}{9x} \). This simplification process helps when solving equations or comparing fractions, as it makes the expressions easier to work with.