Problem 19 - Intermediate Algebra

Question

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{8}{3 x}+\frac{5}{3 x}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{13}{3x} \).

Step 1: Identify Like Terms

The first step is to notice that both terms, \( \frac{8}{3x} \) and \( \frac{5}{3x} \), have the same denominator: \( 3x \). This means they are like terms and can be added directly.

Step 2: Add the Numerators

Since the denominators are the same, we keep the denominator \( 3x \) and add the numerators: \( 8 + 5 = 13 \).

Step 3: Write the Result Fraction

Combine the result from the addition of numerators over the common denominator. The expression becomes \( \frac{13}{3x} \).

Step 4: Simplify if Possible

Check if \( \frac{13}{3x} \) can be simplified further. Since 13 is a prime number and doesn't divide evenly into the denominator, the expression is already in its simplest form.

Key Concepts

Like Terms

When working with fractions, like terms are fractions that have precisely the same denominator. In this case, the denominators of the fractions are identical, both being \(3x\). This similarity allows you to add the fractions directly. Recognizing like terms allows you to focus only on the numerators, greatly simplifying the process of adding fractions.

• Helps easily identify which fractions can be combined.
• Makes the addition of fractions straightforward.

Common Denominator

A common denominator is essential for the addition or subtraction of fractions. It refers to a shared denominator between fractions, which in this case is \(3x\). Having a common denominator means each fraction is expressed as being out of the same whole or part, allowing you to simply add the numerators without any adjustments.

• Avoids converting fractions, simplifying them immediately.
• Only exists if denominators match or are made to match.

Numerator Addition

Once you have identified like terms and established a common denominator, the next step is to add the numerators. This step is where the magic happens for fraction addition. You simply add the numbers on top of the fraction bars while retaining the common denominator. For the exercise, adding the numerators gives: \(8 + 5 = 13\). Place this sum over the original denominator, \(3x\), giving \(\frac{13}{3x}\).

• Keeps calculations simple and easy.
• Directly leads to the result fraction.