Problem 28 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{9 a}{7 b}+\frac{3 a}{7 b} $$

Step-by-Step Solution

Answer

Question: Add the following rational expressions with the same denominator: $$\frac{9a}{7b} + \frac{3a}{7b}$$ Answer: $$\frac{12a}{7b}$$

Step 1: Identify the denominators

Since you are given two rational expressions with the same denominator (7b), no further manipulation is needed before adding them.

Step 2: Add the numerators

Add the numerators of the given fractions while keeping the denominator the same: $$ \frac{9a}{7b}+\frac{3a}{7b} = \frac{ 9a + 3a }{7b} $$

Step 3: Simplify the result

Combine the terms in the numerator: $$ \frac{ 9a + 3a }{7b} = \frac{12a}{7b} $$ The resulting rational expression is: $$ \frac{12a}{7b} $$

Key Concepts

Addition and Subtraction of Fractions

Adding and subtracting fractions is a fundamental concept in algebra that helps us work with different parts of the whole. When we deal with rational expressions, which are essentially fractions with polynomials in the numerator and/or the denominator, the process is quite similar. Let's break it down:

• Identify the Denominators: This is your first step. Notice if the denominators are the same or different. If they are the same, the process is simpler, and we can directly add or subtract the numerators. If they are different, we'll need to find a common denominator before proceeding.
• Focus on the Numerators: Once the denominators are the same, we add or subtract the numerators as needed. Just like with simpler numeric fractions, these operations are straightforward when the denominators are aligned.

Using the given exercise, the expressions \(\frac{9a}{7b}\) and \(\frac{3a}{7b}\) clearly have the same denominator (7b), allowing us to simply concentrate on the numerators: \(9a + 3a\). We keep the denominator unchanged, which is crucial to maintaining the value of the expression.

Common Denominators

A common denominator is essential when adding or subtracting fractions because it ensures that the fractions represent parts of the same whole. Without a common denominator, fractions cannot be directly added or subtracted. The importance of common denominators is highlighted even more when dealing with rational expressions.

• Same Denominator Case: If the denominators are the same, as in our problem, the task becomes straightforward because we can directly add the numerators, leaving the common denominator untouched.
• Different Denominator Case: When working with different denominators, finding the least common denominator (LCD) is key. This involves determining the least multiple that all denominators can divide evenly into. This alignment allows us to add or subtract the numerators.

In the context of rational expressions like \(\frac{9a}{7b}\) and \(\frac{3a}{7b}\), having the same denominator, 7b, simplifies the process, eliminating additional steps that would be necessary if denominators were different.

Simplifying Expressions

Simplifying expressions is the process of making them easier to understand or work with by eliminating unnecessary complexity. In mathematics, particularly when working with rational expressions, simplification often involves reducing the expression to its simplest form.

• Combine Like Terms: The first step in simplifying usually involves combining like terms. In our example, after adding the numerators, we get \(9a + 3a\). These are like terms and can be combined to give \(12a\).
• Reduce if Possible: Another key aspect of simplification is reducing expressions to their lowest terms. This involves dividing the numerator and the denominator by their greatest common factor (GCF). For \(\frac{12a}{7b}\), there is no common factor other than 1 between 12a and 7b, so the expression is already in its simplest form.

The goal of simplification is to present expressions in a more compact, comprehensible manner, which in turn makes further algebraic operations more manageable and less error-prone.