Problem 99 - Introductory Algebra for College Students

Question

Will help you prepare for the materia$l covered in the next section. Add: $$\frac{2 x}{3}+\frac{x}{3}$$

Step-by-Step Solution

Answer

\(x\)

Step 1: Identifying Like Terms

Since both terms in the expression \(\frac{2 x}{3} + \frac{x}{3}\) have the denominator \(3\) and variable \(x\), they are similar and can be added together.

Step 2: Add the Numerators

The result of adding the numerators \(2x\) and \(x\) is \(3x\). Hence, the expression becomes \(\frac{3x}{3}\).

Step 3: Simplifying the Result

The expression \(\frac{3x}{3}\) simplifies to \(x\). The variables canceling out, leaving just \(x\).

Key Concepts

Understanding Like Terms

When working with algebraic expressions, it’s crucial to recognize like terms. Like terms are terms that share the same variable raised to the same power. In our exercise, the terms \( \frac{2x}{3} \) and \( \frac{x}{3} \) are considered like terms because they both contain the variable \( x \) and share the same denominator of 3. This commonality ensures that we can combine them in the process of addition.

These shared characteristics allow us to focus solely on the coefficients of the terms for addition. In general, when terms are considered like terms, their coefficients can be added, making the addition of algebraic fractions more manageable. By understanding this principle, you'll be equipped to tackle similar problems efficiently.

Mastering the Numerator

The numerator is the top part of a fraction, and it plays a significant role when adding fractions. In the expression \( \frac{2x}{3} + \frac{x}{3} \), after identifying the like terms, we can turn our attention to the numerators: \(2x\) and \(x\).

Adding the numerators involves straightforward arithmetic with the coefficients: \(2x + x = 3x\). This step underscores the importance of recognizing like terms, as it ensures that the process of addition applies correctly only to coefficients and variables of the same type.

The numerator essentially conveys the "how much" part of each term, adjusting according to the addition performed. Ensuring accuracy at this step is pivotal for maintaining the integrity of our solution.

Embracing Simplification

After adding the numerators and forming a new fraction \( \frac{3x}{3} \), simplification becomes the final and crucial step. Simplification in mathematics means reducing a fraction to its simplest form. In this example, we can simplify \( \frac{3x}{3} \) by canceling out common factors present in both the numerator and the denominator.

The fraction simplifies to \( x \) because \( 3x \) divided by 3 results in \( x \), illustrating the beauty and efficiency of simplification. This process not only makes our answer cleaner but also easier to interpret and use in further calculations or problem-solving scenarios.

Learning to simplify quickly and accurately will assist in honing your algebraic skills and in gaining confidence when tackling more complex expressions.