Simplifying (x+9)/(3y) + (2x)/(3y) A Step-by-Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill that underpins more advanced concepts. This article delves into the process of simplifying the algebraic expression (x+9)/(3y) + (2x)/(3y), a common problem encountered in algebra. We will meticulously break down each step, ensuring clarity and comprehension for readers of all levels. This exploration will not only provide the solution but also enhance your understanding of fraction manipulation and algebraic simplification. The ability to simplify expressions like (x+9)/(3y) + (2x)/(3y) is crucial for various mathematical applications, from solving equations to calculus. This guide serves as a valuable resource for students, educators, and anyone seeking to sharpen their algebraic skills. By mastering these techniques, you'll be well-equipped to tackle more complex mathematical challenges. Let's embark on this mathematical journey, unraveling the intricacies of this expression and solidifying your understanding of algebraic simplification. We'll explore the underlying principles and demonstrate the step-by-step process, empowering you to confidently approach similar problems in the future. So, let's dive in and unlock the secrets of simplifying (x+9)/(3y) + (2x)/(3y).

Understanding the Basics: Fractions with Common Denominators

Before tackling the specific expression, it's essential to grasp the fundamental concept of adding fractions with common denominators. When fractions share the same denominator, the addition process becomes straightforward. You simply add the numerators while keeping the denominator constant. This principle forms the bedrock of our simplification journey. To illustrate, consider the fractions a/c and b/c. Their sum is (a+b)/c. This rule is paramount when simplifying algebraic expressions involving fractions. The expression (x+9)/(3y) + (2x)/(3y) perfectly exemplifies this scenario, as both fractions share the same denominator, 3y. Understanding this foundational rule simplifies the process significantly. This is not just a mathematical rule; it's a principle that guides our simplification journey. Grasping this concept is the key to unlocking the simplification of (x+9)/(3y) + (2x)/(3y) and similar expressions. Think of it as the cornerstone upon which we build our solution. This concept is crucial not only for this specific problem but also for a multitude of algebraic challenges involving fractions. With a firm understanding of this principle, you'll be well-prepared to confidently navigate the steps involved in simplifying the given expression. So, let's carry this understanding forward as we delve into the simplification process.

Step-by-Step Simplification of (x+9)/(3y) + (2x)/(3y)

Now, let's apply the principle of adding fractions with common denominators to simplify the expression (x+9)/(3y) + (2x)/(3y). Since both fractions have the same denominator, 3y, we can directly add the numerators. This involves combining (x+9) and (2x). The sum of the numerators is (x + 9 + 2x). Combining like terms, we get 3x + 9. Therefore, the expression becomes (3x + 9)/(3y). This is a significant step forward, but our simplification isn't complete yet. We can further simplify the numerator by factoring out the greatest common factor (GCF). In this case, the GCF of 3x and 9 is 3. Factoring out 3 from the numerator, we get 3(x + 3). Now, our expression looks like 3(x + 3)/(3y). This step is crucial as it reveals an opportunity for further simplification. Notice that both the numerator and the denominator share a common factor of 3. We can now cancel out this common factor to arrive at the simplest form of the expression. This step highlights the elegance of algebraic simplification, where strategic factoring unveils hidden opportunities for reduction. By meticulously following these steps, we transform the original expression into its most concise form. This not only provides the solution but also reinforces the importance of methodical algebraic manipulation. Let's proceed to the final step, where we eliminate the common factor and present the simplified expression.

Final Simplification: Cancelling Common Factors

Building upon the previous step, we now have the expression 3(x + 3)/(3y). As we observed, both the numerator and the denominator share a common factor of 3. This allows us to perform a crucial simplification: canceling out the common factor. Dividing both the numerator and the denominator by 3, we eliminate the common factor. This process transforms the expression into (x + 3)/y. This is the simplified form of the original expression, (x+9)/(3y) + (2x)/(3y). We have successfully navigated the simplification process, arriving at the most concise representation of the expression. This final step underscores the power of identifying and eliminating common factors in algebraic expressions. It's a fundamental technique that significantly simplifies complex expressions. The simplified form, (x + 3)/y, is not only easier to understand but also more convenient for further mathematical operations. This demonstrates the practical value of algebraic simplification. By mastering this process, you gain the ability to manipulate expressions effectively and efficiently. This is a crucial skill for various mathematical applications, including solving equations, calculus, and more. The journey from the original expression to its simplified form highlights the elegance and power of algebraic manipulation. So, let's celebrate this achievement and solidify your understanding of the process.

The Answer and Conclusion

After meticulously simplifying the expression (x+9)/(3y) + (2x)/(3y), we have arrived at the simplified form: (x + 3)/y. Therefore, the correct answer is C) (x+3)/y. This journey has not only provided us with the solution but has also reinforced our understanding of key algebraic principles. We started by understanding the basics of adding fractions with common denominators. Then, we meticulously applied this principle to the given expression, combining the numerators and keeping the denominator constant. We further simplified the expression by factoring out the greatest common factor from the numerator and then canceling out the common factor between the numerator and the denominator. This step-by-step approach highlights the importance of methodical algebraic manipulation. It also underscores the power of identifying and eliminating common factors in simplifying expressions. The simplified form, (x + 3)/y, is not only concise but also easier to work with in further mathematical operations. This exemplifies the practical value of algebraic simplification. By mastering these techniques, you equip yourself with essential skills for tackling more complex mathematical challenges. This problem serves as a valuable exercise in algebraic manipulation and simplification. It reinforces the fundamental principles and empowers you to approach similar problems with confidence. So, let's carry this understanding forward and continue to explore the fascinating world of mathematics.