Solving The Equation (3n+3)/5 = (5n-1)/9 A Step-by-Step Guide

Algebraic equations form the bedrock of mathematics, and the ability to solve them is a fundamental skill that opens doors to more advanced mathematical concepts. In this comprehensive guide, we will dissect a specific equation, (3n+3)/5 = (5n-1)/9, to not only find the correct value of n but also to illustrate the step-by-step process involved in solving such equations. Whether you're a student grappling with algebra or simply looking to brush up on your math skills, this article will provide a clear and thorough explanation. Solving equations like this is crucial because it strengthens your understanding of mathematical principles, enhances your problem-solving abilities, and prepares you for more complex mathematical challenges. The equation (3n+3)/5 = (5n-1)/9 might seem daunting at first glance, but by breaking it down into manageable steps, we can reveal its solution and solidify your confidence in tackling algebraic problems. Each step in the solution process serves a purpose, and understanding these purposes is key to mastering algebra. From clearing fractions to isolating the variable, we will cover each technique in detail, ensuring you grasp not only the 'how' but also the 'why' behind each manipulation. By the end of this guide, you will not only know the correct value of n but also have a deeper appreciation for the elegance and logic of algebra.

Understanding the Equation: (3n+3)/5 = (5n-1)/9

To effectively solve the equation (3n+3)/5 = (5n-1)/9, it is essential to first understand its structure and the mathematical principles at play. This equation involves fractions, variables, and constants, all of which need to be carefully considered. The variable n represents an unknown value that we aim to find, and our goal is to isolate n on one side of the equation. The presence of fractions, namely (3n+3)/5 and (5n-1)/9, adds a layer of complexity, but we can tackle this by employing techniques that clear these fractions. Understanding the properties of equality is also crucial. The fundamental principle here is that any operation performed on one side of the equation must also be performed on the other side to maintain balance. This ensures that the value of n remains consistent throughout the solution process. For example, if we multiply one side of the equation by a certain number, we must multiply the other side by the same number. Similarly, adding or subtracting terms from both sides keeps the equation balanced. The constants in the equation, such as the numbers 3, 5, and 9, play a role in determining the value of n. By carefully manipulating these constants and the terms involving n, we can gradually simplify the equation and move closer to the solution. Each term in the equation has a specific role, and by understanding how these terms interact, we can develop a strategic approach to solving the equation. The left side of the equation, (3n+3)/5, represents a quantity that is dependent on the value of n. Similarly, the right side, (5n-1)/9, also represents a quantity that varies with n. The equation as a whole states that these two quantities are equal, and our task is to find the specific value of n that makes this equality true. Before diving into the algebraic manipulations, it is helpful to visualize the equation and consider what it represents. This initial understanding can guide our steps and help us avoid common pitfalls. By breaking down the equation into its components and understanding the underlying principles, we set the stage for a successful solution.

Step-by-Step Solution: Isolating the Variable n

Now, let's walk through the step-by-step solution to find the value of n in the equation (3n+3)/5 = (5n-1)/9. This process involves several key algebraic techniques, each designed to simplify the equation and bring us closer to the solution. The first step in solving this equation is to eliminate the fractions. To do this, we need to find the least common multiple (LCM) of the denominators, which are 5 and 9. The LCM of 5 and 9 is 45. We then multiply both sides of the equation by 45 to clear the fractions. Multiplying both sides by 45 gives us: 45 * [(3n+3)/5] = 45 * [(5n-1)/9]. This simplifies to 9(3n+3) = 5(5n-1). The next step is to distribute the numbers on both sides of the equation. Distributing the 9 on the left side gives us 27n + 27, and distributing the 5 on the right side gives us 25n - 5. So, our equation now looks like this: 27n + 27 = 25n - 5. Now, we need to isolate the terms involving n on one side of the equation and the constants on the other side. To do this, we can subtract 25n from both sides of the equation: 27n - 25n + 27 = 25n - 25n - 5. This simplifies to 2n + 27 = -5. Next, we subtract 27 from both sides of the equation to isolate the term with n: 2n + 27 - 27 = -5 - 27. This simplifies to 2n = -32. Finally, to solve for n, we divide both sides of the equation by 2: (2n)/2 = -32/2. This gives us the solution n = -16. Each step in this process is crucial for arriving at the correct solution. By clearing the fractions, distributing the numbers, and isolating the variable, we systematically simplify the equation until we can determine the value of n. This methodical approach is a cornerstone of algebra and can be applied to a wide range of equations.

Verifying the Solution: Ensuring Accuracy

After finding a solution to an equation, it is crucial to verify its accuracy. This step ensures that our calculations are correct and that the value we found for n indeed satisfies the original equation. To verify the solution n = -16 for the equation (3n+3)/5 = (5n-1)/9, we substitute -16 for n in the original equation and check if both sides are equal. First, let's substitute n = -16 into the left side of the equation: (3(-16) + 3)/5. This simplifies to (-48 + 3)/5, which further simplifies to -45/5. Thus, the left side of the equation evaluates to -9. Now, let's substitute n = -16 into the right side of the equation: (5(-16) - 1)/9. This simplifies to (-80 - 1)/9, which further simplifies to -81/9. Thus, the right side of the equation also evaluates to -9. Since both sides of the equation equal -9 when n = -16, we have verified that our solution is correct. This verification step is not just a formality; it is a vital part of the problem-solving process. It helps us catch any errors we might have made during the algebraic manipulations and ensures that we have a reliable answer. In many real-world applications, accuracy is paramount, and verifying our solutions is a way to guarantee that we are making sound decisions based on our calculations. Furthermore, the process of verification reinforces our understanding of the equation and the solution. By substituting the value back into the original equation, we are essentially retracing our steps and confirming that each step was logically sound. This can help solidify our grasp of the underlying mathematical principles and improve our problem-solving skills in the long run. In addition to numerical verification, it can also be helpful to consider the reasonableness of the solution in the context of the problem. Does the value of n make sense given the original equation? Are there any constraints or conditions that the solution must satisfy? These types of checks can provide additional confidence in our answer.

Conclusion: The Power of Algebraic Problem-Solving

In conclusion, we have successfully solved the equation (3n+3)/5 = (5n-1)/9, demonstrating the power and precision of algebraic problem-solving. By following a systematic approach, we were able to isolate the variable n and determine its value to be -16. We began by understanding the structure of the equation, identifying the fractions, variables, and constants involved. This initial analysis laid the groundwork for our solution strategy. We then employed the crucial step of clearing fractions by multiplying both sides of the equation by the least common multiple of the denominators. This eliminated the fractions and simplified the equation, making it easier to work with. Next, we distributed the numbers on both sides of the equation, which further simplified the terms and prepared us for isolating the variable. The process of isolating n involved a series of algebraic manipulations, including adding and subtracting terms from both sides of the equation. Each step was carefully executed to maintain the balance of the equation and ensure the accuracy of our solution. Once we found the value of n, we took the important step of verifying our solution. By substituting n = -16 back into the original equation, we confirmed that both sides were equal, thus validating our answer. This verification process not only gives us confidence in our solution but also reinforces our understanding of the equation and the algebraic techniques used. The ability to solve algebraic equations is a valuable skill that extends beyond the classroom. It is a fundamental tool in many fields, including science, engineering, economics, and computer science. By mastering these techniques, we equip ourselves to tackle a wide range of problems and make informed decisions based on quantitative analysis. Moreover, the problem-solving process itself is a valuable skill. It teaches us how to break down complex problems into smaller, manageable steps, how to apply logical reasoning, and how to persevere in the face of challenges. These skills are transferable to many aspects of life and can contribute to our success in various endeavors. Therefore, the journey of solving the equation (3n+3)/5 = (5n-1)/9 is not just about finding a numerical answer; it is about developing our mathematical prowess and enhancing our problem-solving abilities.