Solving 6n - 47 = 9 - 8n A Step-by-Step Guide
In the realm of mathematics, solving linear equations is a fundamental skill. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations are crucial as they form the basis for more complex mathematical concepts and are widely applied in various fields, including science, engineering, economics, and computer science. The primary goal when solving a linear equation is to isolate the variable on one side of the equation to determine its value. This process involves applying algebraic operations such as addition, subtraction, multiplication, and division while maintaining the equality of both sides. In this article, we will delve into a step-by-step solution of the linear equation 6n - 47 = 9 - 8n. We will explore the underlying principles, the specific techniques used, and the reasoning behind each step. By understanding this process, you will gain a solid foundation for solving a wide range of linear equations. Whether you are a student learning algebra or someone looking to refresh your mathematical skills, this guide will provide a clear and comprehensive understanding of how to tackle linear equations effectively. The ability to solve linear equations is not just an academic exercise; it is a practical skill that empowers you to solve real-world problems. From calculating financial budgets to designing engineering structures, linear equations play a pivotal role. This guide aims to equip you with the necessary tools and knowledge to confidently approach and solve these equations, enhancing your problem-solving capabilities. We will break down the solution into manageable steps, ensuring clarity and ease of understanding. By the end of this article, you will have a thorough grasp of the methods involved and be able to apply them to other similar equations. So, let's embark on this mathematical journey and unravel the intricacies of solving linear equations.
Step 1: Understanding the Equation
Before we dive into the solution, it's crucial to understand the given equation: 6n - 47 = 9 - 8n. This equation is a linear equation because the highest power of the variable 'n' is 1. It consists of two expressions separated by an equals sign (=), indicating that both expressions have the same value. The left side of the equation, 6n - 47, involves the variable 'n' multiplied by the coefficient 6, and a constant term -47. The right side of the equation, 9 - 8n, also involves the variable 'n' multiplied by the coefficient -8, and a constant term 9. Understanding the structure of the equation is the first step in solving it. Recognizing the terms, coefficients, and constants helps in planning the steps required to isolate the variable. We need to rearrange the equation in such a way that all terms involving 'n' are on one side, and all constant terms are on the other side. This rearrangement is achieved by applying algebraic operations that maintain the equality of the equation. For instance, adding the same value to both sides or subtracting the same value from both sides does not change the solution. Similarly, multiplying or dividing both sides by the same non-zero value preserves the equality. The goal is to simplify the equation step by step until we have 'n' isolated on one side, giving us the solution. Before proceeding, let's recap the key components of a linear equation. A variable, in this case 'n', represents an unknown value that we are trying to find. A coefficient is a number that multiplies the variable, such as 6 and -8 in our equation. A constant is a term that does not involve the variable, such as -47 and 9. With a clear understanding of these components, we can now proceed to the next step, which involves rearranging the equation to group like terms together. This foundational step is essential for a smooth and accurate solution.
Step 2: Combining Like Terms
The next step in solving the equation 6n - 47 = 9 - 8n is to combine like terms. Like terms are terms that contain the same variable raised to the same power, or constants. In this equation, the terms involving 'n' are 6n and -8n, and the constant terms are -47 and 9. To combine like terms, we need to move all the 'n' terms to one side of the equation and all the constant terms to the other side. This is achieved by adding or subtracting terms from both sides of the equation. Let's start by moving the -8n term from the right side to the left side. To do this, we add 8n to both sides of the equation: 6n - 47 + 8n = 9 - 8n + 8n. This simplifies to 14n - 47 = 9. Adding 8n to both sides ensures that the equation remains balanced, as we are performing the same operation on both sides. Now, we need to move the constant term -47 from the left side to the right side. To do this, we add 47 to both sides of the equation: 14n - 47 + 47 = 9 + 47. This simplifies to 14n = 56. By adding 47 to both sides, we have successfully isolated the term with 'n' on the left side and the constant term on the right side. At this point, the equation is significantly simplified, making it easier to solve for 'n'. The equation 14n = 56 tells us that 14 times 'n' is equal to 56. The process of combining like terms is a crucial step in solving linear equations. It simplifies the equation and brings us closer to isolating the variable. By carefully adding or subtracting terms from both sides, we maintain the equality of the equation while rearranging the terms in a more manageable form. Now that we have combined like terms, the next step is to isolate the variable 'n' completely, which we will address in the following section.
Step 3: Isolating the Variable
After combining like terms, the equation is now in the simplified form 14n = 56. The next critical step is to isolate the variable 'n'. Isolating the variable means getting 'n' by itself on one side of the equation. In this case, 'n' is being multiplied by 14. To isolate 'n', we need to undo this multiplication. The inverse operation of multiplication is division. Therefore, we divide both sides of the equation by 14. This step is crucial because it maintains the balance of the equation while isolating the variable. Dividing both sides by 14, we get: (14n) / 14 = 56 / 14. On the left side, the 14 in the numerator and the 14 in the denominator cancel each other out, leaving us with just 'n'. On the right side, 56 divided by 14 is 4. So, the equation simplifies to: n = 4. This is the solution to the equation. We have successfully isolated the variable 'n' and found its value. The process of isolating the variable is the final step in solving the equation. It involves applying the inverse operation to undo any mathematical operations that are being performed on the variable. In this case, we used division to undo multiplication. If the variable were being added to a number, we would use subtraction to isolate it. If it were being subtracted, we would use addition. The key is to perform the same operation on both sides of the equation to maintain balance. Once the variable is isolated, we have the solution to the equation. In our case, n = 4 is the value of 'n' that satisfies the original equation 6n - 47 = 9 - 8n. To ensure the solution is correct, it is always a good practice to check the solution by substituting it back into the original equation, which we will discuss in the next section.
Step 4: Checking the Solution
After solving for the variable, it is essential to check the solution to ensure accuracy. This step helps to verify that the value we found for 'n' satisfies the original equation. Substituting the solution back into the original equation will confirm whether both sides of the equation are equal. Our solution for the equation 6n - 47 = 9 - 8n is n = 4. To check this solution, we substitute n = 4 back into the original equation: 6(4) - 47 = 9 - 8(4). Now, we simplify both sides of the equation separately. On the left side, 6(4) is 24, so we have 24 - 47. 24 minus 47 is -23. So, the left side of the equation simplifies to -23. On the right side, 8(4) is 32, so we have 9 - 32. 9 minus 32 is -23. So, the right side of the equation simplifies to -23. Now we compare both sides of the equation: -23 = -23. Since both sides are equal, our solution n = 4 is correct. Checking the solution is a crucial step in problem-solving. It helps to identify any errors made during the solving process. By substituting the solution back into the original equation, we can verify whether the value satisfies the equation. If the two sides of the equation are not equal, it indicates that there is an error in our calculations, and we need to review the steps and correct the mistake. In this case, the left side and the right side of the equation are equal when n = 4, which confirms that our solution is accurate. This process not only validates our answer but also reinforces our understanding of the equation and the steps involved in solving it. It is a valuable practice that builds confidence and ensures precision in mathematical problem-solving. The importance of checking the solution cannot be overstated. It is a fundamental aspect of mathematical rigor and a key skill for anyone working with equations.
Conclusion
In this comprehensive guide, we have walked through the step-by-step process of solving the linear equation 6n - 47 = 9 - 8n. We began by understanding the equation and recognizing its components: variables, coefficients, and constants. This foundational understanding is crucial for planning the solution strategy. Next, we combined like terms by moving all terms involving 'n' to one side of the equation and all constant terms to the other side. This involved adding and subtracting terms from both sides while maintaining the equation's balance. After combining like terms, we isolated the variable 'n' by dividing both sides of the equation by the coefficient of 'n'. This step led us to the solution: n = 4. Finally, we checked our solution by substituting n = 4 back into the original equation. This verification step confirmed that our solution was correct, as both sides of the equation were equal. The ability to solve linear equations is a fundamental skill in mathematics. It forms the basis for more advanced topics and has wide-ranging applications in various fields. By mastering the steps outlined in this guide, you can confidently approach and solve a variety of linear equations. Remember, the key to success in mathematics is practice. The more you practice solving linear equations, the more proficient you will become. Each equation you solve strengthens your understanding of the underlying principles and techniques. This guide provides a solid foundation for further mathematical exploration. As you continue your mathematical journey, you will encounter more complex equations and problems. However, the skills and knowledge you have gained from solving linear equations will serve as a valuable tool in your problem-solving arsenal. Mastering the art of solving linear equations is a significant achievement that opens doors to further mathematical understanding and application.
