Solving N + 1 = 4(n - 8) A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of solving the linear equation n + 1 = 4(n - 8). We will explore the step-by-step methods, providing a clear and comprehensive understanding of how to arrive at the solution. Mastering the art of solving linear equations like this one is crucial for various mathematical applications and real-world problem-solving scenarios. This equation, n + 1 = 4(n - 8), while seemingly simple, exemplifies the core principles of algebraic manipulation. We will break down each step, ensuring clarity and building a strong foundation for tackling more complex equations in the future. This guide aims to equip you with the knowledge and skills necessary to confidently solve similar problems and apply these techniques in diverse mathematical contexts. Understanding the underlying concepts of linear equations is essential not only for academic success but also for practical applications in fields such as engineering, finance, and computer science. By mastering the techniques presented here, you will be well-prepared to tackle a wide range of mathematical challenges. Remember, practice is key, so work through the example carefully and try applying the same methods to other similar equations. This will solidify your understanding and build your confidence in solving linear equations. Let's embark on this journey together and unlock the power of algebra!

Step 1: Distribute the 4

The first crucial step in solving the equation n + 1 = 4(n - 8) is to eliminate the parentheses. This is achieved by distributing the 4 across the terms inside the parentheses. Distributing involves multiplying the 4 by both n and -8. This application of the distributive property is a cornerstone of algebraic manipulation and is frequently encountered when solving equations. By correctly distributing, we simplify the equation and pave the way for subsequent steps towards isolating the variable. This initial step is vital, as it transforms the equation into a more manageable form, making it easier to isolate the variable and determine its value. The distributive property, which states that a(b + c) = ab + ac, is a fundamental concept in algebra. Applying it correctly ensures that we maintain the equality of the equation while simplifying its structure. In our case, we are applying the distributive property in reverse, expanding the expression 4(n - 8) to remove the parentheses. This expansion is a critical step in solving for n, as it allows us to combine like terms and isolate the variable. By understanding and correctly applying the distributive property, you'll gain a powerful tool for solving a wide range of algebraic equations. Remember, precision in this step is crucial, as any error here will propagate through the rest of the solution process. So, let's carefully apply the distributive property and move on to the next stage of solving for n.

Applying the distributive property, we get:

n + 1 = 4 * n - 4 * 8
n + 1 = 4n - 32

Step 2: Rearrange the Equation

Having successfully distributed the 4, the next pivotal step in solving n + 1 = 4n - 32 is to strategically rearrange the equation. Our primary goal here is to group like terms together, specifically, to gather the terms containing 'n' on one side of the equation and the constant terms on the other side. This rearrangement is achieved through the fundamental principle of maintaining equality by performing the same operations on both sides of the equation. By systematically moving terms, we're essentially isolating the variable 'n', bringing us closer to the solution. This process of rearranging is not just about moving terms; it's about strategically simplifying the equation to make it easier to solve. The key is to perform operations that maintain the balance of the equation, ensuring that both sides remain equal. Think of it like a balancing scale – whatever you do to one side, you must do to the other. This principle is fundamental to solving any algebraic equation and is crucial for avoiding errors. The rearrangement process involves using inverse operations. For example, if we want to move a term that is being added on one side, we subtract it from both sides. Conversely, if a term is being subtracted, we add it to both sides. This systematic approach ensures that we maintain the integrity of the equation while simplifying it. In our specific case, we'll focus on moving the 'n' terms and the constant terms to their respective sides, setting the stage for the final steps in solving for 'n'. So, let's carefully rearrange the equation, keeping the principle of equality in mind, and move closer to our solution.

To group the 'n' terms on one side, subtract 'n' from both sides:

n + 1 - n = 4n - 32 - n
1 = 3n - 32

Next, to group the constant terms, add 32 to both sides:

1 + 32 = 3n - 32 + 32
33 = 3n

Step 3: Isolate n

After strategically rearranging the equation, we arrive at a critical juncture in solving 33 = 3n: isolating the variable 'n'. This step is paramount as it directly leads to the solution of the equation. Isolating 'n' means getting 'n' by itself on one side of the equation. To achieve this, we must undo any operations that are being performed on 'n'. In our case, 'n' is being multiplied by 3. The inverse operation of multiplication is division. Therefore, to isolate 'n', we will divide both sides of the equation by 3. This principle of using inverse operations is a cornerstone of solving algebraic equations. It ensures that we maintain the balance of the equation while systematically simplifying it. By dividing both sides by 3, we are effectively canceling out the multiplication by 3 on the right side, leaving 'n' isolated. This process of isolation is not just about performing a mathematical operation; it's about strategically manipulating the equation to reveal the value of the unknown variable. The key is to identify the operation being performed on the variable and then apply the inverse operation to both sides of the equation. This consistent application of inverse operations is what allows us to unravel the equation and find the solution. In our quest to isolate 'n', we must remember the fundamental principle of equality: whatever we do to one side of the equation, we must do to the other. This ensures that the equation remains balanced and that our solution is accurate. So, let's confidently divide both sides by 3, isolate 'n', and reveal the solution to our equation.

To isolate 'n', divide both sides by 3:

33 / 3 = 3n / 3
11 = n

Therefore, n = 11.

Step 4: Verify the Solution

Having arrived at a potential solution, the crucial final step in solving n + 1 = 4(n - 8) is to meticulously verify the answer. This step is not merely a formality; it is an essential safeguard against potential errors made during the solving process. Verification involves substituting the calculated value of 'n' back into the original equation and checking if both sides of the equation are equal. This process ensures that our solution satisfies the initial conditions and that no algebraic errors were committed along the way. By substituting 'n = 11' back into the original equation, we can confirm its validity. This substitution transforms the equation into a numerical statement that we can evaluate to check for equality. If both sides of the equation evaluate to the same value, we can confidently conclude that our solution is correct. However, if the two sides are not equal, it indicates that an error was made somewhere in the solving process, and we need to revisit our steps to identify and correct the mistake. The verification step is not just about checking the answer; it's about reinforcing our understanding of the equation and the solution process. It allows us to gain confidence in our algebraic skills and develop a habit of accuracy. By making verification a standard part of our problem-solving routine, we can minimize errors and ensure the reliability of our solutions. So, let's take the time to carefully substitute 'n = 11' back into the original equation and verify that our solution is indeed correct.

Substitute n = 11 into the original equation:

11 + 1 = 4(11 - 8)
12 = 4(3)
12 = 12

The left side equals the right side, so the solution n = 11 is correct.

In conclusion, the journey of solving the equation n + 1 = 4(n - 8) has provided us with a comprehensive understanding of the steps involved in tackling linear equations. From distributing the constant to rearranging terms and finally isolating the variable, each step is crucial in arriving at the correct solution. Moreover, the importance of verifying the solution cannot be overstated, as it ensures the accuracy and reliability of our answer. Mastering these techniques is not just about solving this particular equation; it's about building a solid foundation for tackling more complex mathematical problems in the future. The principles and methods we've explored here are applicable across a wide range of algebraic equations and mathematical contexts. By understanding the underlying concepts and practicing these skills, you can confidently approach any linear equation and systematically solve for the unknown variable. The ability to solve linear equations is a valuable asset in various fields, from science and engineering to finance and economics. It's a fundamental skill that empowers us to analyze problems, make informed decisions, and solve real-world challenges. So, embrace the challenge of solving equations, practice diligently, and continue to expand your mathematical horizons. With a solid understanding of linear equations and the techniques we've discussed, you'll be well-equipped to tackle a wide range of mathematical problems and excel in your academic and professional endeavors. Remember, mathematics is a journey of continuous learning and discovery, and mastering the art of solving equations is a significant step forward on that path.