Finding The Value Of N In The Equation (1/2)(n-4)-3=3-(2n+3)
Introduction
In this article, we delve into the realm of algebra to solve for the unknown variable 'n' in the equation $\frac{1}{2}(n-4)-3=3-(2 n+3)$. Algebraic equations form the backbone of numerous mathematical and scientific disciplines, and mastering the techniques to solve them is crucial for problem-solving and analytical thinking. This guide provides a step-by-step approach to deciphering the equation, ensuring a clear understanding of each maneuver. Our focus is to provide a comprehensive explanation that is suitable for both students and enthusiasts looking to sharpen their algebraic skills. Algebraic equations, like the one we're tackling, often appear complex at first glance. However, by breaking them down into smaller, manageable steps, we can systematically isolate the variable and arrive at the solution. The beauty of algebra lies in its ability to represent real-world problems in a symbolic form, allowing us to manipulate these symbols and gain insights into the underlying relationships. Whether you're preparing for an exam, brushing up on your algebra basics, or simply curious about mathematical problem-solving, this guide is tailored to meet your needs. We will explore the fundamental principles of algebraic manipulation, including the distributive property, combining like terms, and the importance of maintaining balance on both sides of the equation. By the end of this exploration, you'll be equipped with the knowledge and confidence to tackle similar algebraic challenges with ease. So, let's embark on this mathematical journey together and unlock the value of 'n' in the given equation.
Step 1: Distribute and Simplify
Our first step in solving the equation $\frac1}{2}(n-4)-3=3-(2 n+3)$ is to simplify both sides by distributing any coefficients and dealing with parentheses. This involves applying the distributive property, which states that a(b + c) = ab + ac. On the left-hand side of the equation, we have the term $\frac{1}{2}(n-4)$. To distribute the $\frac{1}{2}$, we multiply it by both terms inside the parentheses2} * n = \frac{1}{2}n$ and $\frac{1}{2} * -4 = -2$. Therefore, $\frac{1}{2}(n-4)$ simplifies to $\frac{1}{2}n - 2$. Now, let's consider the right-hand side of the equation, which is $3-(2n+3)$. The negative sign in front of the parentheses acts as a -1 coefficient that needs to be distributed. So, we multiply -1 by both terms inside the parentheses2}n - 2 - 3 = 3 - 2n - 3$. The next step is to combine like terms on both sides of the equation. On the left-hand side, we have -2 and -3, which combine to -5. On the right-hand side, we have 3 and -3, which cancel each other out, leaving us with 0. So, the equation simplifies further to{2}n - 5 = -2n$. By applying the distributive property and combining like terms, we've successfully reduced the complexity of the equation, bringing us closer to isolating the variable 'n'. This step is crucial because it streamlines the equation, making it easier to manipulate in subsequent steps. Remember, the goal is to maintain balance on both sides of the equation while performing these operations. This foundation sets the stage for the next phase of solving for 'n'.
Step 2: Combine 'n' Terms
Having simplified the equation to $\frac1}{2}n - 5 = -2n$, our next objective is to consolidate all terms containing 'n' onto one side of the equation. This maneuver allows us to isolate the variable and move closer to finding its value. To achieve this, we need to eliminate the -2n term from the right-hand side. The key principle in algebra is that any operation performed on one side of the equation must also be performed on the other side to maintain balance. Therefore, to eliminate -2n from the right-hand side, we add 2n to both sides of the equation. This gives us2}n - 5 + 2n = -2n + 2n$. On the right-hand side, -2n and +2n cancel each other out, resulting in 0. On the left-hand side, we need to combine the 'n' terms2}n + 2n$. To add these terms, it's helpful to express them with a common denominator. We can rewrite 2n as $\frac{4}{2}n$. Now, we can add the fractions2}n + \frac{4}{2}n = \frac{5}{2}n$. So, the left-hand side of the equation becomes $\frac{5}{2}n - 5$. Putting it all together, our equation now reads{2}n - 5 = 0$. By adding 2n to both sides, we've successfully combined the 'n' terms on one side of the equation. This is a significant step forward in isolating the variable. The equation is now in a form that's easier to manipulate, and we're one step closer to finding the value of 'n'. The careful application of algebraic principles, such as maintaining balance, is crucial in this process. In the next step, we'll continue to isolate 'n' by addressing the constant term on the left-hand side.
Step 3: Isolate the 'n' Term
With the equation now in the form $\frac5}{2}n - 5 = 0$, our next goal is to further isolate the term containing 'n'. Currently, we have the constant term -5 on the same side as the 'n' term. To isolate the $\frac{5}{2}n$ term, we need to eliminate this -5. As we've emphasized, maintaining balance is paramount in algebraic manipulations. Therefore, to eliminate -5 from the left-hand side, we add 5 to both sides of the equation. This gives us2}n - 5 + 5 = 0 + 5$. On the left-hand side, -5 and +5 cancel each other out, leaving us with just $\frac{5}{2}n$. On the right-hand side, 0 + 5 equals 5. So, the equation simplifies to{2}n = 5$. We've successfully isolated the term containing 'n' on one side of the equation. This is a crucial step because it brings us closer to directly solving for the value of 'n'. By adding 5 to both sides, we've effectively neutralized the constant term that was hindering our progress. The equation is now in a much simpler form, with the variable term standing alone on the left-hand side. This sets the stage for the final step in solving for 'n', which involves undoing the multiplication by $\frac{5}{2}$. The careful and consistent application of algebraic principles, such as adding the same value to both sides, is what allows us to systematically solve equations like this one. In the next step, we'll complete the process by finding the value of 'n'.
Step 4: Solve for 'n'
Having successfully isolated the term with 'n', our equation now stands as $\frac5}{2}n = 5$. The final step is to solve for 'n' by undoing the multiplication by the fraction $\frac{5}{2}$. To do this, we need to multiply both sides of the equation by the reciprocal of $\frac{5}{2}$, which is $\frac{2}{5}$. Remember, multiplying a fraction by its reciprocal results in 1, effectively isolating the variable. So, we multiply both sides of the equation by $\frac{2}{5}${5} * \frac{5}{2}n = 5 * \frac{2}{5}$. On the left-hand side, $\frac{2}{5}$ and $\frac{5}{2}$ cancel each other out, leaving us with just 'n'. On the right-hand side, we have $5 * \frac{2}{5}$. We can think of 5 as $\frac{5}{1}$, so the multiplication becomes $\frac{5}{1} * \frac{2}{5}$. Multiplying the numerators gives us 10, and multiplying the denominators gives us 5, resulting in the fraction $\frac{10}{5}$. This fraction simplifies to 2. Therefore, the right-hand side of the equation is equal to 2. Putting it all together, we have: $n = 2$. We have successfully solved for 'n'! By multiplying both sides of the equation by the reciprocal of the coefficient of 'n', we were able to isolate the variable and determine its value. This final step demonstrates the power of using inverse operations to unravel algebraic equations. The solution, n = 2, is the value that satisfies the original equation. To verify this, we can substitute 2 back into the original equation and check if both sides are equal. In the next section, we'll perform this verification step to ensure the accuracy of our solution.
Step 5: Verification
To ensure the accuracy of our solution, n = 2, it's essential to verify it by substituting this value back into the original equation: $\frac1}{2}(n-4)-3=3-(2 n+3)$. Replacing 'n' with 2, we get2}(2-4)-3=3-(2(2)+3)$. Now, let's simplify both sides of the equation separately. On the left-hand side, we first evaluate the expression inside the parentheses{2}(-2) - 3$. Next, we multiply $\frac{1}{2}$ by -2, which gives us -1. Thus, the left-hand side further simplifies to: $-1 - 3$. Finally, we subtract 3 from -1, resulting in -4. So, the left-hand side of the equation simplifies to -4. Now, let's move to the right-hand side of the equation: $3-(2(2)+3)$. First, we perform the multiplication inside the parentheses: $2 * 2 = 4$. So, the right-hand side becomes: $3 - (4 + 3)$. Next, we add 4 and 3 inside the parentheses: $4 + 3 = 7$. Thus, the right-hand side simplifies to: $3 - 7$. Finally, we subtract 7 from 3, which gives us -4. So, the right-hand side of the equation simplifies to -4. Comparing both sides, we have: $-4 = -4$. Since both sides of the equation are equal when n = 2, our solution is verified. This verification step is crucial in problem-solving as it confirms that our calculations and algebraic manipulations were performed correctly. It provides confidence in the accuracy of the solution and demonstrates a thorough understanding of the problem-solving process. In conclusion, we have successfully solved the equation and verified the solution, reinforcing the value of n = 2.
Conclusion
In this comprehensive guide, we have successfully navigated the process of solving for 'n' in the equation $\frac{1}{2}(n-4)-3=3-(2 n+3)$. We began by simplifying both sides of the equation using the distributive property and combining like terms. This initial step streamlined the equation, making it more manageable for subsequent operations. Next, we consolidated all terms containing 'n' onto one side of the equation by adding 2n to both sides. This allowed us to isolate the variable and move closer to finding its value. We then isolated the 'n' term further by adding 5 to both sides, effectively eliminating the constant term on the same side. Finally, we solved for 'n' by multiplying both sides of the equation by the reciprocal of the coefficient of 'n', which was $\frac{2}{5}$. This led us to the solution n = 2. To ensure the accuracy of our solution, we performed a verification step by substituting n = 2 back into the original equation. This confirmed that both sides of the equation were equal, solidifying our confidence in the correctness of the solution. The process of solving algebraic equations involves a systematic application of fundamental principles, such as maintaining balance on both sides of the equation and using inverse operations to isolate the variable. By breaking down the problem into smaller, manageable steps, we can effectively tackle even complex equations. The skills and techniques discussed in this guide are valuable tools for anyone seeking to enhance their algebraic abilities and problem-solving prowess. Mastering these concepts opens doors to a deeper understanding of mathematics and its applications in various fields. We hope this guide has provided a clear and insightful journey into solving for 'n', empowering you to confidently approach similar algebraic challenges in the future. Remember, practice is key to proficiency, so continue to explore and apply these techniques to further refine your skills.
