Solving For N In The Equation 1/2(n-4)-3=3-(2n+3)

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In this article, we will delve into solving a mathematical equation to find the value of n. Equations are the backbone of algebra, and the ability to manipulate and solve them is a crucial skill. We'll dissect the given equation step-by-step, applying the fundamental principles of algebra to isolate n and arrive at the solution. This process will not only help in solving this particular problem but also strengthen your understanding of algebraic techniques applicable to a wide range of mathematical scenarios. So, let's embark on this journey of equation-solving and uncover the value of n in the equation 12(n−4)−3=3−(2n+3){\frac{1}{2}(n-4)-3=3-(2 n+3)}. The journey through this article will enhance your problem-solving capabilities and provide a solid foundation for more advanced mathematical concepts. Remember, practice is the key to mastering these skills, and this article serves as a valuable guide in that endeavor. So, stay focused, and let's begin!

The Problem

We are tasked with finding the value of n in the following equation:

12(n−4)−3=3−(2n+3)\frac{1}{2}(n-4)-3=3-(2 n+3)

This equation involves fractions, parentheses, and variables, making it a good exercise in applying the order of operations and algebraic manipulation. Our goal is to isolate n on one side of the equation, which will reveal its value. The process will involve distributing, combining like terms, and performing inverse operations. By carefully following each step, we can accurately determine the value of n that satisfies the equation. This is a fundamental skill in algebra and is essential for solving more complex mathematical problems. The step-by-step approach we will take in this article will not only provide the solution but also help develop a systematic way of tackling similar equations in the future. So, let's get started and break down the equation to find the value of n.

Step-by-Step Solution

Step 1: Distribute

The first step in solving the equation is to eliminate the parentheses by distributing the constants. On the left side, we distribute the 12{\frac{1}{2}} across the terms inside the parenthesis (n−4){(n - 4)}. On the right side, we distribute the negative sign (which is equivalent to multiplying by -1) across the terms inside the parenthesis (2n+3){(2n + 3)}. This process simplifies the equation and prepares it for further manipulation. The distribution step is crucial as it removes the grouping symbols, allowing us to combine like terms later on. By carefully applying the distributive property, we ensure that each term inside the parentheses is correctly multiplied by the constant outside, maintaining the equation's balance. This step is a fundamental algebraic technique and is frequently used in solving equations and simplifying expressions. Let's perform the distribution:

12∗n+12∗(−4)−3=3−1∗(2n)−1∗(3)\frac{1}{2} * n + \frac{1}{2} * (-4) - 3 = 3 - 1 * (2n) - 1 * (3)

Simplifying this gives us:

12n−2−3=3−2n−3\frac{1}{2}n - 2 - 3 = 3 - 2n - 3

Step 2: Combine Like Terms

Now that we've distributed and eliminated the parentheses, the next step is to combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power (or are constants). On the left side, we can combine the constants -2 and -3. On the right side, we can combine the constants 3 and -3. Combining like terms simplifies the equation further, making it easier to isolate the variable. This step is a fundamental part of algebraic manipulation and is essential for solving equations efficiently. By grouping and combining like terms, we reduce the number of terms in the equation, making it more manageable and less prone to errors in subsequent steps. Let's combine the like terms:

On the left side:

−2−3=−5-2 - 3 = -5

On the right side:

3−3=03 - 3 = 0

So, the equation becomes:

12n−5=−2n\frac{1}{2}n - 5 = -2n

Step 3: Isolate the Variable Term

To isolate the variable term, we need to get all the terms containing n on one side of the equation. A common strategy is to add the term with the variable from one side to the other side. In this case, we can add 2n to both sides of the equation. This will eliminate the -2n term on the right side and move it to the left side, where it can be combined with the 12n{\frac{1}{2}n} term. Isolating the variable term is a crucial step in solving equations, as it brings us closer to finding the value of the variable. By performing the same operation on both sides of the equation, we maintain the balance and ensure that the equation remains valid. This step demonstrates the importance of using inverse operations to manipulate equations and isolate variables. Let's add 2n to both sides:

12n−5+2n=−2n+2n\frac{1}{2}n - 5 + 2n = -2n + 2n

Simplifying this gives us:

12n+2n−5=0\frac{1}{2}n + 2n - 5 = 0

Now, we need to combine the n terms. To do this, we can rewrite 2n as 42n{\frac{4}{2}n}:

12n+42n−5=0\frac{1}{2}n + \frac{4}{2}n - 5 = 0

Combining the terms gives us:

52n−5=0\frac{5}{2}n - 5 = 0

Step 4: Isolate the Variable

Now that we have the variable term isolated on one side, we need to isolate the variable n itself. To do this, we first need to eliminate the constant term on the left side, which is -5. We can do this by adding 5 to both sides of the equation. This will cancel out the -5 on the left side and move the constant to the right side. Isolating the variable is the final step in solving for n, and it involves undoing any operations that are being performed on the variable. By performing inverse operations, we gradually strip away the terms and coefficients surrounding the variable until we are left with n by itself. This step highlights the importance of understanding inverse operations in solving equations. Let's add 5 to both sides:

52n−5+5=0+5\frac{5}{2}n - 5 + 5 = 0 + 5

Simplifying this gives us:

52n=5\frac{5}{2}n = 5

Next, to isolate n, we need to get rid of the coefficient 52{\frac{5}{2}}. We can do this by multiplying both sides of the equation by the reciprocal of 52{\frac{5}{2}}, which is 25{\frac{2}{5}}. This will cancel out the fraction on the left side and leave us with n by itself. Multiplying by the reciprocal is a common technique for solving equations involving fractions. It effectively undoes the multiplication by the fraction and isolates the variable. This step demonstrates a practical application of the properties of fractions and their inverses. Let's multiply both sides by 25{\frac{2}{5}}:

25∗52n=5∗25\frac{2}{5} * \frac{5}{2}n = 5 * \frac{2}{5}

Simplifying this gives us:

n=2n = 2

Final Answer

Therefore, the value of n in the equation 12(n−4)−3=3−(2n+3){\frac{1}{2}(n-4)-3=3-(2 n+3)} is 2. This corresponds to option B. We have successfully solved the equation by systematically applying the principles of algebra, including distribution, combining like terms, and using inverse operations to isolate the variable. The step-by-step approach we followed ensures accuracy and provides a clear understanding of the solution process. This problem demonstrates the importance of algebraic manipulation in solving equations and highlights the fundamental skills required for more advanced mathematical concepts. By mastering these techniques, you can confidently tackle a wide range of mathematical challenges. Remember to always double-check your work and ensure that your solution satisfies the original equation. In this case, substituting n = 2 back into the original equation confirms that it is indeed the correct solution. The final answer is:

B. n = 2