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

Hey there, math enthusiasts! Ever stumbled upon an equation that looks like a tangled mess but secretly holds a simple solution? Today, we are going to untangle one such equation together. Our mission? To find the value of n in the equation 1/2(n-4)-3=3-(2n+3). It might look intimidating at first glance, but trust me, with a bit of algebraic finesse, we'll crack it open. So, grab your thinking caps, and let's dive into the world of equation solving!

Decoding the Equation 1/2(n-4)-3=3-(2n+3)

The Initial Setup: Understanding the Equation

Alright, let's break down this equation step by step. We've got 1/2(n-4)-3=3-(2n+3). The first thing that probably catches your eye is the fractions and parentheses, right? But don't worry; we're going to handle them like pros. In this algebraic equation, our goal is to isolate n on one side of the equation. This means we need to simplify both sides, combine like terms, and eventually get n all by itself. Think of it as a mathematical puzzle where each step brings us closer to the final answer. Remember, the key to solving equations is to keep them balanced. Whatever operation we perform on one side, we must perform on the other. This ensures that the equality remains true throughout our solving process.

Step 1: Distribute and Conquer

Our first move is to get rid of those parentheses. We do this by distributing the numbers outside the parentheses to the terms inside. On the left side, we have 1/2 multiplying (n-4). So, we multiply 1/2 by n and 1/2 by -4. This gives us (1/2)*n - (1/2)*4, which simplifies to n/2 - 2. Now, let's tackle the right side of the equation: 3-(2n+3). Here, we're subtracting the entire expression (2n+3). This is the same as distributing a -1 across the parentheses. So, we get 3 - 2n - 3. See how we changed the signs of the terms inside the parentheses? This is super important! After distributing, our equation looks like this: n/2 - 2 - 3 = 3 - 2n - 3. We've taken the first major step in simplifying our equation. By distributing correctly, we've opened the door to combining like terms and further simplifying.

Step 2: Combine Like Terms

Now that we've distributed and removed the parentheses, it's time to tidy things up by combining like terms. Like terms are terms that have the same variable raised to the same power (or no variable at all, which we call constants). On the left side of our equation, n/2 - 2 - 3, we can combine the constants -2 and -3. This gives us -5. So, the left side simplifies to n/2 - 5. On the right side, 3 - 2n - 3, we can combine the constants 3 and -3. Guess what? They cancel each other out, leaving us with 0. So, the right side simplifies to -2n. Our equation now looks much cleaner: n/2 - 5 = -2n. We're making progress! By combining like terms, we've reduced the number of terms in our equation, making it easier to work with. This step is all about making the equation more manageable and setting us up for the next steps in solving for n.

Step 3: Eliminate the Fraction

Fractions can sometimes make equations look more complicated than they are. So, let's get rid of the fraction in our equation. We have n/2 - 5 = -2n. The fraction we want to eliminate is n/2. To do this, we can multiply every term in the equation by the denominator of the fraction, which is 2. This will effectively cancel out the fraction. When we multiply n/2 by 2, we get n. When we multiply -5 by 2, we get -10. And when we multiply -2n by 2, we get -4n. So, our equation transforms into: n - 10 = -4n. Notice how the fraction is gone? That's the power of this step! By multiplying through by the denominator, we've cleared the fraction and made our equation much easier to solve. We're one step closer to isolating n and finding its value.

Step 4: Isolate the Variable Term

Our next goal is to get all the terms with n on one side of the equation. We have n - 10 = -4n. Currently, we have n on the left side and -4n on the right side. To get them together, we can add 4n to both sides of the equation. This will cancel out the -4n on the right side and add to the n on the left side. So, we add 4n to both sides: n + 4n - 10 = -4n + 4n. This simplifies to 5n - 10 = 0. Now, all the terms with n are on the left side, which is exactly what we wanted! By isolating the variable term, we've simplified the equation further and brought ourselves closer to solving for n. The next step will involve isolating n completely by dealing with the constant term.

Step 5: Isolate the Variable

We're almost there! We now have 5n - 10 = 0. Our goal is to get n all by itself on one side of the equation. To do this, we first need to get rid of the constant term, which is -10. We can do this by adding 10 to both sides of the equation. This will cancel out the -10 on the left side and add a positive 10 to the right side. So, we add 10 to both sides: 5n - 10 + 10 = 0 + 10. This simplifies to 5n = 10. Now, we have 5 times n equals 10. To isolate n, we need to undo the multiplication by 5. We can do this by dividing both sides of the equation by 5. This will give us n on its own. So, we divide both sides by 5: (5n)/5 = 10/5. This simplifies to n = 2. We've done it! We've successfully isolated n and found its value. The solution to our equation is n = 2. High five!

Verifying the Solution n=2

Plugging n=2 Back into the Original Equation

Alright, mathletes, we've arrived at a potential solution: n = 2. But before we declare victory, it's crucial to verify our answer. Think of it as the final boss battle in our equation-solving game. To verify, we'll take our solution, n = 2, and plug it back into the original equation: 1/2(n-4)-3=3-(2n+3). If our solution is correct, then substituting 2 for n should make both sides of the equation equal. This process ensures that we haven't made any errors along the way and that our solution is indeed the correct one.

Evaluating the Left-Hand Side (LHS)

Let's start by tackling the left-hand side (LHS) of the equation: 1/2(n-4)-3. We'll replace n with 2: 1/2(2-4)-3. Now, we follow the order of operations (PEMDAS/BODMAS). First, we deal with the parentheses: 2 - 4 = -2. So, we have 1/2(-2)-3. Next, we perform the multiplication: 1/2 multiplied by -2 is -1. Our expression now looks like this: -1 - 3. Finally, we subtract: -1 - 3 = -4. So, the left-hand side of the equation evaluates to -4 when n = 2. We've successfully simplified the LHS and are ready to tackle the right-hand side. By carefully following the order of operations, we've ensured an accurate evaluation of the LHS.

Evaluating the Right-Hand Side (RHS)

Now, let's move on to the right-hand side (RHS) of the equation: 3-(2n+3). Again, we'll substitute n with 2: 3-(2(2)+3). Following the order of operations, we first deal with the parentheses. Inside the parentheses, we have 2(2) + 3. We start with the multiplication: 2 multiplied by 2 is 4. So, we have 4 + 3, which equals 7. Now, our expression looks like this: 3 - 7. Finally, we subtract: 3 - 7 = -4. Guess what? The right-hand side also evaluates to -4 when n = 2! We've successfully simplified the RHS and found that it matches the value of the LHS. This is a great sign that our solution is correct.

The Grand Finale: LHS = RHS

Drumroll, please! We've evaluated both sides of the equation, and guess what we found? When n = 2, the left-hand side (LHS) equals -4, and the right-hand side (RHS) also equals -4. This means that LHS = RHS, which is exactly what we need to confirm that our solution is correct. We've successfully verified that n = 2 is indeed the solution to the equation 1/2(n-4)-3=3-(2n+3). Give yourselves a pat on the back, math champions! We've not only solved the equation but also rigorously checked our answer. This verification step is a crucial part of problem-solving, ensuring accuracy and building confidence in our mathematical abilities.

Conclusion: Celebrating Our Mathematical Victory

Woo-hoo! We did it, guys! We successfully navigated the equation 1/2(n-4)-3=3-(2n+3) and discovered that the value of n is indeed 2. We didn't just stop at finding a solution; we went the extra mile and verified it, ensuring our answer was spot-on. This journey through algebra highlights the power of breaking down complex problems into smaller, manageable steps. Remember, equations might look intimidating at first, but with a systematic approach, a dash of algebraic skill, and a sprinkle of perseverance, they're no match for us! So, the next time you encounter a tricky equation, remember our adventure today and tackle it with confidence. You've got this!

FAQs About Solving Equations

What is the key to solving complex equations?

The key to solving complex equations lies in breaking them down into smaller, more manageable steps. Start by simplifying each side of the equation separately. This often involves distributing any multiplication over parentheses and combining like terms. Once both sides are simplified, focus on isolating the variable you're trying to solve for. This usually involves performing inverse operations (addition/subtraction, multiplication/division) to get the variable by itself on one side of the equation. Remember, whatever operation you perform on one side of the equation, you must also perform on the other side to maintain balance. Consistency and attention to detail are crucial when tackling complex equations. Don't rush, and double-check your work at each step to minimize errors. With practice and a systematic approach, even the most daunting equations can be conquered.

Why is it important to verify the solution in an equation?

Verifying the solution in an equation is a critical step because it ensures the accuracy of your answer. Think of it as the ultimate safety check in your problem-solving process. When you solve an equation, you're essentially manipulating it through a series of steps to isolate the variable. While each step might seem correct on its own, it's possible to make a mistake along the way, such as an incorrect distribution, sign error, or miscalculation. Verifying your solution involves plugging the value you found for the variable back into the original equation. If the left-hand side (LHS) of the equation equals the right-hand side (RHS) after the substitution, then your solution is correct. If the LHS and RHS are not equal, it indicates that an error was made during the solving process, and you need to revisit your steps to identify and correct the mistake. Verifying not only confirms the correctness of your answer but also helps you build confidence in your problem-solving skills. It's a best practice that every math student should adopt.

Can all equations be solved?

No, not all equations can be solved, at least not in the traditional sense of finding a numerical value for the variable. There are a few scenarios where an equation might not have a solution. One such scenario is when the equation leads to a contradiction. For example, an equation like x + 1 = x + 2 has no solution because it implies that 1 = 2, which is false. Another scenario is when the equation results in an undefined operation, such as division by zero. Additionally, some equations might have solutions, but those solutions are complex numbers (involving the imaginary unit i, where i² = -1) rather than real numbers. Furthermore, certain types of equations, like transcendental equations (equations involving transcendental functions such as sine, cosine, exponential, and logarithmic functions), might not have solutions that can be expressed in terms of elementary functions. These equations often require numerical methods or approximations to find solutions. So, while many equations can be solved using algebraic techniques, it's important to recognize that there are cases where solutions either don't exist or require more advanced mathematical tools to find.

What are common mistakes to avoid when solving equations?

When solving equations, several common mistakes can derail your progress and lead to incorrect solutions. One of the most frequent errors is making mistakes with signs, especially when distributing a negative sign across parentheses or combining like terms. Always double-check the signs of your terms and ensure you're applying the distributive property correctly. Another common pitfall is not performing the same operation on both sides of the equation. Remember, the golden rule of equation solving is to maintain balance; whatever you do to one side, you must do to the other. Failing to do so will disrupt the equality and lead to a wrong answer. Additionally, students often make mistakes when combining like terms, either by combining terms that are not alike or by incorrectly adding or subtracting coefficients. Pay close attention to the variables and their exponents when combining terms. Forgetting to verify your solution is another critical mistake. Always plug your solution back into the original equation to check for accuracy. Lastly, rushing through the steps can increase the likelihood of making errors. Take your time, write neatly, and double-check each step to minimize mistakes and ensure you arrive at the correct solution. By being mindful of these common errors, you can significantly improve your equation-solving skills.