Solving (x+8)/2 = 8 - (3x)/2 A Step-by-Step Guide

Jul 19, 2025 by ADMIN 50 views

Solving the Equation (x+8)/2 = 8 - (3x)/2: A Comprehensive Guide

Introduction

Hey guys! Today, we're diving into the exciting world of algebra to tackle a specific equation: (x+8)/2 = 8 - (3x)/2. Don't worry if it looks a bit intimidating at first glance. We're going to break it down step-by-step, making sure you understand the logic behind each move. Think of it like solving a puzzle – each step brings us closer to the final answer. This isn't just about getting the right answer; it's about understanding the process, which will help you tackle similar problems with confidence. In this guide, we'll not only solve this equation but also explore the fundamental principles of algebraic manipulation. We'll cover everything from clearing fractions to isolating the variable, ensuring you have a solid grasp of these essential concepts. So, grab your pencils and paper, and let's get started! We'll make this journey through the world of equations fun and engaging, turning what might seem like a daunting task into an enjoyable challenge. Remember, mathematics is not just about numbers and symbols; it's about critical thinking and problem-solving, skills that are valuable in all aspects of life. This equation, (x+8)/2 = 8 - (3x)/2, is a perfect example of how we can use algebraic tools to unravel complex expressions and find solutions. By the end of this guide, you'll not only be able to solve this particular equation but also have a toolkit of techniques to conquer a wide range of algebraic challenges. So, let's embark on this mathematical adventure together!

1. Clearing the Fractions: The First Step to Simplicity

Okay, so the first thing that probably jumps out at you in the equation (x+8)/2 = 8 - (3x)/2 is the fractions. Fractions can sometimes make equations look more complicated than they actually are. So, our initial goal is to get rid of them. To clear these fractions, we're going to use a clever trick: multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, we only have one denominator, which is 2. This makes things nice and straightforward! The LCM of 2 is simply 2. So, we're going to multiply both the left-hand side (LHS) and the right-hand side (RHS) of the equation by 2. This is a crucial step because it maintains the balance of the equation. Remember, whatever we do to one side, we must do to the other to keep things equal. This principle is the bedrock of algebraic manipulation. When we multiply the LHS, (x+8)/2, by 2, the 2 in the numerator and the 2 in the denominator cancel each other out, leaving us with just (x+8). On the RHS, we have 8 - (3x)/2. We need to distribute the multiplication by 2 to both terms. So, 2 multiplied by 8 gives us 16, and 2 multiplied by (3x)/2 results in the 2s canceling, leaving us with just 3x. Now, our equation looks much simpler: x + 8 = 16 - 3x. See how much cleaner it is without the fractions? This is a classic technique in algebra, and mastering it will make solving equations significantly easier. Clearing fractions is often the first step towards simplifying complex equations, and it's a skill you'll use again and again. By understanding why this step works – the principle of maintaining balance and the cancellation of common factors – you're building a strong foundation in algebraic problem-solving. So, let's move on to the next step in our journey to solve this equation.

2. Gathering the 'x' Terms: Bringing the Variables Together

Now that we've successfully cleared the fractions, our equation stands as x + 8 = 16 - 3x. The next logical step is to gather all the terms containing 'x' on one side of the equation. This is a key strategy in solving for a variable – we want to isolate 'x' so we can determine its value. Currently, we have 'x' on the left-hand side and '-3x' on the right-hand side. To bring them together, we'll perform an operation that cancels out the '-3x' on the RHS. The inverse operation of subtraction is addition, so we'll add 3x to both sides of the equation. Remember, maintaining balance is crucial, so whatever we add to one side, we must add to the other. When we add 3x to the LHS, we get x + 3x, which simplifies to 4x. So, the LHS now becomes 4x + 8. On the RHS, adding 3x to -3x cancels them out, leaving us with just 16. Our equation now looks like this: 4x + 8 = 16. We've successfully gathered the 'x' terms on the left side, making progress towards isolating 'x'. This step demonstrates the power of using inverse operations to manipulate equations. By adding 3x to both sides, we effectively moved the 'x' term from the right to the left. This technique is a fundamental tool in algebra, allowing us to rearrange equations into a form that's easier to solve. Gathering like terms is a common practice in simplifying expressions and equations, and it's a skill that will serve you well in more advanced mathematical studies. So, let's continue our journey and isolate 'x' even further in the next step.

3. Isolating the 'x' Term: Moving the Constants

We've made good progress and now have the equation 4x + 8 = 16. Our goal is to isolate 'x', meaning we want to get 'x' by itself on one side of the equation. The next obstacle in our way is the '+ 8' on the left-hand side. To get rid of it, we need to perform the inverse operation of addition, which is subtraction. We'll subtract 8 from both sides of the equation, again maintaining that crucial balance. When we subtract 8 from the LHS, the '+ 8' and '- 8' cancel each other out, leaving us with just 4x. On the RHS, we subtract 8 from 16, which gives us 8. So, our equation now simplifies to 4x = 8. We're getting closer and closer to the solution! This step highlights the importance of using inverse operations to isolate variables. By strategically subtracting 8 from both sides, we effectively moved the constant term from the LHS to the RHS. This is a common technique in solving equations, and it's essential for isolating the variable we're trying to find. Isolating the variable is like peeling away the layers of an onion – each step removes a term that's hindering us from getting to the core, which is the value of 'x'. This process of systematically isolating the variable is a hallmark of algebraic problem-solving. Now that we have 4x = 8, we're just one step away from the final answer. Let's see how we can finally solve for 'x' in the next section.

4. Solving for 'x': The Final Division

Alright, we've reached the final stage! Our equation is now 4x = 8. We're so close to finding the value of 'x'. The last step is to get 'x' completely by itself. Currently, 'x' is being multiplied by 4. To undo this multiplication, we need to perform the inverse operation, which is division. We'll divide both sides of the equation by 4. Dividing both sides by the same number keeps the equation balanced, which is our golden rule in algebra. When we divide the LHS, 4x, by 4, the 4s cancel out, leaving us with just 'x'. On the RHS, we divide 8 by 4, which gives us 2. Therefore, our final answer is x = 2. Congratulations! We've successfully solved the equation. This final step underscores the power of inverse operations in isolating variables. By dividing both sides by 4, we effectively undid the multiplication and revealed the value of 'x'. This technique is a cornerstone of algebraic problem-solving, and it's crucial for solving a wide range of equations. Finding the solution is not just about getting the right number; it's about understanding the process that leads to that number. By understanding each step – clearing fractions, gathering like terms, isolating the variable – you're developing a deep understanding of algebraic principles. So, let's recap the entire process to solidify our understanding.

5. Verifying the Solution: Ensuring Accuracy

We've arrived at the solution x = 2, but before we declare victory, it's always a good idea to verify our answer. This step is crucial to ensure we haven't made any errors along the way. To verify our solution, we'll substitute x = 2 back into the original equation: (x+8)/2 = 8 - (3x)/2. Let's start with the left-hand side (LHS). Substituting x = 2, we get (2+8)/2, which simplifies to 10/2, which equals 5. Now, let's look at the right-hand side (RHS). Substituting x = 2, we get 8 - (3*2)/2, which simplifies to 8 - 6/2, which further simplifies to 8 - 3, which also equals 5. Since the LHS (5) equals the RHS (5), our solution x = 2 is indeed correct! Verifying the solution is a powerful technique for ensuring accuracy in problem-solving. It's like a safety check that confirms our work and gives us confidence in our answer. This step is especially important in exams or assessments, where a small error can lead to an incorrect result. By substituting the solution back into the original equation, we're essentially reversing the steps we took to solve it. If the equation holds true, we can be confident that our solution is accurate. Verification is not just about getting the right answer; it's about developing a habit of checking our work and ensuring that our reasoning is sound. So, always remember to verify your solutions whenever possible.

Conclusion: Mastering Algebraic Equations

Awesome! We've successfully navigated the equation (x+8)/2 = 8 - (3x)/2 and found the solution: x = 2. We not only solved the equation but also explored the underlying principles of algebraic manipulation. We learned how to clear fractions, gather like terms, isolate the variable, and verify our solution. These are essential skills that will serve you well in your mathematical journey. Remember, mathematics is not just about memorizing formulas; it's about understanding the logic and reasoning behind each step. By breaking down complex problems into smaller, manageable steps, we can conquer even the most challenging equations. This equation served as a great example of how we can use algebraic tools to unravel complex expressions and find solutions. We started by clearing fractions to simplify the equation, then gathered the 'x' terms on one side and the constant terms on the other. We used inverse operations to isolate 'x' and finally arrived at the solution. And, importantly, we verified our solution to ensure its accuracy. So, keep practicing, keep exploring, and keep building your mathematical skills. The world of algebra is vast and exciting, and the more you practice, the more confident you'll become. You've got this! We hope this guide has been helpful in your understanding of solving algebraic equations. Remember, the key is to practice consistently and break down problems into smaller, manageable steps. Keep up the great work, and you'll be mastering algebraic equations in no time!