Solving The Equation: (x + 78) - 47 = 93

Hey everyone, let's dive into solving the equation (x + 78) – 47 = 93! Don't worry, it's not as scary as it might look at first glance. We're going to break it down step-by-step, making it super easy to understand. This is a fundamental concept in algebra, and once you get the hang of it, you'll be solving similar equations in your sleep. So, grab your pencils, and let's get started. We'll be using basic arithmetic operations to isolate the variable x and find its value. Remember, the goal is always to get x by itself on one side of the equation. We'll achieve this by performing the same operations on both sides of the equation, ensuring that the equality is maintained throughout the process. This principle of maintaining balance is crucial in algebra. Think of the equation as a balanced scale; whatever you do to one side, you must do to the other to keep it balanced. This approach ensures that the solution remains valid and accurate. We'll start by simplifying the left side and then move towards isolating x. Let's tackle this problem together, and you'll see how straightforward it can be. Understanding these concepts will give you a solid foundation for more complex mathematical problems. This equation is a great example of applying the order of operations in reverse. So, let’s begin our mathematical journey and make sure that this is super simple and easy to understand.

Step-by-Step Solution: Unveiling the Mystery

Alright guys, let's get down to the nitty-gritty and solve this equation! Here's the breakdown, one step at a time: The original equation is (x + 78) – 47 = 93. First, we need to simplify things a bit. We've got (x + 78) and then we're subtracting 47. We can combine the constants on the left side. So, we have the equation (x + 78) – 47 = 93. Now, subtract 47 from 78. This gives us x + 31 = 93. Now that we have the simplified equation, our goal is to isolate x. To do that, we need to get rid of the + 31 next to the x. The opposite of adding 31 is subtracting 31. So, we subtract 31 from both sides of the equation. This is super important: whatever you do to one side, you must do to the other to keep the equation balanced. So, we'll do: x + 31 – 31 = 93 – 31. On the left side, +31 – 31 cancels out, leaving us with just x. On the right side, 93 – 31 equals 62. So now we're left with x = 62. And boom! We've found the solution. It's really that simple! Let's summarize and write the solution in a more simplified way so that this is much easier to understand. The key to this problem is using the reverse order of operations. This helps ensure that we do not encounter any problems while solving and helps solve problems much more efficiently. Remember, we are trying to isolate x. We subtracted 31 from both sides to cancel out the positive 31 on the left side. And we found out that x = 62. That means the value of x is 62. Now, let’s verify our answer!

Verification: Proof of the Pudding

Alright, folks, we've got our answer, x = 62. But how do we know if we're right? Always a great idea to verify, right? Let's substitute the value of x back into the original equation to check if it holds true. The original equation was (x + 78) – 47 = 93. Now, we're going to replace x with 62. So, our equation becomes (62 + 78) – 47 = 93. Let's simplify the terms inside the parentheses first. 62 + 78 equals 140. So now we have 140 – 47 = 93. Okay, what's 140 – 47? It’s 93! So, we have 93 = 93. Since both sides of the equation are equal, our solution is correct! This verification step is a crucial part of problem-solving. It not only confirms the accuracy of our solution but also helps to build confidence in our abilities. If the equation does not hold true after substituting the value of x, then we would know that we need to re-evaluate our steps and look for any potential errors in our calculation. However, in our case, the equation balances out perfectly, which provides solid evidence that x = 62 is indeed the correct solution. It also shows us that we can tackle these kinds of problems and not be intimidated. Remember to always verify your answers to ensure accuracy and build a deeper understanding of the concepts. Keep practicing these types of problems, and you'll find yourself getting faster and more accurate with each attempt. This way of validating your work not only helps you avoid errors but also boosts your confidence in your math skills. So, always make it a habit to check your solutions.

Key Concepts: Grasping the Essentials

Let's take a moment to recap the core concepts we used to solve this equation. Understanding these concepts is like having a toolbox filled with the right tools for any algebraic problem. This is going to help you in the long run. The first key concept is the order of operations (often remembered by the acronym PEMDAS/BODMAS). While we didn't explicitly use the order of operations in this exact form, we were essentially working in reverse. We first simplified within the parentheses, then addressed the subtraction. Secondly, the properties of equality are super important. This means that if you perform the same operation on both sides of an equation, the equality remains true. In our case, we subtracted 31 from both sides to isolate x. This ensures that the equation stays balanced. Think of it like a seesaw; you need to keep both sides equal to maintain balance. And finally, isolating the variable is the ultimate goal. The whole point of solving an equation is to find the value of the unknown variable. In our case, we isolated x by performing inverse operations. We reversed the addition and subtraction to get x alone on one side. These concepts are fundamental in algebra. Make sure you understand them. It's like having the key to unlock the solutions. Grasping these concepts will provide a solid foundation for tackling more complex algebraic problems. Keep these principles in mind as you progress in your mathematical journey, and you'll find that solving equations becomes much more manageable and enjoyable. Remember, practice is key. The more you work through these problems, the more confident and proficient you will become. And do not hesitate to ask for help or clarification whenever needed; it's a great way to solidify your understanding. Each concept builds upon the previous one. And so, mastering these basic principles will allow you to solve more complex equations with ease and confidence. So keep going, and you'll see great progress!

Common Mistakes: Avoiding Pitfalls

Alright, let's talk about some common mistakes that people make when solving equations like this, so you can avoid them! One of the biggest mistakes is forgetting to perform the same operation on both sides of the equation. Remember the seesaw analogy? If you only subtract from one side, the equation is no longer balanced, and your solution will be wrong. Another common error is mixing up the signs. For example, when moving terms across the equal sign, make sure you correctly apply the inverse operation. If you are moving a positive number, you must subtract it from both sides. This is a very common mistake. Be extra careful with the negative signs, especially when distributing them in more complex equations. Pay attention to the details. Accuracy is key in math, and a small error can lead to the wrong answer. Take your time, write your steps clearly, and double-check your calculations. It's better to go slow and get it right than to rush and make a mistake. Be organized. Keep your work neat. Organizing your work helps you to identify and fix mistakes. Also, it’s much easier to follow your steps. Remember to substitute your answer back into the original equation to verify your solution. This will help you to catch any errors you might have made along the way. Be mindful of these common pitfalls, and you'll significantly increase your chances of solving equations correctly. Practice makes perfect. So, keep working through different types of problems, and don't be afraid to ask for help if you get stuck. These issues are very common and very easy to fix. So, always remember this when you try to solve an equation.

Practice Problems: Test Your Skills

Ready to put your newfound knowledge to the test? Here are a few practice problems for you to solve on your own. Try these equations, and make sure to show your work step-by-step. Remember to verify your answers! It's super important. Here are some equations you can try: (x + 55) – 22 = 88, (x + 33) – 11 = 44, and (x + 90) – 60 = 70. The most important thing here is to remain calm and approach each problem systematically. Break down each equation step by step. Try to stay organized, write down each step carefully. Always double-check your calculations and, of course, verify your answers by substituting them back into the original equation. The goal is not just to get the right answer, but to understand the process. The more you practice, the more confident you'll become in your problem-solving abilities. Don’t worry if you don’t get it right the first time; it's all about learning and improving. Each problem you solve will reinforce the concepts we’ve discussed and help you build a solid foundation in algebra. Keep practicing and applying the techniques we've learned, and you will begin to see a significant improvement in your skills. Always remember the importance of checking your solutions and learning from any errors you make. Feel free to come back and review the steps we've covered, or look up additional resources online. With each problem, you're building a stronger understanding of algebraic concepts. This helps you develop critical thinking skills. Keep at it, and you'll find that solving equations becomes easier and more enjoyable over time! This continuous practice will not only enhance your mathematical skills but also improve your overall problem-solving abilities.

Conclusion: You've Got This!

Awesome work, everyone! You've successfully solved the equation (x + 78) – 47 = 93! You’ve taken a journey through algebra. Remember that solving these types of equations is all about understanding the underlying principles and following a logical process. We've broken down the problem step-by-step, verified our answer, and identified some common pitfalls to avoid. More importantly, you now have the tools and the confidence to tackle similar problems. So keep practicing, stay curious, and never be afraid to ask questions. Math can be fun and rewarding when you approach it with the right mindset. Always remember the key concepts we discussed, and keep practicing these methods! Embrace the challenge and celebrate your successes. Each equation solved is a step forward in your mathematical journey. This builds confidence in your skills. You are now equipped with the fundamental knowledge and skills needed to tackle more complex algebraic problems. Keep exploring, keep learning, and keep growing! You've got this! We've made it through this math problem together. Well done!