Solving For X: A Step-by-Step Guide

Hey everyone! Today, we're diving into a classic algebra problem: If 7x - 12 = x + 48, then what is the value of x? Don't worry, it's not as scary as it might look at first glance. We'll break it down step by step, making it super easy to understand. This is a fundamental concept in mathematics, and mastering it will give you a solid foundation for more complex problems down the road. So, grab your pencils and let's get started! We'll use a clear and concise approach, ensuring you grasp every step of the process. This isn't just about finding the answer; it's about understanding the 'why' behind each action, empowering you to tackle similar problems with confidence. By the end of this guide, you'll be able to solve this type of equation without breaking a sweat, and you might even start enjoying algebra! Ready to unlock the secrets of this equation? Let's go!

This kind of problem is a cornerstone of algebra, and understanding how to isolate a variable is key to success in higher-level math. So, let's transform what appears complex into something simple and manageable. Our focus will be on clear explanations, ensuring that even if you're new to algebra, you'll be able to follow along. We'll start with the basics, define our terms, and then move step by step, which helps to build a strong foundation. This approach is designed to make learning fun and rewarding, because once you understand these concepts, you'll realize just how powerful they are. Whether you're a student preparing for a test, or someone just looking to brush up on your skills, this guide is made for you. No prior knowledge of algebra is required, which means that you can jump right in. We will cover the core principles and techniques you need to conquer equations like this one. So get ready, because you're about to become an algebra whiz! Now, let us begin the adventure and see how it works.

Step 1: Combining Like Terms

Alright, guys, our first move is to combine like terms. This means we want to get all the 'x' terms on one side of the equation and all the constant numbers (the ones without 'x') on the other side. This is like sorting your socks – you want all the matching ones together! Start by subtracting 'x' from both sides of the equation. This will eliminate the 'x' on the right side. The equation now looks like this: 7x - x - 12 = 48. Now, it's easier to see the terms, as it is clearer to combine them. When we do the operation, it turns to 6x - 12 = 48. See? We're already making progress!

This step is crucial because it simplifies the equation and makes it easier to solve. It's like decluttering your room before you start cleaning – it makes everything more manageable. This operation maintains the equation's balance, ensuring that both sides remain equal. Subtracting 'x' from both sides doesn't change the truth of the equation; it just rearranges it to get us closer to our goal. Understanding this principle is fundamental to solving any algebraic equation. Keep in mind that every step you take must maintain the equation's balance. Always remember this principle as it is a core concept that guides you through the process.

By systematically combining terms, we make the equation less cluttered and more intuitive. Think of it like organizing your thoughts before writing an essay – it helps you to structure your ideas logically. The result is a cleaner, more manageable equation that's easier to solve. With each step, we’re moving closer to isolating 'x' and revealing its value. This step is about preparation, ensuring that the main part of the solving process is done efficiently and accurately.

Step 2: Isolating the Variable

Now, we need to get that 'x' all by itself. To do this, we're going to get rid of the '- 12'. We do this by adding 12 to both sides of the equation. Remember, whatever you do to one side, you must do to the other to keep things balanced. So, our equation becomes 6x - 12 + 12 = 48 + 12. This simplifies to 6x = 60. See how we're isolating 'x' bit by bit? It is much clearer, and easier to solve the problem this way!

This step is all about getting 'x' alone, like isolating a suspect in an investigation – you want to know what it is doing without any interference. By adding 12 to both sides, we cancel out the -12 on the left side, leaving us with a term containing only 'x'. This is a critical step because it focuses our equation directly on the variable we're trying to solve for. It also brings us one step closer to our goal, by making the value of 'x' directly available to us. By understanding this principle, you'll be able to confidently solve more complex equations. Always remember the goal: to isolate the variable. This approach ensures you remain on track until you find the solution. The balance of the equation should be maintained in this part as well, to make sure it is accurate.

Think of it this way: we’re slowly peeling away the layers around 'x' until we get to its core value. This step sets us up perfectly for the final calculation. Now, we are much closer to solving for 'x'. It is as if we are removing the obstacles and reducing the problem. By doing the operation to both sides, the core meaning of the equation remains untouched.

Step 3: Solving for X

We're in the home stretch now, folks! We have 6x = 60. To find the value of 'x', we need to get rid of that '6'. Since '6x' means '6 multiplied by x', we do the opposite: divide both sides by 6. This gives us 6x / 6 = 60 / 6, which simplifies to x = 10. And there you have it! We've found the value of 'x'.

This step is the culmination of our efforts, where we finally reveal the value of 'x'. It's the moment of truth, the point where we uncover the solution to the equation. Remember the fundamental concept that we introduced previously? This is also the core idea for solving the equation. The equation must be balanced on both sides for it to be true. It's like cracking the code, which is an amazing feeling! Dividing both sides by 6 isolates 'x' and gives us its exact value. We're removing the last barrier and revealing the solution. This is the goal we have been striving for, from the very beginning. Remember, we are trying to find the value of 'x', and now we have it.

This step is about the final calculation, where you get to see how the mathematical puzzle unfolds. It's like putting the final piece into a jigsaw puzzle, and watching the bigger picture come into view. We've gone from a complex equation to a clear answer: x = 10. And it’s not just the answer, but the journey of understanding that counts.

Conclusion: Your Algebra Success

So, there you have it, guys! We've successfully solved for 'x' in the equation 7x - 12 = x + 48. By following these simple steps – combining like terms, isolating the variable, and solving for x – you can tackle similar problems with confidence. Remember, practice makes perfect. The more you work through these problems, the more comfortable and proficient you'll become. Keep practicing, and you'll be an algebra superstar in no time!

This is more than just solving an equation; it's about building your problem-solving skills. Remember that this principle can be applied to different equations and fields. These skills are invaluable for all areas of life. Whether you're a student or someone just looking to sharpen your mathematical skills, this process is an asset. Never stop exploring and challenging yourself, and remember that with practice and persistence, you can master algebra and so much more. This is just the beginning of your journey into the world of mathematics. Every step you take makes you become even better at these concepts, and you will become stronger. Great job, and congratulations! You did a fantastic job!

I hope you found this guide helpful. If you have any questions or want to try some more examples, feel free to ask. Keep learning and keep growing! You've got this!