Solve 3x + 4 = X + 6: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the world of algebra to tackle a classic equation: 3x + 4 = x + 6. Don't worry if equations seem daunting – we're going to break it down step by step, making it super easy to understand. This isn't just about finding the answer; it's about understanding the process, so you can confidently solve similar problems in the future. So, grab your pencils and paper, and let's get started!

Understanding the Basics of Algebraic Equations

Before we jump into solving our specific equation, let's quickly recap some fundamental concepts of algebraic equations. Think of an equation as a balanced scale. The equals sign (=) is the fulcrum, and both sides of the equation must remain balanced. Our goal in solving an equation is to isolate the variable (in this case, 'x') on one side of the equation. This means we want to get 'x' all by itself, so we know what value it represents. To do this, we use inverse operations – operations that undo each other. For example, addition and subtraction are inverse operations, and so are multiplication and division. The golden rule is: whatever you do to one side of the equation, you must do to the other side to maintain the balance. This ensures that the equation remains true throughout the solving process. Many students find that writing out each step clearly helps them avoid mistakes and keep track of their progress. It's like creating a roadmap for your solution! Remember, the key to mastering algebra is practice, practice, practice. The more equations you solve, the more comfortable and confident you'll become. And don't be afraid to make mistakes – they're part of the learning process. Just try to understand where you went wrong and learn from it. Solving equations is like building a puzzle – each step brings you closer to the final solution. And the satisfaction of finding that solution is definitely worth the effort!

Step 1: Grouping Like Terms

Okay, guys, let's get into the nitty-gritty of solving 3x + 4 = x + 6. The first thing we want to do is group the like terms. Like terms are terms that contain the same variable (in this case, 'x') or are constants (numbers without variables). Looking at our equation, we have '3x' and 'x' as like terms, and '4' and '6' as like terms. Our goal here is to bring all the 'x' terms to one side of the equation and all the constant terms to the other side. This makes the equation easier to simplify and solve. To do this, we'll use inverse operations, remembering to apply the same operation to both sides of the equation. Let's start by getting rid of the 'x' term on the right side. We can do this by subtracting 'x' from both sides of the equation. This gives us: 3x + 4 - x = x + 6 - x. Simplifying this, we get 2x + 4 = 6. Notice that the 'x' on the right side has disappeared, and we've successfully moved the 'x' term to the left side. Now, let's focus on the constant terms. We want to move the '4' from the left side to the right side. We can do this by subtracting '4' from both sides of the equation. This gives us: 2x + 4 - 4 = 6 - 4. Simplifying this, we get 2x = 2. Great! We've now grouped the like terms, and our equation looks much simpler. We have '2x' on the left side and '2' on the right side. This sets us up perfectly for the next step, which is isolating the variable.

Step 2: Isolating the Variable

Alright, we're making great progress! We've got our equation down to 2x = 2. Now, the next key step is isolating the variable. Remember, isolating the variable means getting 'x' all by itself on one side of the equation. Currently, 'x' is being multiplied by '2'. To undo this multiplication, we need to use the inverse operation, which is division. So, we're going to divide both sides of the equation by '2'. This gives us: (2x) / 2 = 2 / 2. On the left side, the '2' in the numerator and the '2' in the denominator cancel each other out, leaving us with just 'x'. On the right side, 2 divided by 2 is equal to 1. Therefore, our equation simplifies to x = 1. Woohoo! We've successfully isolated the variable and found the value of 'x'. This is the solution to our equation. But before we celebrate too much, there's one more important step we should always take to make sure our answer is correct.

Step 3: Verifying the Solution

Okay, guys, we've found our solution: x = 1. But it's always a good idea to verify the solution to make sure we didn't make any mistakes along the way. This is like double-checking your work to ensure accuracy. To verify our solution, we're going to substitute the value we found for 'x' (which is 1) back into the original equation: 3x + 4 = x + 6. Replacing 'x' with '1', we get: 3(1) + 4 = 1 + 6. Now, we simplify both sides of the equation. On the left side, 3 multiplied by 1 is 3, so we have 3 + 4 = 7. On the right side, 1 plus 6 is also 7. So, we have 7 = 7. This is a true statement! Since both sides of the equation are equal when we substitute 'x = 1', we can confidently say that our solution is correct. Verifying your solution is a crucial step in solving equations. It helps you catch any errors you might have made and ensures that you have the correct answer. It's like having a built-in safety net for your math skills! So, always remember to verify your solutions, and you'll become a much more confident and accurate equation solver.

Conclusion: Mastering Algebraic Equations

Awesome job, everyone! We've successfully solved the equation 3x + 4 = x + 6, and we've learned some valuable skills along the way. We started by understanding the basics of algebraic equations, including the concept of balancing the equation and using inverse operations. Then, we moved on to the specific steps involved in solving our equation: grouping like terms, isolating the variable, and verifying the solution. Remember, solving equations is like learning a new language – it takes practice and patience. Don't get discouraged if you don't understand something right away. Keep practicing, and you'll gradually build your skills and confidence. And most importantly, have fun with it! Math can be challenging, but it can also be incredibly rewarding. The ability to solve equations is a powerful tool that can be applied in many different areas of life, from personal finance to scientific research. So, keep practicing, keep exploring, and keep challenging yourself. You've got this! And remember, there are tons of resources available to help you on your math journey, including textbooks, online tutorials, and even your friendly neighborhood math tutor. So, don't hesitate to seek out help when you need it. With the right tools and the right attitude, you can master algebraic equations and unlock a whole new world of mathematical possibilities.