Solving For 'a': A Step-by-Step Guide

Hey guys! Ever stumbled upon an equation and thought, "Whoa, how do I solve this?" Well, let's dive into the equation a - 2 = (3 + 6a) / 3 and figure out how to find the value of 'a'. Don't worry, it's easier than you might think! We'll break it down into simple, manageable steps. Solving for a variable, especially in a linear equation like this, is a fundamental skill in mathematics. It's like the building block for more complex problems, so understanding it well is super important. We're going to walk through this together, step by step, so you'll feel confident tackling similar problems in the future. This process isn’t just about finding an answer; it’s about understanding the logic behind the solution. Ready to get started? Let's do this!

Step 1: Simplify the Equation

Alright, let's start by simplifying our equation: a - 2 = (3 + 6a) / 3. The first thing we want to do is get rid of that fraction. How do we do that? Well, we can multiply both sides of the equation by 3. Remember, what you do to one side, you must do to the other to keep everything balanced. This is like using a scale – if you add something to one side, you have to add the same to the other to keep it level. Let's see how that looks: 3 * (a - 2) = 3 * ((3 + 6a) / 3). Now, on the right side, the 3s cancel each other out, leaving us with just (3 + 6a). On the left side, we need to distribute the 3 across the a and the -2. So, 3 * a becomes 3a, and 3 * -2 becomes -6. Putting it all together, our equation now looks like this: 3a - 6 = 3 + 6a. See how we've simplified it already? Great job, you're on your way! Simplifying equations is often the first step in making them easier to solve. It helps to reduce the complexity and makes it easier to isolate the variable we're trying to find.

Step 2: Isolate the Variable 'a'

Okay, in this step, our goal is to get all the terms containing 'a' on one side of the equation and the constants (the numbers without 'a') on the other side. To do this, let's start by subtracting 6a from both sides of the equation 3a - 6 = 3 + 6a. This is like moving the 6a term from the right side to the left side. So, we have: 3a - 6 - 6a = 3 + 6a - 6a. Now, let's simplify this. On the left side, 3a - 6a gives us -3a, and we still have the -6. On the right side, the 6a and -6a cancel each other out, leaving us with just 3. So, our equation now becomes: -3a - 6 = 3. Next, we want to get rid of that -6 on the left side. To do that, we'll add 6 to both sides of the equation. Remember, adding 6 to one side keeps the balance. So, we get: -3a - 6 + 6 = 3 + 6. This simplifies to -3a = 9. See how we're getting closer to isolating 'a'? Each step is designed to make the equation simpler and easier to solve. Think of it like peeling away layers to get to the core of the problem. Keeping the equation balanced at each stage is crucial.

Step 3: Solve for 'a'

Almost there, folks! We're now at the final step: solving for 'a'. Our equation is currently -3a = 9. To isolate 'a', we need to get rid of the -3 that's multiplying it. We do this by dividing both sides of the equation by -3. Remember, dividing keeps everything balanced. So, we have: (-3a) / -3 = 9 / -3. On the left side, the -3s cancel out, leaving us with just 'a'. On the right side, 9 divided by -3 gives us -3. So, the solution is: a = -3. And that's it! We've solved for 'a'. You did it! We started with a complex-looking equation, and through a series of simple steps, we found the value of 'a'. This method can be applied to many other similar problems, meaning you can now approach various equations with confidence. Keep in mind, that practice makes perfect. The more you work through these problems, the more comfortable you'll become with the process. Each time you solve an equation like this, you're not just finding an answer; you're strengthening your problem-solving skills. It's a fantastic feeling when you finally get to the solution!

Checking Your Work: Verification

Always a good idea to double-check our work! We've found that a = -3. Let's plug this value back into the original equation: a - 2 = (3 + 6a) / 3. Substitute -3 for 'a': -3 - 2 = (3 + 6*(-3)) / 3. Simplify the left side: -3 - 2 = -5. Simplify the right side: 3 + 6*(-3) = 3 - 18 = -15. Then, -15 / 3 = -5. So, we have -5 = -5. Since both sides of the equation are equal, our solution is correct! It's always a great idea to check your answers, even if you feel confident. Mistakes happen, and it's better to catch them early. This step reinforces the idea that we've accurately solved the equation and reinforces our understanding. It also builds confidence in your abilities to solve similar problems. The ability to verify your answers is a powerful tool in mathematics.

Conclusion: You Did It!

Congratulations, guys! You've successfully solved for 'a' in the equation a - 2 = (3 + 6a) / 3! We've walked through the steps together: simplifying the equation, isolating the variable, and solving for 'a'. Remember, the value of a is -3. We also checked our work, confirming our solution. This process teaches valuable skills in mathematics. Keep practicing, and you'll find these types of problems becoming easier and easier. Keep up the awesome work! You can totally do this! Solving these equations is a great way to develop your problem-solving skills. Understanding how to manipulate equations is fundamental. Now, go forth and conquer more equations! The more you practice these problems, the better you'll become at them. Keep in mind, that math is a subject where understanding the concepts and regularly solving problems is more important than rote memorization. Each step that we've taken here will help you in solving more complex mathematical challenges in the future. So, keep practicing and keep learning. You're doing great!