Hey guys! Let's dive into solving a basic algebraic equation together. We're tackling the equation 6x - 12 = 42 today. Don't worry, it's not as scary as it looks! We'll break it down step-by-step so you can conquer similar problems with confidence. Remember, in mathematics, the key is to understand the process, not just memorize the answer. So, let's get started and solve this equation like pros!
Understanding the Goal
Before we jump into the calculations, let's take a moment to understand what we're trying to achieve. When we solve an equation, we're essentially trying to find the value of the unknown variable, which in our case is 'x'. Think of 'x' as a mystery number that we need to uncover. The equation is a puzzle, and our mission is to isolate 'x' on one side of the equation so we can see its true value. We'll do this by performing operations on both sides of the equation to maintain balance. It's like a seesaw – whatever we do to one side, we must do to the other to keep it level. This ensures that the equation remains valid and the value of 'x' remains accurate. Solving equations is a fundamental skill in algebra, and it's used in countless real-world applications, from calculating finances to designing structures. So, mastering this skill will definitely come in handy!
Step 1: Isolating the Term with 'x'
The first step in solving 6x - 12 = 42 is to isolate the term that contains 'x', which is '6x'. Currently, we have '-12' hanging out on the same side of the equation. Our goal is to move this '-12' to the other side. How do we do that? We use the concept of inverse operations. The inverse operation of subtraction is addition. So, to get rid of '-12', we need to add 12 to both sides of the equation. Remember, what we do to one side, we must do to the other to keep the equation balanced. So, we add 12 to both sides:
6x - 12 + 12 = 42 + 12
This simplifies to:
6x = 54
See how the '-12' disappeared from the left side? We're one step closer to isolating 'x'! This step is crucial because it simplifies the equation and brings us closer to the solution. By adding 12 to both sides, we've effectively canceled out the constant term on the left side, leaving us with only the term containing 'x'. Now, we can move on to the next step and finally solve for 'x'.
Step 2: Solving for 'x'
Now that we have 6x = 54, we're in the home stretch! 'x' is still not completely isolated because it's being multiplied by 6. To get 'x' all by itself, we need to undo this multiplication. Again, we'll use the concept of inverse operations. The inverse operation of multiplication is division. So, to get rid of the 6 that's multiplying 'x', we need to divide both sides of the equation by 6. Remember the seesaw analogy – we must do the same thing to both sides to maintain balance. So, we divide both sides by 6:
6x / 6 = 54 / 6
This simplifies to:
x = 9
And there you have it! We've successfully solved for 'x'. The value of 'x' that makes the equation 6x - 12 = 42 true is 9. It's like finding the missing piece of the puzzle. This step is the culmination of all our efforts, and it gives us the final answer we were looking for. By dividing both sides by 6, we've isolated 'x' and revealed its value. Now, let's move on to the crucial step of checking our answer to make sure we've got it right.
Step 3: Checking the Solution
Okay, we've found our solution: x = 9. But before we celebrate, it's super important to check our answer. Think of it as double-checking your work on a test – you want to make sure you didn't make any silly mistakes. To check our solution, we'll substitute 'x = 9' back into the original equation, 6x - 12 = 42, and see if it holds true. This is like plugging the missing piece back into the puzzle to see if it fits. Let's substitute:
6 * 9 - 12 = 42
Now, let's simplify the left side of the equation:
54 - 12 = 42
42 = 42
Bingo! The left side of the equation equals the right side. This means our solution, x = 9, is correct. Checking our solution is a vital step because it helps us catch any errors we might have made during the solving process. It's a way to ensure that our answer is accurate and that we've truly solved the equation. So, always remember to check your solutions, guys!
Conclusion: We Solved It!
Awesome! We've successfully solved the equation 6x - 12 = 42 and found that x = 9. We followed a step-by-step process: isolating the term with 'x', solving for 'x', and, most importantly, checking our solution. Remember, solving equations is a fundamental skill in mathematics, and the more you practice, the better you'll become. So, keep up the great work, and don't be afraid to tackle more algebraic challenges! You've got this!
Key Takeaways for Solving Equations
Before we wrap up, let's recap some key takeaways that will help you solve equations like a pro:
- Understand the goal: Know that you're trying to isolate the variable on one side of the equation.
- Use inverse operations: To undo an operation, use its inverse (addition/subtraction, multiplication/division).
- Maintain balance: Whatever you do to one side of the equation, do to the other.
- Check your solution: Always substitute your answer back into the original equation to ensure it's correct.
- Practice, practice, practice: The more you solve equations, the more comfortable and confident you'll become.
By keeping these key takeaways in mind, you'll be well-equipped to tackle a wide range of algebraic equations. Remember, mathematics is a journey, and every equation you solve is a step forward. So, keep learning, keep practicing, and keep exploring the wonderful world of math!
Additional Practice Problems
Want to sharpen your equation-solving skills even further? Here are a few extra practice problems you can try:
- 3x + 5 = 14
- 2y - 7 = 9
- 4z + 10 = 2
- 5a - 3 = 17
Try solving these problems using the same step-by-step approach we discussed earlier. Remember to check your answers! Solving these additional problems will help solidify your understanding of the equation-solving process and build your confidence in your abilities. Don't be afraid to make mistakes – they're a natural part of the learning process. Just learn from them and keep moving forward. Happy solving!