Solve 50 - [3(2x - 3) - 2(2x - 1)] = 23 Easily

Hey guys! Today, we are going to dive deep into solving a fascinating algebraic equation. Equations might seem daunting at first, but trust me, breaking them down step-by-step makes them super manageable. We'll tackle the equation 50 - [3(2x - 3) - 2(2x - 1)] = 23 together. This isn't just about finding the answer; it's about understanding the process and building your problem-solving skills. So, let's grab our math hats and get started!

Understanding the Equation

Before we jump into solving, let's break down what we’re actually looking at. The equation 50 - [3(2x - 3) - 2(2x - 1)] = 23 is an algebraic equation where we need to find the value of 'x' that makes the equation true. The left side of the equation involves several operations, including multiplication, subtraction, and the use of brackets and square brackets. The square brackets indicate a higher level of grouping, meaning we need to simplify the expressions inside them before we can deal with the outer parts of the equation. Our goal is to isolate 'x' on one side of the equation. This involves performing inverse operations in the correct order to unravel the equation and reveal the value of 'x'. Think of it like peeling back layers of an onion – we're going to carefully remove each layer until we get to the core, which in this case, is 'x'. Why is understanding this important? Because mastering this process will equip you with the skills to tackle more complex equations and problems in the future. You'll start to see patterns, understand the logic behind each step, and feel much more confident in your math abilities. This isn't just about getting the right answer; it's about developing a logical, step-by-step approach to problem-solving, a skill that's valuable in many areas of life. So, let's keep this mindset as we move forward and break down each step of the solution.

Step 1: Distribute Inside the Parentheses

The first step in solving this equation is to get rid of those parentheses. We'll do this by distributing the numbers outside the parentheses to the terms inside. This means multiplying the '3' by both terms inside the first parenthesis (2x - 3) and multiplying the '-2' by both terms inside the second parenthesis (2x - 1). This is a crucial step because it simplifies the equation and makes it easier to work with. Remember, the order of operations (PEMDAS/BODMAS) tells us to handle parentheses first, so this is exactly what we're doing. Let's break it down: 3 * (2x - 3) becomes 3 * 2x - 3 * 3, which simplifies to 6x - 9. Similarly, -2 * (2x - 1) becomes -2 * 2x - (-2) * 1, which simplifies to -4x + 2. Notice the importance of the negative sign! It changes the sign of the terms inside the parenthesis when multiplied. Now, let's rewrite the equation with these simplified terms: 50 - [6x - 9 - 4x + 2] = 23. See how much cleaner it looks already? By distributing correctly, we've eliminated the parentheses and prepared the equation for the next step, which is combining like terms. This step is all about precision and attention to detail. Make sure you multiply the correct numbers and pay close attention to the signs. A small mistake here can throw off the entire solution. So, double-check your work and make sure everything is accurate before moving on. With the parentheses gone, we're one step closer to isolating 'x' and finding our solution!

Step 2: Combine Like Terms

Now that we've distributed and eliminated the parentheses, our equation looks like this: 50 - [6x - 9 - 4x + 2] = 23. The next step is to simplify the expression inside the square brackets by combining like terms. Like terms are those that have the same variable raised to the same power. In this case, we have '6x' and '-4x' as like terms, and '-9' and '+2' as like terms. Combining like terms makes the equation simpler and easier to manipulate. Think of it like organizing your closet – you group similar items together to make everything more manageable. Let’s combine the 'x' terms first: 6x - 4x = 2x. Then, let's combine the constant terms: -9 + 2 = -7. Now, we can rewrite the equation with these simplified terms inside the square brackets: 50 - [2x - 7] = 23. See how much simpler it looks? We've reduced the number of terms inside the brackets, making it easier to deal with the remaining operations. Why is this step important? Because it streamlines the equation and brings us closer to isolating 'x'. By combining like terms, we're essentially tidying up the equation and making it more manageable. This step reduces the chance of making errors in the subsequent steps and helps us see the structure of the equation more clearly. It's like taking a deep breath and organizing your thoughts before tackling a complex problem. So, with the like terms combined, we're ready to move on to the next step and continue our journey towards finding the value of 'x'.

Step 3: Distribute the Negative Sign

Our equation is now looking like this: 50 - [2x - 7] = 23. The next crucial step is to get rid of the square brackets. Notice that there's a negative sign in front of the brackets. This means we need to distribute this negative sign to every term inside the brackets. Think of it as multiplying each term inside the brackets by -1. This is a critical step, because forgetting to distribute the negative sign correctly is a very common mistake that can lead to a wrong answer. So, let’s be extra careful here. When we distribute the negative sign, we get: -1 * (2x) = -2x and -1 * (-7) = +7. Remember, a negative times a negative is a positive! Now, we can rewrite the equation without the square brackets: 50 - 2x + 7 = 23. See how the signs of the terms inside the brackets have changed? The '2x' became '-2x' and the '-7' became '+7'. By distributing the negative sign correctly, we've removed the square brackets and further simplified the equation. This step is like removing another layer of complexity, bringing us closer to isolating 'x'. Why is this step so important? Because it ensures that we're treating the equation correctly according to the rules of algebra. Failing to distribute the negative sign properly can completely change the equation and lead to an incorrect solution. So, always remember to distribute the negative sign to every term inside the parentheses or brackets. With the square brackets gone, we're ready to move on to the next step and continue our quest to solve for 'x'.

Step 4: Combine Like Terms Again

After distributing the negative sign, our equation is: 50 - 2x + 7 = 23. Now, we have another opportunity to simplify the equation by combining like terms. In this case, we have two constant terms, '50' and '+7', that we can combine. Remember, combining like terms makes the equation more manageable and reduces the chance of errors in the following steps. It's like cleaning your workspace before starting a new task – it helps you stay organized and focused. Let's combine the constant terms: 50 + 7 = 57. Now, we can rewrite the equation with this simplification: 57 - 2x = 23. See how much simpler it looks? We've reduced the number of terms on the left side of the equation, making it easier to isolate 'x'. Why is combining like terms so crucial? Because it streamlines the equation and brings us closer to our goal of solving for 'x'. By combining like terms, we're essentially tidying up the equation and making it more manageable. This step reduces the complexity of the equation and allows us to see the relationship between the terms more clearly. It's like decluttering your mind before tackling a problem – it helps you focus on the essential elements. So, with the like terms combined, we're ready to move on to the next step and continue our journey towards finding the value of 'x'.

Step 5: Isolate the Term with 'x'

Our equation now stands as: 57 - 2x = 23. The next step is to isolate the term containing 'x', which in this case is '-2x'. This means we want to get '-2x' by itself on one side of the equation. To do this, we need to get rid of the '57' that's being added to it. We can do this by subtracting 57 from both sides of the equation. Remember, whatever operation you perform on one side of the equation, you must perform on the other side to maintain the balance. This is a fundamental principle of algebra. Subtracting 57 from both sides gives us: 57 - 2x - 57 = 23 - 57. On the left side, 57 - 57 cancels out, leaving us with -2x. On the right side, 23 - 57 equals -34. So, our equation now looks like this: -2x = -34. We're getting closer! By isolating the term with 'x', we've simplified the equation even further and set ourselves up for the final step. Why is isolating the 'x' term so important? Because it brings us one step closer to finding the value of 'x'. By getting the 'x' term by itself, we're essentially peeling away all the other elements of the equation and focusing solely on 'x'. This makes it much easier to determine the value of 'x' that makes the equation true. It's like zeroing in on your target – you focus all your attention on the goal you're trying to achieve. So, with the 'x' term isolated, we're ready to take the final step and solve for 'x'.

Step 6: Solve for 'x'

We've reached the final step! Our equation is currently: -2x = -34. To solve for 'x', we need to get 'x' by itself. Right now, 'x' is being multiplied by -2. To undo this multiplication, we need to divide both sides of the equation by -2. Remember, we always perform the inverse operation to isolate the variable. Dividing both sides by -2 gives us: (-2x) / -2 = (-34) / -2. On the left side, -2 divided by -2 cancels out, leaving us with 'x'. On the right side, -34 divided by -2 equals 17. Remember, a negative divided by a negative is a positive. So, our solution is x = 17. Congratulations! We've successfully solved the equation. We've found the value of 'x' that makes the equation 50 - [3(2x - 3) - 2(2x - 1)] = 23 true. But we're not quite done yet. It's always a good idea to check our answer to make sure we haven't made any mistakes along the way. Why is checking our solution so important? Because it gives us confidence that we've solved the equation correctly. Checking our answer helps us catch any errors we might have made and ensures that our solution is accurate. It's like proofreading your work before submitting it – you want to make sure everything is perfect. So, let's move on to the next step and verify our solution.

Step 7: Check Your Solution

Okay, we've solved the equation and found that x = 17. But before we celebrate too much, let’s make sure our answer is correct. Checking your solution is a crucial step in problem-solving. It’s like having a safety net – it catches any mistakes you might have made along the way. To check our solution, we'll substitute x = 17 back into the original equation: 50 - [3(2x - 3) - 2(2x - 1)] = 23. Let's plug in 17 for x: 50 - [3(2 * 17 - 3) - 2(2 * 17 - 1)] = 23. Now, we need to simplify the equation following the order of operations (PEMDAS/BODMAS). First, let's simplify inside the parentheses: 2 * 17 = 34, so we have 50 - [3(34 - 3) - 2(34 - 1)] = 23. Next, let's continue simplifying inside the parentheses: 34 - 3 = 31 and 34 - 1 = 33, so we have 50 - [3(31) - 2(33)] = 23. Now, let's perform the multiplications: 3 * 31 = 93 and 2 * 33 = 66, so we have 50 - [93 - 66] = 23. Next, let's simplify inside the square brackets: 93 - 66 = 27, so we have 50 - 27 = 23. Finally, let's perform the subtraction: 50 - 27 = 23. And there you have it! 23 = 23. Our solution checks out! This means that x = 17 is indeed the correct solution to the equation. Why is this verification step so satisfying? Because it confirms that all our hard work has paid off. It's like reaching the summit of a mountain after a long and challenging climb – you feel a sense of accomplishment and pride. So, always remember to check your solutions, not just in math, but in any problem-solving situation. It’s a great habit that will help you avoid mistakes and build confidence in your abilities.

Conclusion

And there you have it, guys! We've successfully navigated the equation 50 - [3(2x - 3) - 2(2x - 1)] = 23 and found that x = 17. We didn't just get the answer; we walked through each step, understanding the why behind the what. We talked about distributing, combining like terms, isolating the variable, and most importantly, checking our solution. Remember, solving equations is like building a puzzle. Each step is a piece, and when you put them together correctly, you get the whole picture. The key is to break down the problem into smaller, manageable steps and tackle each one methodically. Don't be afraid to make mistakes – they're part of the learning process. What’s the biggest takeaway from this journey? It's that math isn't just about numbers and formulas; it's about logical thinking and problem-solving. The skills you learn in math can be applied to many areas of life, from balancing your budget to making important decisions. So, keep practicing, keep asking questions, and keep challenging yourself. And remember, every equation you solve is a victory, a step forward on your path to mathematical mastery. Now, go out there and conquer some more equations! You've got this!