Solving 8x - 73 = 1143 A Step-by-Step Guide

Hey guys! Ever get stuck on an equation and feel like you're staring at a confusing mess of numbers and symbols? Don't worry, we've all been there! Today, we're going to break down a common type of algebraic equation and show you exactly how to solve it. We'll use the example 8x - 73 = 1143 and walk through each step in a super clear, easy-to-understand way. So, grab your pencils and let's dive in!

Understanding the Basics of Algebraic Equations

Before we jump into solving 8x - 73 = 1143, let's make sure we're all on the same page with some basic algebra concepts. Think of an algebraic equation like a puzzle. Our goal is to find the value of the unknown, which we usually represent with a letter like 'x'. In our case, 'x' is the mystery number we need to figure out. The equation itself is a statement that two things are equal. The left side of the equals sign (=) has the same value as the right side. To solve the equation, we need to isolate 'x' on one side. This means getting 'x' by itself, with no other numbers or operations hanging around. We do this by performing the same operations on both sides of the equation to keep everything balanced. Imagine it like a seesaw: if you add weight to one side, you need to add the same weight to the other side to keep it level. This principle of balance is key to solving algebraic equations. You might be wondering, why is this so important? Well, algebra is the foundation for so many areas of math and science. From calculating the trajectory of a rocket to designing a bridge, algebraic equations are used everywhere! Mastering these skills will open up a whole new world of problem-solving abilities. So, let’s get comfortable with the fundamentals and tackle our equation with confidence. We'll break down each step, explaining the 'why' behind the 'how' so you're not just memorizing steps, but truly understanding the process. Think of it as building a solid foundation for your mathematical journey. And remember, practice makes perfect! The more you work with algebraic equations, the easier they'll become. So, don't be afraid to make mistakes – they're just opportunities to learn and grow. Now, with our foundation in place, let's get back to our specific equation and see how we can crack the code and find the value of 'x'. We'll take it one step at a time, making sure every move we make is logical and clear. Are you ready to become an equation-solving pro? Let's do this!

Step 1: Isolating the Term with 'x'

Okay, let's get started with isolating the term with 'x' in our equation, 8x - 73 = 1143. This first step is crucial because we want to get the '8x' part all by itself on one side of the equation. Think of it like clearing the clutter around 'x' so we can see it more clearly. To do this, we need to get rid of the '-73' that's hanging out with the '8x'. Remember our seesaw analogy? We need to do the opposite operation to both sides to keep the equation balanced. Since we have '-73', the opposite operation is adding 73. So, we're going to add 73 to both sides of the equation. This looks like this: 8x - 73 + 73 = 1143 + 73. Notice how we're adding 73 to both sides. This is super important! If we only added it to one side, the equation wouldn't be equal anymore. Now, let's simplify. On the left side, '-73 + 73' cancels out, leaving us with just '8x'. On the right side, '1143 + 73' equals '1216'. So, our equation now looks like this: 8x = 1216. See how much simpler it is already? We've successfully isolated the term with 'x'! This is a big step, guys! We're one step closer to solving for 'x'. But why does this work? It's all about using inverse operations. Subtraction and addition are inverse operations – they undo each other. So, by adding 73, we effectively canceled out the -73, leaving us with just the term we wanted. This is a fundamental technique in algebra, and you'll use it all the time. It's like having a superpower – the ability to manipulate equations and reveal their secrets! Now that we've isolated the '8x' term, we're ready to move on to the next step, which will bring us even closer to finding the value of 'x'. Remember, each step we take is a step towards solving the puzzle. So, let's keep going and see what's next!

Step 2: Solving for 'x'

Alright, we've made some great progress! We're at Step 2: Solving for 'x'. Our equation is currently 8x = 1216. This means '8 times x equals 1216'. Now, we need to get 'x' all by itself. What's the operation that's currently attached to 'x'? It's multiplication – '8' is being multiplied by 'x'. To undo multiplication, we need to use its inverse operation: division. So, we're going to divide both sides of the equation by 8. This is crucial for maintaining the balance of the equation, just like we talked about before. It looks like this: (8x) / 8 = 1216 / 8. On the left side, '8x divided by 8' simplifies to just 'x'. This is exactly what we wanted! We're isolating 'x'. On the right side, '1216 divided by 8' equals '152'. So, our equation now looks like this: x = 152. Boom! We've solved for 'x'! The value of 'x' that makes the equation 8x - 73 = 1143 true is 152. How cool is that? We took a seemingly complicated equation and, by following a few simple steps, we found the answer. But why does dividing both sides by 8 work? It's all about inverse operations again. Division is the inverse of multiplication, so dividing by 8 undoes the multiplication by 8. This leaves 'x' isolated and reveals its value. This technique is a cornerstone of algebra and will be incredibly useful as you tackle more complex problems. Now, before we celebrate too much, it's always a good idea to check our work. This is a crucial step in problem-solving, as it helps us catch any mistakes and ensure our answer is correct. So, in the next step, we'll plug our solution back into the original equation and see if it holds true. This will give us confidence that we've not only solved the equation but also solved it correctly. Are you ready to verify our solution? Let's move on to the final step!

Step 3: Verifying the Solution

Okay, we've got our solution: x = 152. But before we declare victory, let's verify the solution to make sure we did everything correctly. This is like the final piece of the puzzle, guys! It's super important to double-check our work to avoid any sneaky errors. To verify, we're going to plug our value of 'x' (which is 152) back into the original equation: 8x - 73 = 1143. So, everywhere we see an 'x', we'll replace it with '152'. This gives us: 8 * 152 - 73 = 1143. Now, we need to simplify the left side of the equation and see if it equals the right side (1143). First, let's do the multiplication: 8 * 152 = 1216. So, our equation now looks like this: 1216 - 73 = 1143. Next, let's do the subtraction: 1216 - 73 = 1143. Look at that! The left side of the equation (1143) does equal the right side of the equation (1143). This means our solution, x = 152, is correct! We nailed it! This verification step is so important because it gives us confidence in our answer. It's like having a built-in safety net. If the left side didn't equal the right side, we'd know we made a mistake somewhere and need to go back and check our work. But in this case, our solution checks out perfectly. Why is verification so crucial? Well, imagine you're building a bridge or designing a critical piece of technology. A small error in the calculations could have serious consequences. By verifying our solutions, we're developing a habit of accuracy and attention to detail, which are essential skills in any field. Plus, it just feels good to know you've solved a problem correctly! So, always remember to verify your solutions, especially in math and science. It's the final touch that ensures your hard work pays off. Now that we've successfully solved and verified our equation, let's recap the steps we took and celebrate our achievement! We've conquered this equation, and we're ready to tackle more challenges in the world of algebra!

Conclusion: Mastering Equations

Woohoo! We did it! We successfully solved the equation 8x - 73 = 1143 and found that x = 152. More importantly, we learned the step-by-step process for solving this type of algebraic equation. Let's quickly recap the key steps we took: First, we isolated the term with 'x' by adding 73 to both sides of the equation. This got rid of the constant term on the left side and brought us closer to isolating 'x'. Then, we solved for 'x' by dividing both sides of the equation by 8. This undid the multiplication and gave us the value of 'x'. Finally, and crucially, we verified the solution by plugging our value of 'x' back into the original equation. This confirmed that our solution was correct and gave us confidence in our work. By understanding these steps, you can tackle similar equations with confidence. The key takeaway here is the concept of inverse operations. We used addition to undo subtraction and division to undo multiplication. This principle is fundamental to solving algebraic equations. And remember, the seesaw analogy is a helpful way to visualize the importance of keeping the equation balanced. Whatever operation you perform on one side, you must perform on the other side to maintain equality. But the journey doesn't end here! The world of algebra is vast and exciting, with many more types of equations to explore and solve. From linear equations to quadratic equations, there's always something new to learn. The skills you've gained today will serve as a strong foundation as you delve deeper into the subject. So, keep practicing, keep asking questions, and keep challenging yourself. The more you work with algebraic equations, the more comfortable and confident you'll become. And who knows, maybe one day you'll be solving equations that help build bridges, design rockets, or even discover new scientific breakthroughs! The possibilities are endless. But for now, let's celebrate our accomplishment. We've mastered another mathematical challenge, and we're one step closer to becoming equation-solving pros! So, pat yourselves on the back, guys, and get ready for the next adventure in the amazing world of math!