Solving Equations: A Step-by-Step Guide

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Hey everyone! Today, we're diving into a classic math problem: 5(2x - 1) = 50. I know, equations can sometimes seem a little intimidating, but trust me, we'll break it down into easy-to-understand steps. We'll find out the answer together, and more importantly, we'll learn how to solve these types of problems. Think of it like a fun puzzle – we're just figuring out what 'x' has to be to make the equation true. Let's get started!

Understanding the Basics: What's an Equation?

Alright, before we jump into the nitty-gritty, let's make sure we're all on the same page. An equation is simply a mathematical statement that shows two things are equal. It's like a balanced scale; whatever's on one side must be equal to what's on the other. The key to solving an equation is to find the value of the unknown, usually represented by a letter like 'x', that makes the equation true. In our case, the equation is 5(2x - 1) = 50. We want to figure out the value of 'x' that satisfies this equation. The goal is to isolate 'x' on one side of the equation. We do this by performing the same operations on both sides to keep the equation balanced.

Now, let's talk about the parts of our equation. We have 5(2x - 1) on the left side and 50 on the right side. The parentheses, (2x - 1), tell us to do something specific first. The number 5 is multiplied by the entire expression inside the parentheses. This is where the distributive property comes in handy, but we'll get to that in a bit. The equal sign (=) is the heart of the equation, telling us that the value on the left is the same as the value on the right. Remember, our ultimate aim is to find the value of 'x' that makes this statement true. By the end of this, you'll be a pro at solving these types of equations. Let's break this down step by step and make it super clear. It's really not as hard as it might look at first glance. We'll be using some fundamental math rules, and I promise, by the time we're done, you'll feel confident tackling similar problems on your own. It's all about understanding the process and practicing a little bit!

Step-by-Step Solution: Unraveling the Equation

Okay, guys, let's roll up our sleeves and solve this equation step-by-step. We'll break it down nice and easy. Remember, our equation is 5(2x - 1) = 50. Here's how we'll solve it:

  1. Distribute the 5: The first step is to get rid of those parentheses. We do this using the distributive property. This means we multiply the 5 by each term inside the parentheses. So, we multiply 5 by 2x, which gives us 10x, and we multiply 5 by -1, which gives us -5. Our equation now becomes: 10x - 5 = 50.
  2. Isolate the term with 'x': Our next goal is to get the 'x' term by itself. To do this, we need to get rid of that -5 on the left side. We do the opposite of subtracting 5, which is adding 5 to both sides of the equation. This keeps the equation balanced. So, we add 5 to both sides: 10x - 5 + 5 = 50 + 5. This simplifies to 10x = 55.
  3. Solve for 'x': Now we have 10x = 55. 'x' is being multiplied by 10, so to isolate 'x', we do the opposite – we divide both sides by 10. This gives us: 10x / 10 = 55 / 10. This simplifies to x = 5.5.

And there you have it, folks! We've found the solution. x = 5.5. See? Not so scary after all! We went from a somewhat complex-looking equation to a straightforward answer by following a few simple steps. The key is to be methodical and remember that whatever you do to one side of the equation, you must do to the other to keep things balanced. Let's quickly recap what we did: we distributed, isolated the x term, and then solved for x. Practice these steps, and you'll become a pro in no time. You can apply these principles to many other similar equations. It's all about practice and understanding the logic behind each step. Now, let's go on to the next part and verify our solution to make sure we're right!

Verification: Checking Our Answer

Alright, now that we've found our answer, it's always a good idea to check our work. This is like a safety net; it makes sure we haven't made any mistakes along the way. To check if x = 5.5 is correct, we'll substitute 5.5 back into the original equation, which was 5(2x - 1) = 50. Let's do it!

  1. Substitute x with 5.5: Replace 'x' with 5.5 in the original equation: 5(2 * 5.5 - 1) = 50.
  2. Simplify within the parentheses: First, we multiply 2 by 5.5, which gives us 11. Then, subtract 1 from 11, which gives us 10. So, we have: 5(10) = 50.
  3. Final Calculation: Multiply 5 by 10, and we get 50. So, the equation becomes: 50 = 50.

And guess what, guys? The equation holds true! Since the left side equals the right side, our solution x = 5.5 is correct! That means all the steps we took to solve the equation were spot on. This process of substituting the value back into the original equation is crucial. It gives us confidence that we've solved the equation correctly. It's always a good habit to verify your solutions, especially in math. This will help you catch any minor errors you might have made along the way. So next time you solve an equation, take a moment to double-check your answer this way. You'll not only confirm your solution, but you'll also build your confidence in your math skills.

Tips and Tricks: Mastering Equation Solving

So, you've solved your first equation. Awesome! Now, let's talk about some tips and tricks that can help you become a real equation-solving ninja. First off, practice makes perfect. The more equations you solve, the more comfortable you'll become with the steps and the different types of problems you might encounter. Start with simple equations and gradually move on to more complex ones. There are plenty of online resources, textbooks, and workbooks with practice problems. Secondly, understand the order of operations. Remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). This tells you the order in which to perform calculations. Always work within parentheses first, then handle exponents, followed by multiplication and division, and finally, addition and subtraction. Always make sure you're distributing correctly; this is a common place where errors happen. When distributing, remember to multiply the term outside the parentheses by every term inside the parentheses. Don't forget to pay attention to the signs – positive and negative signs are super important in equations. A small mistake with a sign can change the entire answer. Take your time, and double-check your work, especially when dealing with negative numbers. Make sure to always keep the equation balanced. Any operation you perform on one side of the equation must be performed on the other side. This ensures that the equality remains true. Lastly, break down complex problems. If you're faced with a more challenging equation, don't get overwhelmed. Break it down into smaller, more manageable steps. Solve one part at a time, and don't try to do too many things at once. By following these tips and practicing regularly, you'll build your equation-solving skills and confidence!

Conclusion: You've Got This!

We did it, guys! We successfully solved the equation 5(2x - 1) = 50. We learned the steps, checked our answer, and even picked up some handy tips and tricks. Remember, solving equations is all about understanding the basic principles and practicing regularly. Don't be afraid to make mistakes; they're a part of the learning process. Each time you solve an equation, you're building your skills and confidence. Keep practicing, stay curious, and you'll find that solving equations becomes easier and more enjoyable. You've got this! Now go out there and tackle some more equations. You are well-equipped to do so. Remember the steps, practice them, and always check your work. And most importantly, have fun with it! Math can be a lot of fun when you start to understand it, and solving equations is just one piece of the puzzle. Keep exploring, keep learning, and keep challenging yourself. You are on your way to math mastery! Congratulations on solving this equation, and best of luck on your future math adventures!