Solving For X In The Equation 10x - 4.5 + 3x = 12x - 1.1
Hey everyone! Today, we're diving into a fun little algebra problem. We've got an equation, and our mission is to find out what value of x makes it true. Don't worry; it's not as scary as it looks! We'll break it down step by step so it's super easy to follow. Think of it like solving a puzzle – each step gets us closer to the final answer. So, grab your thinking caps, and let's get started!
Understanding the Equation
Okay, first things first, let's take a good look at the equation we're dealing with: 10x - 4.5 + 3x = 12x - 1.1. Equations like this might seem a bit intimidating at first glance, but they're really just balanced scales. What we do on one side, we have to do on the other to keep it balanced. Our goal here is to isolate x, which means getting it all by itself on one side of the equation. To do that, we need to simplify things and move terms around. Remember, x is just a mystery number we're trying to uncover. The numbers in front of x (like the 10 and 3) are called coefficients, and the numbers without an x (like the -4.5 and -1.1) are called constants. We'll be working with both of these to solve for x. The key is to take it one step at a time and keep everything organized. So, let's start simplifying!
Step-by-Step Solution
Alright, let's get our hands dirty and solve this equation step-by-step. Remember, the goal is to get x all by itself on one side. First up, we're going to combine like terms. This just means we're going to add or subtract the terms that have x in them and the terms that are just numbers. On the left side of the equation, we have 10x and 3x. If we add those together, we get 13x. So, the left side now looks like 13x - 4.5. The right side of the equation already looks pretty simple: 12x - 1.1. Nothing to combine there just yet! So, our equation now reads: 13x - 4.5 = 12x - 1.1. See? We're already making progress! Next, we want to get all the x terms on one side and all the constant terms on the other. To do this, we'll subtract 12x from both sides. This cancels out the 12x on the right side and leaves us with just a number. On the left side, 13x - 12x is simply x. Now our equation looks like: x - 4.5 = -1.1. We're getting so close! Finally, to get x completely alone, we need to get rid of that -4.5. We do this by adding 4.5 to both sides. This cancels out the -4.5 on the left side. On the right side, -1.1 + 4.5 equals 3.4. And there you have it! Our final answer is x = 3.4. Wasn't that fun?
Detailed Breakdown of Each Step
Let's break down each step we took in solving the equation 10x - 4.5 + 3x = 12x - 1.1. This detailed look will help solidify our understanding and make tackling similar problems a breeze. First, we combined like terms on the left side of the equation. We had 10x and 3x, which both contain the variable x. Adding these together gives us 13x. So, the left side became 13x - 4.5. The right side, 12x - 1.1, remained unchanged at this point. This step is crucial because it simplifies the equation, making it easier to work with. Next, we wanted to get all the x terms on one side of the equation. To do this, we subtracted 12x from both sides. Subtracting the same value from both sides keeps the equation balanced. This gave us 13x - 12x - 4.5 = 12x - 12x - 1.1, which simplifies to x - 4.5 = -1.1. Notice how the 12x term disappeared from the right side, moving us closer to isolating x. Now, we needed to isolate x completely. To do this, we added 4.5 to both sides of the equation. This gets rid of the -4.5 on the left side. We ended up with x - 4.5 + 4.5 = -1.1 + 4.5. The -4.5 and +4.5 on the left cancel each other out, leaving us with just x. On the right side, -1.1 + 4.5 equals 3.4. Therefore, our final solution is x = 3.4. Each step is a careful manipulation to maintain balance and isolate the variable. Practice makes perfect, so try working through similar problems to get the hang of it!
Checking the Solution
Okay, we've got our solution, x = 3.4, but how do we know it's right? The best way to be sure is to check our answer by plugging it back into the original equation. This is like double-checking your work on a test – it gives you peace of mind that you've nailed it! Our original equation was 10x - 4.5 + 3x = 12x - 1.1. Now, we're going to replace every x in the equation with 3.4 and see if both sides come out to be equal. So, let's plug it in! We get 10(3.4) - 4.5 + 3(3.4) = 12(3.4) - 1.1. First, we do the multiplications: 10 * 3.4 = 34 and 3 * 3.4 = 10.2. So, the left side becomes 34 - 4.5 + 10.2. Now, let's do the multiplication on the right side: 12 * 3.4 = 40.8. So, the right side becomes 40.8 - 1.1. Next, let's simplify each side separately. On the left side, 34 - 4.5 = 29.5, and then 29.5 + 10.2 = 39.7. So, the left side equals 39.7. On the right side, 40.8 - 1.1 = 39.7. Guess what? Both sides are equal! 39.7 = 39.7. This means our solution, x = 3.4, is correct. We solved the puzzle! Checking your solution is a super important step in algebra, so always remember to do it.
Tips and Tricks for Solving Equations
Solving equations can be like unlocking a secret code, and there are some tips and tricks that can make the process even smoother. Let's go over a few that will help you tackle any equation that comes your way. First off, always simplify before you start moving terms around. This means combining like terms on each side of the equation, just like we did in our example. Simplifying makes the equation less cluttered and easier to manage. Another important tip is to do the same thing to both sides. Remember, an equation is like a balanced scale. If you add something to one side, you have to add the same thing to the other side to keep it balanced. The same goes for subtraction, multiplication, and division. Keep your work organized. Write each step neatly and clearly. This will help you avoid mistakes and make it easier to check your work later. If you try to do too much in your head, it's easy to get lost or make a small error. Think about the order of operations (PEMDAS/BODMAS). This is crucial when you're simplifying expressions. Remember Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Finally, check your solution. We talked about this earlier, but it's worth repeating. Plugging your answer back into the original equation is the best way to make sure you've got it right. With these tips and tricks in your toolkit, you'll be solving equations like a pro in no time!
Common Mistakes to Avoid
When you're solving equations, it's easy to make a little slip-up here and there. But don't worry, everyone does it! Knowing the common mistakes can help you avoid them and make your equation-solving journey a lot smoother. One of the most common errors is not distributing correctly. If you have something like 2(x + 3), you need to multiply the 2 by both the x and the 3. It's easy to forget to multiply by the second term, so always double-check. Another frequent mistake is combining unlike terms. You can only add or subtract terms that have the same variable and exponent. For example, you can combine 3x and 5x, but you can't combine 3x and 5x² or 3x and 5. Remember the balanced scale? A big mistake is not doing the same thing to both sides of the equation. If you add a number to one side, you have to add it to the other side too. If you multiply one side by something, you have to multiply the other side by the same thing. Otherwise, the equation becomes unbalanced, and your solution will be wrong. Sign errors are another common pitfall. Pay close attention to positive and negative signs, especially when you're adding or subtracting terms. A small sign mistake can throw off the entire solution. Finally, forgetting to check your solution is a mistake that can cost you. Always plug your answer back into the original equation to make sure it works. By being aware of these common mistakes, you'll be well-equipped to avoid them and solve equations with confidence!
Practice Problems
Okay, now that we've gone through the solution, the steps, the tips, and the common mistakes, it's time to put your skills to the test! Practice is the key to mastering any math concept, so let's tackle a few practice problems similar to the one we just solved. Grab a pencil and paper, and let's dive in! Problem 1: Solve for y in the equation 5y + 2.5 - 2y = 8y - 5.5. Remember to combine like terms, get the y terms on one side, and the constants on the other. Don't forget to check your answer! Problem 2: What is the value of z in the equation 9z - 6 + 4z = 15z + 3? Take your time, follow the steps we discussed, and see if you can find the solution. Problem 3: Find a if 7a - 3.2 + 2a = 10a - 1.7. This one is just like the example we did together. You've got this! Problem 4: Solve for b: 11b + 1.8 - 5b = 6b - 0.6. Keep an eye on those signs and remember to simplify! Problem 5: If 4c - 5.1 + c = 7c + 2.9, what is the value of c? This is your final challenge! Give it your best shot. Working through these problems will help you build confidence and solidify your understanding of how to solve equations. Remember, the more you practice, the easier it gets. Good luck, and have fun solving!
Conclusion
Great job, guys! We've journeyed through solving the equation 10x - 4.5 + 3x = 12x - 1.1, and you've learned some super valuable skills along the way. We started by understanding the equation, breaking it down into its components, and then we tackled it step-by-step. We combined like terms, moved terms around to isolate x, and finally arrived at our solution: x = 3.4. But we didn't stop there! We went the extra mile and checked our solution to make sure it was correct. Then, we dove into some tips and tricks for solving equations, common mistakes to avoid, and even tackled some practice problems. You've now got a solid foundation for solving all sorts of algebraic equations. Remember, the key to success in math is practice, practice, practice! Keep working at it, and you'll be amazed at how much you can achieve. Equations might have seemed intimidating at first, but now you know you have the tools and the knowledge to conquer them. So, keep up the fantastic work, and never stop learning!
