Solving For X: A Step-by-Step Guide

Hey guys! Let's dive into solving for x in the equation 4.5(8 - x) + 36 = 102 - 2.5(3x + 24). It might look a little intimidating at first, but trust me, we'll break it down into easy-to-follow steps. This is a common algebra problem, and understanding how to solve it is super important for your math journey. We'll go through each step carefully, explaining what we're doing and why. By the end of this, you'll be a pro at solving this type of equation. Ready to get started?

Step 1: Distribute and Simplify - The First Steps to Solving for x

Alright, the first thing we're gonna do is tackle those parentheses. Remember the distributive property? It's where we multiply the number outside the parentheses by each term inside. On the left side of the equation, we have 4.5(8 - x). Let's distribute that 4.5: 4.5 * 8 which equals 36, and 4.5 * -x which gives us -4.5x. So, the left side of the equation becomes 36 - 4.5x + 36. Cool, right? We've simplified a chunk already!

Now, let's do the same thing on the right side. We have -2.5(3x + 24). Distributing the -2.5, we get -2.5 * 3x which is -7.5x, and -2.5 * 24 which equals -60. Therefore, the right side of the equation becomes 102 - 7.5x - 60. We've got it! We've successfully distributed and simplified both sides. Now our equation looks like this: 36 - 4.5x + 36 = 102 - 7.5x - 60. We're making great progress, and everything is starting to fall into place. Always remember to double-check your calculations in this step. A small mistake can mess up the whole process. Make sure you're multiplying correctly and that you keep track of those negative signs. And if you're ever unsure, don't hesitate to write everything out step by step. This way, you won't get lost, and you'll find the errors easily, and that is a great way to stay organized.

Now that we've distributed, let's simplify further by combining like terms on each side of the equation. On the left side, we have two constants: 36 and 36. Adding them together gives us 72. So, the left side simplifies to 72 - 4.5x. On the right side, we also have two constants: 102 and -60. Adding these together yields 42. So, the right side simplifies to 42 - 7.5x. Our equation is now: 72 - 4.5x = 42 - 7.5x. This is significantly cleaner and easier to work with! The key to this step is to identify those terms that can be combined. Remember, like terms are terms that have the same variable raised to the same power. This simplification makes the equation much more manageable. You will also minimize the number of steps that you will need to take by simplifying the equation. It will also help you to see what you need to isolate to find x. It's really that simple!

Step 2: Isolating the Variable - Getting x by Itself

Alright, now that we've simplified, it's time to get all the x terms on one side of the equation and the constants on the other side. This is like herding cats, but it's super important to solve for x. Let's start by adding 7.5x to both sides of the equation. Why? Because this cancels out the -7.5x on the right side. Remember, whatever you do to one side of the equation, you have to do to the other to keep it balanced. So, our equation becomes 72 - 4.5x + 7.5x = 42 - 7.5x + 7.5x. Simplifying, we get 72 + 3x = 42. See how we moved all the x terms to the left side?

Now, we need to get rid of that 72 on the left side. To do that, we subtract 72 from both sides: 72 + 3x - 72 = 42 - 72. This leaves us with 3x = -30. We're getting closer to solving for x! We're essentially rearranging the equation to isolate the variable. This step is about using inverse operations. Addition and subtraction are inverse operations. Multiplication and division are inverse operations. By using these, you can move terms around and simplify the equation. When you perform these operations, always double-check your work. You don't want to make a simple math error. Be organized, write everything out clearly, and you'll be just fine.

Step 3: Solve for x - The Final Push

We're almost there! We have 3x = -30. To solve for x, we need to isolate it completely. Right now, x is being multiplied by 3. The opposite of multiplication is division, so we'll divide both sides of the equation by 3. This gives us 3x / 3 = -30 / 3. Simplifying this, we get x = -10. Boom! We've found the value of x!

So, x equals -10. But before we get too excited, let's check our answer to make sure it's correct. We can do this by plugging -10 back into the original equation and seeing if both sides are equal. This is a crucial step! It is a way to ensure that you have the right answer.

Here’s how we can check it. Substitute -10 for x in the original equation: 4.5(8 - (-10)) + 36 = 102 - 2.5(3*(-10) + 24). Let's break this down: 4.5(8 + 10) + 36 = 102 - 2.5(-30 + 24). Then, we have 4.5(18) + 36 = 102 - 2.5(-6). Which simplifies to: 81 + 36 = 102 + 15. Further simplifying: 117 = 117.

Since both sides are equal, we know that our answer, x = -10, is correct! High five! You have successfully solved for x. Remember, practice makes perfect. The more you work through these problems, the easier it will become. Keep practicing and keep up the great work! And just like that, we've solved for x!

Conclusion: Mastering the Art of Solving for x

We did it, guys! We successfully solved for x in the equation 4.5(8 - x) + 36 = 102 - 2.5(3x + 24). We went through each step, from distributing and simplifying to isolating the variable and solving for x. We even checked our answer to make sure we got it right. Remember, the key is to break down the problem into smaller, more manageable steps. Don't rush, and always double-check your work. Practice makes perfect when solving equations!

So, what did we learn? We learned how to use the distributive property, combine like terms, and use inverse operations to isolate the variable. We also learned the importance of checking our answer. These are fundamental skills that you'll use throughout your math journey. Keep practicing and applying these steps, and you'll become a pro at solving equations like this. Feel free to try more examples on your own. You can find tons of problems online or in your textbook. And remember, if you get stuck, don't worry. Go back to the steps, review the concepts, and try again. You got this! Keep practicing, and you'll become a math whiz in no time. If you got any questions, feel free to ask!