Solving (18x=30)6=90-67: A Distributive Approach

Hey everyone! Today, we're diving into solving the equation (18x = 30)6 = 90 - 67 using the distributive property. This might look a bit intimidating at first, but don't worry, we'll break it down step-by-step so it's super easy to understand. The distributive property is a powerful tool in algebra, and mastering it can make solving complex equations a breeze. So, let's get started and see how we can tackle this problem together! We will focus on understanding the core concepts and applying them practically. Remember, math isn't just about numbers and symbols; it's about logic and problem-solving. Think of it as a puzzle where each step brings you closer to the solution. And guys, trust me, once you get the hang of it, you'll feel like a math whiz! So, let’s dive in and make math our playground.

Understanding the Distributive Property

First, let's quickly recap what the distributive property actually means. In simple terms, it's a way to multiply a number by a group of numbers added together. The property states that a(b + c) = ab + ac. What this means is that you multiply the number outside the parentheses (a) by each number inside the parentheses (b and c) and then add the results. This property is super useful because it allows us to simplify expressions and solve equations that would otherwise be quite tricky. Think of it like this: you're distributing the 'a' to both 'b' and 'c'. It's like sharing the love, or in this case, the multiplication! Understanding this basic principle is key to tackling more complex problems. We often use this property without even realizing it in everyday life. For example, if you're buying 3 bags of apples, and each bag has 5 red apples and 2 green apples, you can calculate the total number of apples by multiplying 3 by (5 + 2), which is the same as (3 * 5) + (3 * 2). See? Math is everywhere! This foundational understanding will help us as we move forward to solve the main equation. We'll be using the distributive property to simplify our equation and make it easier to handle. So, keep this definition in mind, and let's move on to the next step.

Breaking Down the Equation (18x = 30)6 = 90 - 67

Now, let's take a closer look at our equation: (18x = 30)6 = 90 - 67. The first thing we need to do is simplify both sides of the equation. On the left side, we have (18x = 30)6. It's important to note that the initial part '18x = 30' is likely a typo or a misunderstanding of the problem. The distributive property applies to expressions of the form a(b + c) or a(b - c). It seems there might be a missing operator inside the parenthesis. Let’s assume for a moment that the equation is meant to be (18x - 30)6 = 90 - 67, this allows us to actually apply the distributive property. So, with this adjustment, we will proceed to solve the problem step by step. Simplifying the right side of the equation is much more straightforward: 90 - 67 equals 23. So, the right side of our equation is now 23. Remember, guys, it's crucial to double-check the equation and ensure it makes sense mathematically. Small errors in the initial setup can lead to significant problems later on. So, before we move further, let's take a moment to appreciate the importance of accuracy in math. Now that we've simplified the right side and addressed the potential issue on the left side, we're ready to move on to the next step, which involves applying the distributive property to the left side of the equation. Let's keep going and unravel this mathematical puzzle!

Applying the Distributive Property: (18x - 30)6

Okay, guys, here's where the magic happens! We're going to apply the distributive property to the left side of our adjusted equation: (18x - 30)6. Remember, the distributive property tells us that a(b + c) = ab + ac, and similarly, a(b - c) = ab - ac. In our case, 'a' is 6, 'b' is 18x, and 'c' is 30. So, we need to multiply 6 by both 18x and 30. Let's do it step by step. First, we multiply 6 by 18x, which gives us 108x. Then, we multiply 6 by 30, which gives us 180. Since we have a subtraction inside the parentheses (18x - 30), we subtract the second result from the first. So, (18x - 30)6 becomes 108x - 180. This is a crucial step, so make sure you understand how we got here. Distributing correctly is key to solving the equation accurately. It's like following a recipe – if you add the ingredients in the wrong order, the final dish won't taste right! So, now we've successfully distributed the 6, and our equation is starting to look a lot simpler. We've transformed a potentially confusing expression into a more manageable one. Let’s keep this momentum going as we approach the final stages of solving for x. The next step involves setting up our simplified equation and isolating the variable.

Setting Up and Solving the Simplified Equation

Alright, now that we've distributed the 6, our equation looks like this: 108x - 180 = 23. We've made significant progress, haven't we? The goal now is to isolate 'x' on one side of the equation. To do this, we need to get rid of the -180. The golden rule of equations is that whatever you do to one side, you must do to the other. So, to get rid of -180, we add 180 to both sides of the equation. This gives us: 108x - 180 + 180 = 23 + 180. Simplifying this, we get 108x = 203. We're almost there, guys! Now, we just need to get 'x' by itself. Currently, 'x' is being multiplied by 108. To undo this multiplication, we divide both sides of the equation by 108. So, we have: 108x / 108 = 203 / 108. This simplifies to x = 203 / 108. This is our solution! We can leave it as a fraction, or we can convert it to a decimal if we prefer. Either way, we've successfully solved for 'x'. Remember, solving equations is like unwrapping a present – each step reveals a little more until you get to the final answer. And now, we've unwrapped our mathematical present! Let’s take a moment to recap what we’ve done and appreciate the journey we’ve taken to solve this equation.

Checking the Solution

Before we celebrate our victory, it's always a good idea to check our solution. This ensures that we haven't made any mistakes along the way. To check our solution, we substitute x = 203/108 back into our adjusted original equation: (18x - 30)6 = 90 - 67. So, we have (18 * (203/108) - 30)6 = 23. Let's simplify the expression inside the parentheses first. 18 * (203/108) simplifies to (203/6). Now we have (203/6 - 30)6 = 23. To subtract 30 from 203/6, we need to convert 30 to a fraction with a denominator of 6. So, 30 becomes 180/6. Now we have (203/6 - 180/6)6 = 23. Subtracting the fractions, we get (23/6)6 = 23. Finally, we multiply (23/6) by 6, which gives us 23. So, we have 23 = 23, which is absolutely correct! This confirms that our solution, x = 203/108, is indeed the correct answer. Checking our solution is like proofreading an essay – it helps us catch any errors and ensures our final answer is accurate. So, always take the time to check your work, guys! It's a small step that can make a big difference. With our solution checked and verified, we can confidently move on and apply these skills to other equations.

Conclusion: Mastering the Distributive Property

So, guys, we've successfully solved the equation (18x = 30)6 = 90 - 67 (with a slight adjustment to make it solvable using the distributive property) by using the distributive property. We started by understanding the basic principle of the distributive property, then we broke down the equation, applied the property, simplified, solved for 'x', and finally, we checked our solution. That's quite a journey! The key takeaway here is that the distributive property is a powerful tool for simplifying complex expressions and solving equations. By mastering this property, you can tackle a wide range of algebraic problems with confidence. Remember, math isn't about memorizing formulas; it's about understanding the underlying concepts and applying them logically. So, keep practicing, keep exploring, and keep challenging yourselves. Math is like a muscle – the more you use it, the stronger it gets! And the more problems you solve, the more confident you'll become. So, go out there and conquer those equations! You've got this! And always remember, every problem solved is a step forward in your mathematical journey. Keep up the great work, everyone!