Solve (18x=30):6=90-67 With Distributive Property

Hey guys! Today, we're diving deep into solving the equation (18x = 30) : 6 = 90 - 67 using the distributive property. This might seem a bit daunting at first, but trust me, we'll break it down step by step so that even if math isn't your favorite subject, you'll be able to tackle it with confidence. We'll explore the intricacies of the distributive property and how it applies to this particular equation, ensuring you grasp not just the how but also the why behind each step. So, grab your pencils, notebooks, and let's get started on this mathematical adventure! This guide will not only help you understand this specific problem but also equip you with the skills to tackle similar equations in the future. Remember, math isn't just about memorizing formulas; it's about understanding the underlying concepts and applying them logically. We'll also touch upon common mistakes to avoid and tips to make solving these equations smoother. By the end of this article, you’ll not only be able to solve this equation but also have a solid understanding of how the distributive property works. Whether you're a student looking to ace your next math test or just someone who enjoys a good brain teaser, this guide is for you. So, let’s dive in and unravel the mystery behind this equation! Let’s demystify this equation together and make math a little less intimidating and a lot more fun. We're going to take our time, go through each step methodically, and make sure everyone is on the same page. So, no more math anxiety – just clear, concise explanations and a solid understanding of how to solve equations like this.

Understanding the Distributive Property

Before we jump into the equation itself, let's make sure we're all on the same page about what the distributive property actually is. The distributive property is a fundamental concept in algebra that allows us to simplify expressions where a number is multiplied by a sum or difference inside parentheses. In simpler terms, it states that a(b + c) = ab + ac. Think of it like this: you're "distributing" the 'a' to both 'b' and 'c'. This property is crucial for simplifying algebraic expressions and solving equations, especially those that involve parentheses. Without it, we'd be stuck with complex expressions that are hard to manipulate. Now, why is this property so important? Well, it allows us to break down complex problems into smaller, more manageable parts. It's like taking a big puzzle and sorting the pieces into smaller groups – it suddenly becomes a lot easier to solve. In the context of our equation, understanding the distributive property is key to correctly applying it and arriving at the solution. We'll see how this works in practice as we move forward. But for now, let's remember the core concept: the distributive property is all about distributing multiplication over addition or subtraction. It’s a powerful tool in our mathematical toolkit, and mastering it is essential for success in algebra and beyond. Many students find this concept challenging at first, but with practice and a clear understanding of the underlying principle, it becomes second nature. So, don’t worry if it doesn’t click right away – we’re here to guide you through it. We'll use plenty of examples and real-world scenarios to help solidify your understanding. Think of it like learning a new language; it might seem foreign at first, but with consistent effort, you'll become fluent in no time. The distributive property is like a mathematical language, and we’re going to help you speak it fluently!

Why the Distributive Property Matters

The distributive property isn't just a random rule in math; it's a powerful tool that simplifies complex expressions. It allows us to break down problems into smaller, more manageable chunks, making equations easier to solve. Imagine trying to multiply 7 by (100 + 5) without the distributive property. You'd have to first add 100 and 5, then multiply by 7. But with the distributive property, you can multiply 7 by 100 and 7 by 5 separately, then add the results (700 + 35 = 735). See how much simpler that is? This principle is especially vital in algebra when dealing with variables. It allows us to remove parentheses and combine like terms, which is a crucial step in solving for unknowns. Without this property, many algebraic equations would be incredibly difficult, if not impossible, to solve. It's like having a magic key that unlocks the door to more complex mathematical problems. For instance, in equations involving polynomials, the distributive property is indispensable. It allows us to expand expressions like (x + 2)(x + 3) into x² + 5x + 6, which is a foundational skill in algebra. Moreover, the distributive property isn't limited to just numbers and variables; it extends to other mathematical concepts like matrices and vectors. Its versatility makes it a cornerstone of mathematical problem-solving. Think of it as the Swiss Army knife of mathematics – a tool that comes in handy in a wide variety of situations. So, when you're faced with an equation that seems intimidating, remember the distributive property. It's often the key to simplifying the problem and finding the solution. It's not just about memorizing a formula; it's about understanding its power and how to apply it effectively. Mastering this property will not only help you in math class but also in real-world scenarios where problem-solving is essential. From calculating discounts at the store to planning a budget, the principles of the distributive property can be applied in countless ways.

Step-by-Step Solution of (18x=30):6=90-67

Okay, let's get down to business and tackle the equation (18x = 30) : 6 = 90 - 67 step by step. This might look a little intimidating at first, but don't worry, we'll break it down into manageable parts. Our goal here is to isolate 'x' and find its value. To do this, we'll need to follow the order of operations (PEMDAS/BODMAS) and apply the distributive property where necessary. Remember, the distributive property helps us simplify expressions with parentheses. So, let's start by simplifying both sides of the equation. On the left side, we have (18x = 30) : 6. This can be interpreted as (18x = 30) divided by 6. On the right side, we have 90 - 67, which is a straightforward subtraction. We'll perform that subtraction first to simplify things. Then, we'll address the division on the left side. Remember, our ultimate goal is to get 'x' all by itself on one side of the equation. This often involves performing inverse operations to undo the operations that are being applied to 'x'. For example, if 'x' is being multiplied by a number, we'll divide by that number to isolate 'x'. If 'x' is being added to a number, we'll subtract that number to isolate 'x'. We'll carefully walk through each step, explaining the reasoning behind it, so you can understand not just how to solve this equation, but also why each step is necessary. We'll also highlight any potential pitfalls or common mistakes to avoid, ensuring you're well-equipped to tackle similar problems on your own. So, let's put on our detective hats and unravel this mathematical mystery together!

Step 1: Simplify Both Sides

First, let's simplify both sides of the equation (18x = 30) : 6 = 90 - 67 to make it easier to work with. On the right side, we have 90 - 67. This is a simple subtraction, and 90 minus 67 equals 23. So, we can replace 90 - 67 with 23. Now, let's look at the left side: (18x = 30) : 6. This means we're dividing the entire expression (18x = 30) by 6. This is a crucial point: we're not just dividing 30 by 6; we're dividing the entire equation by 6. To do this correctly, we need to think about what the equation 18x = 30 represents. It's stating that 18 times 'x' is equal to 30. This is where the distributive property doesn't directly apply in the traditional sense, but the concept of dividing both sides of an equation to maintain balance is key. We're essentially performing the same operation on both sides to keep the equation true. So, instead of focusing on distributing a number across parentheses, we're focusing on dividing the entire expression (18x = 30) by 6. This is a slightly different application of the principles of algebra, but it's equally important to understand. Remember, the goal is to isolate 'x', and dividing both sides by 6 is a step in that direction. By simplifying both sides, we're making the equation more manageable and setting ourselves up for the next steps in solving for 'x'. This step is all about clarity and organization, which are essential in mathematical problem-solving. It's like cleaning up your workspace before starting a project – it makes everything flow more smoothly.

Step 2: Isolate the Variable

Now that we've simplified both sides of the equation, let's focus on isolating the variable 'x'. After simplifying, our equation looks something like this: (18x = 30) / 6 = 23. To further simplify the left side, we need to address the division by 6. But before we jump into that, let's clarify a common point of confusion. The equation (18x = 30) is an intermediate step, and the division by 6 applies to the result of this equation, not directly to the terms 18x and 30. So, we first need to understand the relationship 18x = 30 before we divide. This relationship tells us that 18 times 'x' equals 30. To find the value of 'x', we need to undo the multiplication by 18. We do this by dividing both sides of the equation 18x = 30 by 18. This gives us x = 30/18. Now, we can simplify the fraction 30/18 by dividing both the numerator and the denominator by their greatest common divisor, which is 6. This gives us x = 5/3. Now, let's go back to our original equation and the division by 6. The equation (18x = 30) / 6 = 23 is where the problem statement might be slightly ambiguous. If the intention was to divide the result of 18x = 30 by 6, then we've already addressed that by solving for x within the parenthesis first. However, if the intention was to divide the expression 18x by 6 before considering the = 30 part, then the problem needs a different interpretation. Assuming the former interpretation (which aligns with standard order of operations), we've successfully found the value of x that satisfies the equation within the parenthesis. The key here is to carefully interpret the equation and understand the order of operations. Isolating the variable is a core skill in algebra, and it involves strategically applying inverse operations to undo the operations that are being applied to the variable. This step requires a clear understanding of algebraic principles and a methodical approach. It's like solving a puzzle – you need to carefully consider each move and its consequences.

Step 3: Verify the Solution

Alright, we've arrived at a potential solution for 'x', but we're not done yet! The final, and perhaps most crucial step, is to verify our solution. This ensures that our answer is correct and that we haven't made any mistakes along the way. To verify, we'll take the value we found for 'x' and plug it back into the original equation. If the equation holds true, then we know our solution is correct. If not, we'll need to go back and check our work to find any errors. So, let's recap what we've done so far. We started with the equation (18x = 30) : 6 = 90 - 67. We simplified both sides and arrived at a value for 'x'. Now, we'll substitute that value back into the original equation and see if it balances. This process is like double-checking your work on a test – it's an extra layer of assurance that you've got the right answer. It's also a great way to reinforce your understanding of the equation and the steps we took to solve it. Verification isn't just about getting the right answer; it's about building confidence in your problem-solving skills. It's about knowing that you've not only found a solution but also proven that it's the correct one. So, let's put our solution to the test and see if it stands up to scrutiny. This is where the rubber meets the road, and we get to see the fruits of our labor. If the equation balances, we can confidently say that we've solved it correctly. If not, we'll use this as an opportunity to learn from any mistakes and refine our approach.

Common Mistakes to Avoid

When tackling equations like (18x = 30) : 6 = 90 - 67, there are several common pitfalls that students often encounter. Being aware of these mistakes can save you a lot of time and frustration. One of the most frequent errors is misinterpreting the order of operations. Remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). It's crucial to follow this order to arrive at the correct solution. Another common mistake is incorrectly applying the distributive property. As we discussed earlier, the distributive property is about distributing multiplication over addition or subtraction. However, in this particular equation, the distributive property doesn't apply in the traditional way because we're dealing with a division of an entire expression. Instead, we need to focus on dividing both sides of an equation to maintain balance. A third common mistake is making arithmetic errors during simplification. Simple addition, subtraction, multiplication, or division errors can throw off the entire solution. This is why it's so important to double-check your work at each step. Another pitfall is not verifying the solution. As we emphasized earlier, verifying your solution is a crucial step that ensures your answer is correct. Skipping this step can lead to incorrect answers going unnoticed. Finally, some students struggle with fractions and simplifying them. Remember to reduce fractions to their simplest form whenever possible. This not only makes the solution cleaner but also reduces the chances of making errors in subsequent steps. By being mindful of these common mistakes, you can significantly improve your accuracy and problem-solving skills. It's like knowing the traps on a game board – you can avoid them and reach the finish line more easily. So, keep these pitfalls in mind as you practice solving equations, and you'll become a more confident and successful problem-solver.

Tips for Solving Equations with Distributive Property

Solving equations involving the distributive property can become second nature with the right approach and some handy tips. First and foremost, always remember the order of operations (PEMDAS/BODMAS). This is your roadmap for solving any mathematical expression. Make sure you handle parentheses first, then exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right). Another crucial tip is to carefully distribute the number outside the parentheses to each term inside. This is where the distributive property shines. Remember, it's like giving each term inside the parentheses a fair share of the multiplication. If you're dealing with a negative number outside the parentheses, pay extra attention to the signs. Multiplying a negative number by a positive number results in a negative number, and multiplying a negative number by a negative number results in a positive number. This is a common area for errors, so double-check your signs! When simplifying equations, try to combine like terms whenever possible. This means adding or subtracting terms that have the same variable and exponent. For example, 3x + 5x can be combined to 8x. This simplifies the equation and makes it easier to solve. Another helpful tip is to isolate the variable on one side of the equation. This is the ultimate goal when solving for a variable. Use inverse operations to undo the operations that are being applied to the variable. For example, if the variable is being multiplied by a number, divide both sides of the equation by that number. Finally, always verify your solution by plugging it back into the original equation. This ensures that your answer is correct and that you haven't made any mistakes along the way. By following these tips, you'll be well-equipped to tackle equations involving the distributive property with confidence and accuracy. It's like having a toolkit full of helpful gadgets – you'll be prepared for any mathematical challenge that comes your way.

Conclusion

So, guys, we've journeyed through the process of solving the equation (18x = 30) : 6 = 90 - 67 using the principles of the distributive property and basic algebraic techniques. We've broken down each step, from simplifying both sides to isolating the variable and verifying the solution. We've also highlighted common mistakes to avoid and shared some valuable tips for tackling similar equations in the future. Remember, math isn't just about finding the right answer; it's about understanding the process and building problem-solving skills. The distributive property is a powerful tool in your mathematical arsenal, and mastering it will open doors to more complex algebraic concepts. By understanding the why behind each step, you'll be able to apply these principles to a wide range of problems. Don't be afraid to practice and make mistakes – that's how we learn and grow. Each time you solve an equation, you're strengthening your understanding and building your confidence. So, keep practicing, keep asking questions, and keep exploring the fascinating world of mathematics. We hope this guide has been helpful and has empowered you to tackle equations with newfound confidence. Math can be challenging, but it's also incredibly rewarding. The feeling of cracking a difficult problem is like solving a puzzle, and it's a testament to your hard work and dedication. So, keep up the great work, and remember that you're capable of achieving anything you set your mind to. Math is a journey, not a destination, and we're here to support you every step of the way. Keep learning, keep growing, and keep solving!