Verifying The Distributive Property 18 × [17 + (-3)] = [18 × 17] + [18 × (-3)]
Introduction
Hey guys! Today, we're diving into a fun math problem that involves the distributive property. We're going to verify if the equation 18 × [17 + (-3)] = [18 × 17] + [18 × (-3)] holds true. This is a classic example of how multiplication distributes over addition, and it's a super useful concept in algebra and beyond. We will meticulously walk through each step to ensure clarity and understanding, so grab your pencils and let's get started! Understanding the distributive property is crucial not just for solving equations but also for simplifying complex expressions, which you'll encounter frequently in higher-level mathematics. This property allows us to break down problems into smaller, more manageable parts, making calculations easier and less prone to errors. Moreover, the distributive property is not just a theoretical concept; it has practical applications in various fields, including finance, engineering, and computer science. For instance, in finance, it can be used to calculate the total cost of purchasing multiple items at different prices. In engineering, it can help in analyzing forces and stresses in structures. In computer science, it's used in algorithms for data manipulation and optimization. By mastering this property, you're not just learning math; you're gaining a powerful tool that can be applied across various disciplines. So, let's jump in and see how it works in action with our specific problem. Remember, math isn't just about finding the right answer; it's about understanding the process and the underlying principles. So, let's explore this together and make math a little less intimidating and a lot more fun!
Understanding the Distributive Property
Before we jump into the verification, let's quickly recap the distributive property. In simple terms, it states that a(b + c) = ab + ac. This means that multiplying a number by the sum of two other numbers is the same as multiplying the number by each of the two numbers individually and then adding the products. This principle forms the backbone of our verification process. It's like saying if you have a group of items, and each item consists of two different parts, the total value can be calculated either by first adding the parts together and then multiplying by the number of items, or by multiplying each part by the number of items separately and then adding the results. The distributive property is incredibly powerful because it allows us to simplify complex calculations and break them down into smaller, more manageable steps. Imagine you're buying 18 sets of items, each set containing 17 apples and -3 oranges (yes, we're using a negative number to illustrate a point!). You can either first find the total number of fruits in a set (17 + -3 = 14) and then multiply by 18 (18 * 14), or you can multiply the number of apples by 18 (18 * 17) and the number of oranges by 18 (18 * -3) and then add the results. The distributive property tells us that both methods will give us the same answer. This is not just a mathematical trick; it's a fundamental concept that's used extensively in various fields. In algebra, it's used to expand expressions and solve equations. In calculus, it's used in differentiation and integration. In real life, it can be used to calculate costs, analyze data, and make informed decisions. So, understanding the distributive property is not just about acing your math exams; it's about developing a critical thinking skill that will serve you well in various aspects of life. Let's keep this in mind as we proceed with our verification, and we'll see how this property elegantly solves our problem.
Step-by-Step Verification
Okay, let’s break down the equation 18 × [17 + (-3)] = [18 × 17] + [18 × (-3)] step by step to verify it. We'll tackle the left-hand side (LHS) first. Remember the order of operations (PEMDAS/BODMAS)? Parentheses/Brackets come first! So, inside the brackets, we have 17 + (-3). Adding a negative number is the same as subtracting, so 17 + (-3) becomes 17 - 3, which equals 14. Now, we have 18 × 14 on the left-hand side. Multiplying 18 by 14 gives us 252. So, the left-hand side of the equation simplifies to 252. Now, let's move on to the right-hand side (RHS). Here, we have [18 × 17] + [18 × (-3)]. We need to perform the multiplications inside the brackets first. 18 multiplied by 17 equals 306. And 18 multiplied by -3 equals -54. So, we have 306 + (-54). Again, adding a negative number is the same as subtracting, so 306 + (-54) becomes 306 - 54, which equals 252. Guess what? Both the left-hand side and the right-hand side equal 252! This beautifully demonstrates that the equation holds true and that the distributive property is indeed working as expected. This step-by-step approach is crucial for understanding the underlying logic and ensuring accuracy in our calculations. Each step builds upon the previous one, leading us to the final result. By carefully following this process, we not only verify the equation but also reinforce our understanding of the distributive property and the order of operations. This method can be applied to various mathematical problems, making it a valuable skill to master. So, let's continue to practice and refine this approach, and we'll become math whizzes in no time! Remember, math is not about memorizing formulas; it's about understanding the process and applying the principles to solve problems. And that's exactly what we're doing here. Let's keep the momentum going and explore more exciting mathematical concepts!
Calculating the Left-Hand Side (LHS)
Let's start with the left-hand side (LHS) of the equation: 18 × [17 + (-3)]. As mentioned earlier, we need to follow the order of operations, which means tackling the expression inside the brackets first. We have 17 + (-3). Remember, adding a negative number is the same as subtracting. So, this becomes 17 - 3. 17 minus 3 is a straightforward calculation, and it equals 14. Now, the LHS simplifies to 18 × 14. This is a simple multiplication problem. To solve it, we can either use long multiplication or a calculator. 18 multiplied by 14 is 252. Therefore, the left-hand side of the equation equals 252. It's crucial to show each step in the calculation to avoid errors and to ensure a clear understanding of the process. This methodical approach is particularly important when dealing with more complex equations. By breaking down the problem into smaller, manageable steps, we reduce the chances of making mistakes and gain a deeper insight into the underlying mathematical principles. The order of operations is not just a rule; it's a fundamental concept that ensures consistency and accuracy in mathematical calculations. Without it, the same expression could yield different results depending on the order in which the operations are performed. This would lead to confusion and chaos in the world of mathematics. So, by diligently following the order of operations, we're not just solving the problem; we're also reinforcing a crucial mathematical principle. Now that we've successfully calculated the left-hand side, let's move on to the right-hand side and see if it matches. If it does, we'll have verified the equation and demonstrated the distributive property in action. Let's keep the momentum going and tackle the right-hand side with the same level of precision and clarity.
Calculating the Right-Hand Side (RHS)
Now, let's tackle the right-hand side (RHS) of the equation: [18 × 17] + [18 × (-3)]. Again, we'll follow the order of operations, which means performing the multiplications inside the brackets first. Let's start with the first set of brackets: 18 × 17. This is a slightly larger multiplication, but we can handle it. 18 multiplied by 17 equals 306. You can verify this using long multiplication or a calculator. Now, let's move on to the second set of brackets: 18 × (-3). Here, we're multiplying a positive number by a negative number. Remember, when you multiply a positive number by a negative number, the result is always negative. So, 18 multiplied by -3 equals -54. Now, we have 306 + (-54). As we've seen before, adding a negative number is the same as subtracting. So, this becomes 306 - 54. 306 minus 54 is 252. Therefore, the right-hand side of the equation also equals 252. Just like the left-hand side, we've carefully calculated the right-hand side, showing each step clearly. This meticulous approach ensures that we've avoided any errors and that we fully understand the process. The fact that we obtained the same result on both sides of the equation is a testament to the distributive property and the accuracy of our calculations. It's like a mathematical puzzle where all the pieces fit perfectly together. This satisfaction of arriving at the correct answer is one of the many joys of mathematics. Now that we've calculated both sides of the equation, we're just one step away from verifying the equation. We've shown that the left-hand side equals 252 and the right-hand side equals 252. The final step is to compare these two results and draw our conclusion. Let's move on to the final verification step and celebrate our mathematical victory!
Verification Result
Alright, guys, we've reached the final step! We calculated the left-hand side (LHS) of the equation 18 × [17 + (-3)] and found it to be 252. We then calculated the right-hand side (RHS), [18 × 17] + [18 × (-3)], and guess what? It also came out to be 252! Since LHS = 252 and RHS = 252, we can confidently say that LHS = RHS. This confirms that the equation 18 × [17 + (-3)] = [18 × 17] + [18 × (-3)] is indeed true. We have successfully verified the equation using the distributive property. This is a great example of how a mathematical principle can be applied to solve a problem and how careful, step-by-step calculations can lead to a definitive answer. The satisfaction of verifying an equation like this comes from the understanding that we've not just found a solution; we've also demonstrated the validity of a mathematical concept. This is the essence of mathematical thinking: to explore, to question, and to verify. The distributive property, which we've used in this verification, is a fundamental concept in algebra and is used extensively in various mathematical contexts. By understanding and applying this property, we can simplify complex expressions, solve equations, and tackle a wide range of mathematical problems. So, let's take a moment to appreciate the elegance and power of the distributive property and the satisfaction of a successful verification. We've not only solved a problem; we've also reinforced our understanding of a key mathematical principle. Let's keep this spirit of exploration and verification as we continue our mathematical journey. There are many more exciting concepts and challenges ahead, and with a solid foundation like this, we're well-equipped to tackle them.
Conclusion
In conclusion, we have successfully verified that 18 × [17 + (-3)] = [18 × 17] + [18 × (-3)]. By meticulously calculating both the left-hand side and the right-hand side of the equation, we demonstrated that they are equal, thus confirming the validity of the equation. This exercise serves as a practical application of the distributive property, a fundamental concept in mathematics. The distributive property allows us to simplify expressions and solve equations by distributing a factor across a sum or difference. It's a powerful tool that is used extensively in algebra and beyond. By understanding and mastering this property, we can tackle a wide range of mathematical problems with confidence. Moreover, this verification process highlights the importance of following the order of operations (PEMDAS/BODMAS) to ensure accuracy in our calculations. Each step in the calculation was carefully executed, and the order of operations was strictly adhered to, leading us to the correct answer. This methodical approach is crucial for success in mathematics and in any field that requires precise calculations and logical reasoning. The joy of verifying a mathematical equation lies not just in finding the correct answer but also in the process of exploration and discovery. By breaking down the problem into smaller, manageable steps, we gained a deeper understanding of the underlying mathematical principles and developed our problem-solving skills. So, let's continue to embrace the challenges and rewards of mathematics and strive to deepen our understanding of this fascinating subject. With a solid foundation in fundamental concepts like the distributive property and a commitment to careful, step-by-step calculations, we can unlock the beauty and power of mathematics and apply it to solve real-world problems.