Simplify $-9(4p-6q)+8q-7(3q-3p)$: A Step-by-Step Guide
Hey guys! Ever felt lost in the maze of algebraic expressions? Don't worry, you're not alone! Algebraic expressions can seem daunting at first glance, but with a systematic approach, they become surprisingly manageable. In this guide, we'll break down the expression into its simplest form. We'll walk through each step, explaining the underlying principles and techniques involved. Whether you're a student grappling with algebra for the first time or just looking to brush up on your skills, this article is for you. Let's dive in and conquer this algebraic challenge together!
Understanding the Basics: Order of Operations and the Distributive Property
Before we jump into the specific problem, let's quickly revisit two crucial concepts: the order of operations (often remembered by the acronym PEMDAS/BODMAS) and the distributive property. These are the fundamental tools we'll use to unravel the expression. Understanding the order of operations is paramount in simplifying any mathematical expression. It ensures that we perform operations in the correct sequence, leading to the accurate result. PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right), provides a roadmap for simplifying expressions. Neglecting this order can lead to a cascade of errors, turning a simple problem into a complex mess. So, always keep PEMDAS/BODMAS in mind as your guiding principle. Similarly, the distributive property is a cornerstone of algebraic manipulation. It allows us to multiply a single term by multiple terms within parentheses, effectively expanding the expression. This property is crucial for eliminating parentheses and combining like terms, which are essential steps in simplification. The distributive property states that a(b + c) = ab + ac. This seemingly simple rule unlocks the potential to transform complex expressions into more manageable forms. Mastering the distributive property is like having a key to unlock the secrets of algebraic simplification. Without it, many expressions would remain impenetrable. So, make sure you understand this property inside and out; it's your best friend in algebra.
Step 1: Applying the Distributive Property
The first hurdle in simplifying is the presence of parentheses. To eliminate them, we'll employ the distributive property. Remember, this property allows us to multiply the term outside the parentheses by each term inside. This is where the magic happens, guys! We're going to take that -9 and distribute it across the (4p - 6q). This means multiplying -9 by 4p and then -9 by -6q. Let's break it down: -9 * 4p = -36p. Notice the negative sign! It's crucial to keep track of those. And -9 * -6q = +54q. A negative times a negative gives us a positive! So, the first part of the expression, -9(4p - 6q), simplifies to -36p + 54q. Now, let's tackle the second set of parentheses. We have -7(3q - 3p). Again, we distribute the -7: -7 * 3q = -21q and -7 * -3p = +21p. So, -7(3q - 3p) simplifies to -21q + 21p. See how we carefully handled the signs each time? That's the key to avoiding errors. By applying the distributive property, we've successfully removed the parentheses, paving the way for the next step in simplification. This is like clearing the first level of a challenging game – we're making progress! Remember, practice makes perfect. The more you apply the distributive property, the more natural it will become. You'll start seeing opportunities to use it everywhere in algebraic expressions. So, keep practicing, and you'll become a master of distribution in no time!
Step 2: Rewriting the Expression
Now that we've conquered the parentheses, let's rewrite the entire expression with the distributed terms. This will give us a clearer picture of what we're working with. Think of it as organizing your workspace before tackling a big project. Having everything laid out in front of you makes the task seem less daunting. So, after distributing, our expression -9(4p - 6q) + 8q - 7(3q - 3p) transforms into -36p + 54q + 8q - 21q + 21p. Notice how we've simply replaced the expressions in parentheses with their expanded forms. The +8q in the middle stays exactly as it is – we haven't touched it yet. Rewriting the expression in this way is a crucial step because it allows us to see all the terms clearly and identify the ones we can combine. It's like taking a deep breath and getting a bird's-eye view of the problem. This clarity is essential for avoiding mistakes and ensuring that we simplify the expression correctly. Without rewriting, it's easy to miss terms or make errors in the next step. So, don't skip this step! It's a small investment of time that pays off big in terms of accuracy. It's like making sure you have all the ingredients for a recipe before you start cooking – you're setting yourself up for success. And remember, organization is key in algebra, just as it is in many other areas of life. The more organized you are, the easier it will be to solve complex problems. So, take a moment to rewrite the expression, and you'll be well on your way to simplifying it completely.
Step 3: Combining Like Terms
The heart of simplifying algebraic expressions lies in combining like terms. These are terms that have the same variable raised to the same power. Think of it as sorting your socks – you group the pairs together, right? We're doing the same thing here, but with algebraic terms. In our expression, -36p + 54q + 8q - 21q + 21p, we have two types of terms: 'p' terms and 'q' terms. Let's focus on the 'p' terms first. We have -36p and +21p. Combining these is like adding and subtracting numbers: -36 + 21 = -15. So, -36p + 21p simplifies to -15p. Now, let's move on to the 'q' terms. We have +54q, +8q, and -21q. This is where we need to be extra careful with the signs. We can add these terms together in any order, but it's often helpful to group the positive terms first: 54q + 8q = 62q. Then, we subtract the negative term: 62q - 21q = 41q. So, +54q + 8q - 21q simplifies to +41q. By combining like terms, we've significantly reduced the complexity of the expression. This is like pruning a bush – we're getting rid of the unnecessary branches to reveal the core structure. And just as a well-pruned bush looks neater and healthier, a simplified algebraic expression is easier to understand and work with. Remember, the key to combining like terms is to focus on the coefficients (the numbers in front of the variables) and keep the variable and its exponent the same. It's like adding apples to apples – you end up with more apples, not oranges. So, take your time, be careful with the signs, and you'll master the art of combining like terms in no time!
Step 4: The Simplified Expression
After the dust settles from combining like terms, we arrive at our simplified expression. This is the final result, the culmination of all our hard work. It's like reaching the summit after a challenging climb – a moment of satisfaction and accomplishment! In our case, we combined the 'p' terms to get -15p and the 'q' terms to get +41q. So, the simplified expression is -15p + 41q. This is much cleaner and more concise than the original expression, -9(4p - 6q) + 8q - 7(3q - 3p). It's like transforming a tangled mess of wires into a neatly organized cable – everything is in its place, and it's much easier to see what's going on. The simplified expression is not only easier to read but also easier to use in further calculations or problem-solving. It's like having a well-tuned instrument – it's ready to perform at its best. And remember, guys, simplifying expressions is not just about getting the right answer; it's about developing a deeper understanding of algebraic principles and honing your problem-solving skills. Each step we've taken – distributing, rewriting, and combining like terms – has reinforced these crucial skills. So, take a moment to appreciate the journey we've taken and the skills you've gained. You've successfully simplified a complex algebraic expression, and that's something to be proud of! And remember, the more you practice, the more confident and proficient you'll become. So, keep simplifying, keep exploring, and keep challenging yourself!
Conclusion: Mastering Algebraic Simplification
Congratulations! You've successfully navigated the process of simplifying the algebraic expression . We've journeyed from the initial expression to the simplified form of -15p + 41q, and along the way, we've reinforced some essential algebraic skills. This journey is a testament to the power of breaking down complex problems into manageable steps. We started by understanding the fundamental principles of the order of operations and the distributive property. These are the cornerstones of algebraic manipulation, and mastering them is crucial for success. We then applied the distributive property to eliminate parentheses, carefully handling the signs to avoid errors. This step opened the door to combining like terms, which is the heart of simplification. We rewrote the expression to gain clarity and then combined the 'p' terms and 'q' terms separately, arriving at our final simplified form. This step-by-step approach is not just applicable to this specific problem but can be used to tackle a wide range of algebraic expressions. It's like having a roadmap for simplifying any expression that comes your way. And remember, practice is the key to mastery. The more you work with algebraic expressions, the more comfortable and confident you'll become. You'll start recognizing patterns, anticipating steps, and simplifying expressions with ease. So, don't be afraid to tackle challenging problems, and don't get discouraged by mistakes. Each mistake is an opportunity to learn and grow. And finally, remember that algebra is not just a set of rules and procedures; it's a language for expressing mathematical relationships and solving real-world problems. By mastering algebraic simplification, you're not just learning a skill; you're unlocking a powerful tool for understanding the world around you. So, keep exploring, keep learning, and keep simplifying!
