Hey guys! Ever stumbled upon an algebraic expression and felt a bit lost? Don't worry, it happens to the best of us. Today, we're going to break down a seemingly complex problem into easy-to-understand steps. Specifically, we're tackling the expression (7p + 2q) multiplied by 3q. Sounds like a mouthful, right? But trust me, by the end of this guide, you'll be simplifying such expressions like a pro. So, let's dive in and make math a little less intimidating and a lot more fun!

Understanding the Basics: The Distributive Property

Before we jump into the problem, let's quickly revisit a fundamental concept in algebra: the distributive property. This property is the key to simplifying expressions like the one we're dealing with. In simple terms, the distributive property states that for any numbers a, b, and c, a * (b + c) = a * b + a * c. What does this mean? It means that when you're multiplying a single term by a group of terms inside parentheses, you need to multiply that single term by each term inside the parentheses individually. This might sound a bit abstract, so let's put it into practice with some simple examples.

Imagine you have 2 * (x + 3). Using the distributive property, you would multiply 2 by x and then 2 by 3, resulting in 2x + 6. See? Not so scary! Another example could be 5 * (y - 2). In this case, you multiply 5 by y and then 5 by -2, giving you 5y - 10. The distributive property is like a secret weapon that allows us to break down complex expressions into smaller, more manageable pieces. Once you master this property, you'll find that many algebraic problems become much easier to solve. It's all about breaking things down and taking it one step at a time. Remember, practice makes perfect, so don't be afraid to try out different examples and see how the distributive property works in action. With a little bit of practice, you'll be a pro in no time!

Step-by-Step Solution: Multiplying (7p + 2q) by 3q

Okay, now that we've refreshed our understanding of the distributive property, let's apply it to our original problem: (7p + 2q) * 3q. The first step is to distribute 3q to both terms inside the parentheses. This means we need to multiply 3q by 7p and then multiply 3q by 2q. Let's start with the first part: 3q * 7p. When multiplying terms with variables, we multiply the coefficients (the numbers in front of the variables) and then multiply the variables themselves. So, 3 * 7 = 21, and q * p = pq. Therefore, 3q * 7p = 21pq. Now, let's move on to the second part: 3q * 2q. Again, we multiply the coefficients first: 3 * 2 = 6. Then, we multiply the variables: q * q = q². So, 3q * 2q = 6q². Now that we've multiplied 3q by both terms inside the parentheses, we can combine the results: 21pq + 6q². And that's it! We've successfully simplified the expression (7p + 2q) * 3q. The final answer is 21pq + 6q². Remember, the key to solving these types of problems is to take it one step at a time and apply the distributive property correctly. With a little bit of practice, you'll be able to simplify even the most complex expressions with ease. So, keep practicing and don't be afraid to ask for help if you get stuck. We're all in this together!

Breaking Down the Solution: A Closer Look

Let's dissect the solution even further to ensure we grasp every detail. The original expression was (7p + 2q) * 3q. Our mission was to simplify this expression by applying the distributive property. We started by identifying the terms inside the parentheses: 7p and 2q. These are the terms that will be multiplied by 3q. Next, we multiplied 3q by each of these terms individually. First, we multiplied 3q by 7p. Remember, when multiplying terms with variables, we multiply the coefficients and then multiply the variables. So, 3 * 7 equals 21, and q * p equals pq. Therefore, 3q * 7p equals 21pq. It's important to note that the order of the variables doesn't matter, so pq is the same as qp. However, it's common practice to write the variables in alphabetical order for clarity. Next, we multiplied 3q by 2q. Again, we multiply the coefficients first: 3 * 2 equals 6. Then, we multiply the variables: q * q equals q². This is because when you multiply a variable by itself, you're essentially squaring it. So, 3q * 2q equals 6q². Finally, we combined the results of these two multiplications. We added 21pq and 6q² to get the final simplified expression: 21pq + 6q². This is the simplest form of the expression, and we can't simplify it any further because the terms have different variables. And that's it! We've successfully broken down the solution step by step. By understanding each step in detail, you can apply the same principles to simplify other algebraic expressions. Remember, practice is key, so keep working on different examples to master this skill.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's easy to make mistakes if you're not careful. Let's go over some common pitfalls to avoid. One common mistake is forgetting to distribute the term outside the parentheses to all the terms inside. For example, in the expression (7p + 2q) * 3q, some people might only multiply 3q by 7p and forget to multiply it by 2q. This would lead to an incorrect answer. To avoid this mistake, always double-check that you've multiplied the term outside the parentheses by every term inside. Another common mistake is incorrectly multiplying the variables. Remember that when you multiply a variable by itself, you're squaring it. For example, q * q equals q², not 2q. Similarly, when you multiply different variables, you simply write them next to each other. For example, p * q equals pq. A third common mistake is combining terms that cannot be combined. You can only combine terms that have the same variables raised to the same power. For example, you cannot combine 21pq and 6q² because they have different variables. To avoid this mistake, always make sure that the terms you're combining have the same variables and exponents. Finally, be careful with the signs. When you're distributing a negative term, make sure to change the signs of all the terms inside the parentheses accordingly. For example, if you're distributing -2 to (x - 3), the result would be -2x + 6, not -2x - 6. By being aware of these common mistakes and taking the time to double-check your work, you can avoid making errors and simplify algebraic expressions with confidence. Remember, practice makes perfect, so keep working on different examples to hone your skills.

Practice Problems: Test Your Knowledge

Now that we've covered the basics and gone through a step-by-step solution, it's time to put your knowledge to the test! Here are a few practice problems for you to try. Remember to apply the distributive property correctly and avoid the common mistakes we discussed earlier. Problem 1: Simplify the expression (4a + 5b) * 2a. Problem 2: Simplify the expression (3x - 2y) * 4y. Problem 3: Simplify the expression (6m + 7n) * 3m. Take your time to work through each problem, and don't be afraid to refer back to the examples and explanations we've covered. Once you've completed the problems, check your answers against the solutions below. Solution 1: (4a + 5b) * 2a = 8a² + 10ab. Solution 2: (3x - 2y) * 4y = 12xy - 8y². Solution 3: (6m + 7n) * 3m = 18m² + 21mn. How did you do? Did you get all the answers correct? If so, congratulations! You've mastered the art of simplifying algebraic expressions using the distributive property. If you made any mistakes, don't worry. Just review the steps and explanations we've covered, and try the problems again. Remember, practice is key to success, so keep working at it until you feel confident in your ability to solve these types of problems. And don't be afraid to ask for help if you get stuck. We're all here to learn and grow together!

Real-World Applications: Where Will You Use This?

You might be wondering, "Okay, I know how to simplify these expressions, but where will I ever use this in the real world?" Well, you'd be surprised! Algebra, and the distributive property in particular, has many practical applications in various fields. One common application is in business and finance. For example, if you're calculating the total cost of buying a certain number of items at a discounted price, you might use the distributive property to simplify the calculation. Another application is in engineering and physics. Many formulas in these fields involve complex expressions that need to be simplified before they can be used. The distributive property can be a valuable tool for simplifying these expressions. In computer science, the distributive property is used in various algorithms and data structures. For example, it can be used to optimize code and improve performance. Even in everyday life, you might use the distributive property without even realizing it. For example, if you're calculating the total cost of buying multiple items at a store, you might use the distributive property to simplify the calculation. So, while it might not seem obvious at first, the distributive property is a fundamental concept that has many practical applications in various fields. By mastering this concept, you'll be well-equipped to solve a wide range of problems in both your personal and professional life. Remember, math is not just about numbers and equations; it's about problem-solving and critical thinking. And the distributive property is a valuable tool that can help you develop these skills.

Conclusion: Mastering Algebraic Simplification

Alright, guys, we've reached the end of our journey into the world of algebraic simplification! We started with a seemingly complex problem: simplifying the expression (7p + 2q) * 3q. But by breaking it down into manageable steps and applying the distributive property, we were able to conquer it with ease. We learned that the distributive property is a powerful tool that allows us to multiply a single term by a group of terms inside parentheses. We also discussed some common mistakes to avoid, such as forgetting to distribute the term outside the parentheses to all the terms inside, incorrectly multiplying the variables, and combining terms that cannot be combined. And finally, we explored some real-world applications of the distributive property, demonstrating its relevance in various fields, from business and finance to engineering and computer science. By mastering the concepts and techniques we've covered in this guide, you'll be well-equipped to simplify a wide range of algebraic expressions. Remember, practice is key to success, so keep working on different examples to hone your skills. And don't be afraid to ask for help if you get stuck. We're all in this together, and with a little bit of effort, you can become a pro at algebraic simplification! So go forth and conquer those expressions, and remember to have fun along the way. Math can be challenging, but it can also be incredibly rewarding. And with the right tools and techniques, you can unlock its full potential.