Alright, guys, let's dive into how to solve this math problem: 7p - 2q x 3q. 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. Grab your pencils and paper, and let's get started!

Understanding the Basics

Before we jump into the actual calculation, let’s quickly refresh some basic mathematical principles. The most important thing to remember here is the order of operations, often remembered by the acronym PEMDAS (or BODMAS in some regions). This stands for:

• Parentheses (or Brackets)
• Exponents (or Orders)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Following this order ensures that we solve the equation correctly. In our case, we have subtraction and multiplication, so we'll tackle the multiplication first.

Additionally, remember that when we multiply terms with variables (like q), we multiply the coefficients (the numbers in front of the variables) and keep the variables. If the variables are the same, we can combine them using exponent rules. For example, q x q becomes q². Keeping these basics in mind will make solving the problem much smoother.

Breaking Down the Problem

Now, let's apply these concepts to our specific problem: 7p - 2q x 3q. According to PEMDAS, we need to perform the multiplication before the subtraction. So, we'll start by multiplying -2q by 3q.

When multiplying -2q by 3q, we multiply the coefficients (-2 and 3) and the variables (q and q). So, -2 multiplied by 3 is -6, and q multiplied by q is q². Therefore, -2q x 3q equals -6q².

Now our equation looks like this: 7p - 6q². This is much simpler, right? The next step is to understand if we can simplify this further. Since 7p and -6q² are different terms (one has the variable p and the other has q²), we cannot combine them. They are not like terms.

Think of it like trying to add apples and oranges – you can’t just say you have a combined number of “apple-oranges.” Similarly, in algebra, you can only combine terms that have the exact same variable and exponent. Since 7p and -6q² are distinct terms, we leave them as they are.

So, the final simplified expression is 7p - 6q². This is the result of our calculation. Remember, we followed the order of operations, performed the multiplication, and then recognized that we couldn't simplify the expression any further because the terms were different.

Step-by-Step Solution

Let’s go through the solution again, step by step, to make sure we’ve got it all clear:

1. Identify the operation: We have 7p - 2q x 3q.
2. Follow PEMDAS/BODMAS: Multiplication comes before subtraction.
3. Multiply -2q x 3q: (-2 x 3) x (q x q) = -6q².
4. Rewrite the equation: 7p - 6q².
5. Check for like terms: 7p and -6q² are not like terms.
6. Final Result: 7p - 6q².

So, there you have it! The solution to 7p - 2q x 3q is 7p - 6q². Wasn't that fun? Now you can confidently tackle similar algebraic problems.

Common Mistakes to Avoid

When working with algebraic expressions like this, it's easy to make a few common mistakes. Let's go over them so you can avoid these pitfalls:

• Forgetting the Order of Operations: This is the most common mistake. Always remember PEMDAS/BODMAS. Multiply and divide before you add or subtract.
• Combining Unlike Terms: You can only combine terms that have the same variable and exponent. For example, you can combine 3x and 5x to get 8x, but you can’t combine 3x and 5x².
• Sign Errors: Pay close attention to negative signs. Make sure you multiply them correctly. For example, -2q x 3q = -6q², not 6q².
• Incorrectly Applying Exponents: Remember that q x q = q², not 2q. The exponent indicates how many times you multiply the base by itself.
• Skipping Steps: It's tempting to rush through the problem, but taking it one step at a time helps prevent errors.

By keeping these common mistakes in mind, you'll be well on your way to solving algebraic expressions accurately and efficiently.

Real-World Applications

You might be wondering, “Where would I ever use this in real life?” Well, algebraic expressions like 7p - 2q x 3q actually have many practical applications in various fields. Here are a few examples:

• Engineering: Engineers use algebraic expressions to design structures, calculate forces, and model systems. For example, they might use similar equations to determine the stress on a bridge or the flow rate in a pipe.
• Economics: Economists use algebraic expressions to model economic behavior, predict market trends, and analyze data. For instance, they might use equations to calculate the supply and demand equilibrium or to model the growth of a company.
• Computer Science: Computer scientists use algebraic expressions to write algorithms, optimize code, and develop software. For example, they might use similar equations to calculate the efficiency of an algorithm or to model the behavior of a computer network.
• Physics: Physicists use algebraic expressions to describe the laws of nature, model physical phenomena, and make predictions. For example, they might use equations to calculate the trajectory of a projectile or to model the behavior of a quantum particle.
• Everyday Life: Even in everyday life, you might use algebraic thinking without realizing it. For example, when you’re calculating the cost of items at a store, figuring out how much to tip at a restaurant, or planning a budget, you’re using algebraic principles.

So, while it might seem abstract, algebra is a powerful tool that can help you solve problems and make decisions in a wide range of situations. Understanding how to manipulate and simplify algebraic expressions like 7p - 2q x 3q is a valuable skill that can benefit you in many ways.

Practice Problems

To really master this concept, let's try a few practice problems. Grab your pen and paper, and give these a shot:

1. Simplify: 5a + 3b x 2b
2. Simplify: 10x - 4y x y
3. Simplify: 8m + m x 6n
4. Simplify: 12p - 3q x 4q
5. Simplify: 20r + 5s x 2s

Take your time, follow the steps we discussed, and remember to avoid those common mistakes. The answers are below, but try to solve them on your own first!

Solutions to Practice Problems

Here are the solutions to the practice problems:

1. 5a + 6b²
2. 10x - 4y²
3. 8m + 6mn
4. 12p - 12q²
5. 20r + 10s²

How did you do? If you got them all right, congratulations! You've got a solid understanding of how to simplify expressions like 7p - 2q x 3q. If you missed a few, don't worry. Just go back and review the steps, paying close attention to the order of operations and the rules for combining like terms.

Conclusion

In summary, solving 7p - 2q x 3q involves understanding the order of operations (PEMDAS/BODMAS), performing multiplication before subtraction, and recognizing that you can only combine like terms. The final simplified expression is 7p - 6q². Remember to avoid common mistakes, such as forgetting the order of operations or combining unlike terms.

Algebraic expressions like this have many real-world applications in fields such as engineering, economics, computer science, and physics. By mastering these concepts, you'll be well-equipped to tackle more complex problems and make informed decisions in various aspects of life.

Keep practicing, stay curious, and don't be afraid to ask questions. You've got this!