Let's break down the expression 7p - 2q x 3q step by step to understand how to simplify it properly. In mathematics, simplifying expressions involves combining like terms and performing operations in the correct order. This process makes the expression easier to understand and work with, especially when solving equations or evaluating formulas. Whether you're a student learning algebra or just refreshing your math skills, understanding simplification is crucial. We'll use the order of operations (PEMDAS/BODMAS) to guide us: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). By following these rules, we can ensure an accurate and understandable simplification.

Understanding the Initial Expression

The expression we're starting with is 7p - 2q x 3q. This expression involves two variables, p and q, and contains both subtraction and multiplication operations. Before we dive into the steps, let’s make sure we understand what each part of the expression means. The term 7p means 7 times p, and 2q means 2 times q. The multiplication sign x indicates that we need to multiply 2q by 3q. Remember, in algebraic expressions, the order of operations is crucial. Multiplication must be done before subtraction. This order ensures that we follow a consistent and logical approach to simplifying the expression. Ignoring this order would lead to an incorrect result. The variables p and q represent unknown values. Simplifying the expression helps us manipulate these variables in a meaningful way, which is particularly useful when we need to solve for p or q later on. Understanding the basic components of the expression is the first step toward successfully simplifying it.

Step-by-Step Simplification

Multiplication First: 2q x 3q

According to the order of operations, we need to perform the multiplication before the subtraction. So, let's multiply 2q by 3q. When multiplying these terms, we multiply the coefficients (the numbers in front of the variables) and then multiply the variables themselves. Therefore, 2q x 3q becomes (2 x 3) x (q x q), which simplifies to 6q². This is because when you multiply q by q, you get q squared (q²). This step is critical because it reduces the multiplication part of the expression into a single term, making the subsequent subtraction step much easier to manage. Remember, the laws of exponents tell us that when we multiply variables with the same base, we add their exponents. In this case, q has an exponent of 1, so q¹ x q¹ = q¹⁺¹ = q². Understanding how to multiply variables correctly is essential for simplifying algebraic expressions accurately.

Substituting Back into the Expression

Now that we've simplified 2q x 3q to 6q², we substitute this back into our original expression. The original expression was 7p - 2q x 3q. After substituting, it becomes 7p - 6q². This substitution is a key step because it replaces the multiplication operation with a simplified term, which prepares us for the final step: checking if there are any like terms to combine. By accurately substituting the simplified term back into the expression, we ensure that our final simplified expression is correct and mathematically sound. This process of substitution is common in algebra and is used extensively to solve more complex equations and expressions. It's like putting the pieces of a puzzle together – each step brings us closer to the final solution.

Final Simplified Expression

After performing the multiplication and substituting the result back into the original expression, we are left with 7p - 6q². Now, we examine this expression to see if there are any like terms that can be combined. In this case, 7p and 6q² are not like terms because they contain different variables (p and q). Like terms must have the same variable raised to the same power. Since we cannot combine these terms any further, the simplified form of the expression 7p - 2q x 3q is simply 7p - 6q². This is our final answer. Understanding when terms can and cannot be combined is crucial for simplifying expressions correctly. This step ensures that we have reduced the expression to its simplest and most understandable form. This final simplified expression can now be used in further calculations or to solve for specific values of p and q, depending on the context of the problem.

Practical Applications

Understanding how to simplify expressions like 7p - 2q x 3q is extremely useful in various real-world scenarios. For example, in engineering, you might use such expressions to model the behavior of systems or components. By simplifying these expressions, engineers can make calculations more manageable and design more efficient solutions. Similarly, in computer science, simplified expressions can help optimize algorithms and improve the performance of software. In economics, these skills can be applied to simplify complex models and make accurate predictions. Whether you are calculating the cost of materials for a project or analyzing data to make informed decisions, the ability to simplify expressions is a valuable tool. Moreover, in academic settings, mastering simplification techniques is essential for success in algebra, calculus, and other advanced mathematics courses. By practicing and applying these concepts, you can build a solid foundation for future learning and problem-solving in a variety of fields.

Common Mistakes to Avoid

When simplifying expressions like 7p - 2q x 3q, it's easy to make mistakes if you're not careful. One common error is to perform the subtraction before the multiplication. Remember, the order of operations (PEMDAS/BODMAS) dictates that multiplication must be done before subtraction. Doing it the other way around will lead to an incorrect result. Another mistake is incorrectly multiplying the variables. For example, some people might forget that q x q equals q², not just q. Additionally, watch out for sign errors, especially when dealing with negative numbers. Always double-check your calculations to ensure you haven't made any mistakes with the signs. It's also important to remember that you can only combine like terms. Don't try to combine terms that have different variables or different powers of the same variable. To avoid these mistakes, it's helpful to write out each step clearly and double-check your work as you go along. Practicing regularly and reviewing your work can also help you identify and correct any errors. By being aware of these common pitfalls, you can improve your accuracy and confidence when simplifying expressions.

Further Practice

To master the art of simplifying expressions like 7p - 2q x 3q, consistent practice is key. Try working through a variety of similar problems to reinforce your understanding of the order of operations and the rules of algebra. You can find practice problems in textbooks, online resources, or from your teacher. Start with simpler expressions and gradually work your way up to more complex ones. Pay close attention to the steps involved in each problem, and be sure to double-check your work to avoid making mistakes. Another helpful strategy is to work with a study group or tutor. Explaining the concepts to others can help solidify your own understanding, and you can learn from their insights and perspectives. Additionally, consider using online tools or apps that provide step-by-step solutions to algebraic problems. These tools can be a great way to check your work and identify areas where you need more practice. Remember, the more you practice, the more comfortable and confident you will become with simplifying expressions. With dedication and perseverance, you can develop the skills you need to succeed in algebra and beyond. So, keep practicing, and don't be afraid to ask for help when you need it!

Conclusion

In summary, simplifying the expression 7p - 2q x 3q involves understanding and applying the correct order of operations. By first performing the multiplication of 2q x 3q, which results in 6q², and then substituting this back into the original expression, we arrive at the simplified form: 7p - 6q². This final expression cannot be simplified further because 7p and 6q² are not like terms. Remember, the key to simplifying algebraic expressions is to follow the rules of PEMDAS/BODMAS and to combine only like terms. This process is not just a mathematical exercise; it is a fundamental skill that has numerous applications in various fields, from engineering to economics. By mastering these techniques, you can enhance your problem-solving abilities and improve your understanding of the world around you. So, keep practicing, stay curious, and continue to explore the fascinating world of mathematics!