Simplifying Expressions Using The Distributive Property -7(8m+10)

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Introduction

In the realm of mathematics, simplifying expressions is a fundamental skill that paves the way for solving more complex equations and problems. One of the most powerful tools in our arsenal for achieving this simplification is the distributive property. This property allows us to elegantly handle expressions involving parentheses and multiplication, and it's a cornerstone of algebraic manipulation. In this comprehensive guide, we will delve deep into the distributive property, explore its applications, and demonstrate how to use it effectively to simplify expressions. Our specific focus will be on expressions like -7(8m + 10), where we need to distribute a negative number across a sum. Mastering this skill is essential for anyone looking to excel in algebra and beyond.

Understanding the Distributive Property

The distributive property is a mathematical principle that allows us to multiply a single term by two or more terms inside a set of parentheses. The essence of this property lies in its ability to distribute the multiplication operation over addition or subtraction. Mathematically, it can be expressed as follows:

  • a(b + c) = ab + ac
  • a(b - c) = ab - ac

Here, 'a' represents the term being distributed, while 'b' and 'c' are the terms inside the parentheses. The distributive property states that multiplying 'a' by the sum (or difference) of 'b' and 'c' is equivalent to multiplying 'a' by each term individually and then adding (or subtracting) the results. This seemingly simple concept is incredibly powerful, enabling us to break down complex expressions into more manageable parts. For instance, consider the expression 3(x + 2). Applying the distributive property, we multiply 3 by both 'x' and '2', resulting in 3x + 6. This transformed expression is often easier to work with, especially when solving equations or simplifying further expressions. The distributive property is not just a mathematical trick; it's a fundamental principle that reflects the way multiplication interacts with addition and subtraction, making it an indispensable tool in algebra and beyond. To solidify your understanding, let's consider a numerical example: 5(4 + 3). We can either add 4 and 3 first to get 7, then multiply by 5 to get 35. Or, we can distribute the 5: 5 * 4 + 5 * 3 = 20 + 15 = 35. Both methods yield the same result, illustrating the distributive property in action. Understanding this property is crucial for tackling more complex algebraic expressions and equations.

Applying the Distributive Property to -7(8m + 10)

Now, let's apply the distributive property to the expression -7(8m + 10). This expression presents a classic scenario where the distributive property is the key to simplification. The -7 outside the parentheses needs to be multiplied by both terms inside the parentheses: 8m and 10. It's crucial to pay close attention to the negative sign, as it will affect the signs of the resulting terms. Following the distributive property, we multiply -7 by 8m and then -7 by 10. When multiplying -7 by 8m, we multiply the coefficients (-7 and 8) and keep the variable 'm'. This gives us -7 * 8m = -56m. Next, we multiply -7 by 10, which results in -7 * 10 = -70. Now, we combine these two results: -56m and -70. The simplified expression is the sum of these two terms. Therefore, -7(8m + 10) simplifies to -56m - 70. This step-by-step breakdown illustrates how the distributive property allows us to transform a product of a term and a sum into a sum of individual products. The careful handling of the negative sign is paramount to arriving at the correct simplified expression. This example underscores the importance of the distributive property in simplifying algebraic expressions and highlights its role in making complex problems more manageable. By mastering this technique, you'll be well-equipped to tackle a wide range of algebraic challenges.

Step-by-Step Solution for -7(8m + 10)

To further clarify the application of the distributive property, let's break down the solution for -7(8m + 10) into a detailed step-by-step process:

  1. Identify the term to be distributed: In this case, the term to be distributed is -7. It's the number outside the parentheses that needs to be multiplied by each term inside.
  2. Multiply -7 by the first term inside the parentheses (8m): This gives us -7 * 8m. To perform this multiplication, multiply the coefficients: -7 * 8 = -56. The variable 'm' remains as it is. So, -7 * 8m = -56m.
  3. Multiply -7 by the second term inside the parentheses (10): This gives us -7 * 10. Performing this multiplication, we get -7 * 10 = -70. Remember that a negative number multiplied by a positive number results in a negative number.
  4. Combine the results: Now, we combine the results from steps 2 and 3. We have -56m and -70. The simplified expression is the sum of these two terms: -56m + (-70). Since adding a negative number is the same as subtraction, we can write this as -56m - 70.
  5. Final Simplified Expression: Therefore, the simplified expression for -7(8m + 10) is -56m - 70. This step-by-step solution provides a clear and concise roadmap for applying the distributive property. Each step is carefully explained, ensuring a thorough understanding of the process. By following these steps, you can confidently simplify similar expressions and avoid common errors. The key takeaway is to systematically distribute the term outside the parentheses to each term inside, paying close attention to the signs. This methodical approach will lead to accurate and efficient simplification of algebraic expressions.

Common Mistakes to Avoid

While the distributive property is a straightforward concept, there are common mistakes that students often make when applying it. Being aware of these pitfalls can help you avoid them and ensure accurate simplification. One of the most frequent errors is neglecting to distribute the term to all terms inside the parentheses. For example, in the expression -7(8m + 10), some might only multiply -7 by 8m and forget to multiply it by 10. This leads to an incomplete and incorrect simplification. Always remember that the term outside the parentheses must be multiplied by every term inside. Another common mistake involves sign errors. When distributing a negative number, it's crucial to pay close attention to the signs of the resulting terms. For instance, in our example, -7 multiplied by +10 results in -70, not +70. A simple way to avoid this is to treat the negative sign as part of the term being distributed. A third mistake is incorrectly combining terms after distribution. Remember that you can only combine like terms – terms with the same variable raised to the same power. In the simplified expression -56m - 70, -56m and -70 are not like terms because one has the variable 'm' and the other is a constant. Therefore, they cannot be combined further. To avoid these mistakes, it's helpful to write out each step explicitly, paying close attention to the distribution and the signs. Double-checking your work and practicing regularly can also significantly reduce the likelihood of errors. By being mindful of these common pitfalls, you can master the distributive property and simplify expressions with confidence.

Practice Problems

To solidify your understanding of the distributive property, let's work through some practice problems. These examples will allow you to apply the concepts we've discussed and build your confidence in simplifying expressions. Remember to follow the step-by-step approach we outlined earlier, paying close attention to the signs and ensuring that you distribute the term to all terms inside the parentheses. Here are a few problems to get you started:

  1. 5(2x - 3)
  2. -4(3y + 7)
  3. 2(6a - 4b + 1)
  4. -9(-z + 5)
  5. 3(4m + 2n - 8)

For each problem, start by identifying the term to be distributed. Then, multiply that term by each term inside the parentheses. Be careful with the signs! Finally, combine any like terms. Let's take the first problem, 5(2x - 3), as an example. We distribute the 5 to both 2x and -3. 5 * 2x = 10x, and 5 * -3 = -15. So, the simplified expression is 10x - 15. Now, try the other problems on your own. Don't hesitate to refer back to the step-by-step solution and the common mistakes section if you need a refresher. The key to mastering the distributive property is practice, so the more problems you work through, the more comfortable and confident you'll become. After you've attempted these problems, you can seek out additional exercises online or in textbooks to further hone your skills. Remember, the distributive property is a fundamental tool in algebra, and proficiency in its application will serve you well in more advanced mathematical concepts.

Conclusion

The distributive property is an indispensable tool in the world of algebra, empowering us to simplify complex expressions with ease and precision. In this guide, we've explored the essence of this property, demonstrating its application to expressions like -7(8m + 10). We've dissected the process into clear, manageable steps, highlighting the importance of careful distribution and sign management. By understanding the distributive property and practicing its application, you gain a crucial skill that unlocks doors to more advanced mathematical concepts. We've also addressed common pitfalls to avoid, ensuring a smoother path to accurate simplification. Remember, consistent practice is the key to mastery. Work through the provided practice problems and seek out additional exercises to solidify your understanding. The ability to simplify expressions effectively is not just a mathematical skill; it's a problem-solving tool that can be applied in various contexts. As you continue your mathematical journey, the distributive property will serve as a reliable companion, helping you navigate the complexities of algebra and beyond. So, embrace this powerful tool, practice diligently, and watch your mathematical abilities flourish.