Simplify 2(-n-3)-7(5+2n): A Step-by-Step Solution
Algebraic expressions can often appear daunting, but with a systematic approach, even complex expressions can be simplified. In this comprehensive guide, we will delve into the process of simplifying the expression 2(-n-3)-7(5+2n), providing a clear and concise breakdown of each step involved. By the end of this article, you will not only have the solution to this specific expression but also gain a solid understanding of the fundamental principles of simplification, empowering you to tackle similar problems with confidence. Understanding the order of operations, the distributive property, and combining like terms are the cornerstones of simplifying algebraic expressions. These concepts will be thoroughly explained and applied throughout this guide, ensuring that you grasp the underlying logic behind each manipulation. Whether you are a student seeking to improve your algebra skills or simply someone looking to refresh your knowledge, this guide will serve as a valuable resource in your mathematical journey. Let's embark on this simplification adventure and unravel the solution to 2(-n-3)-7(5+2n) together.
The Initial Expression: 2(-n-3)-7(5+2n)
Our starting point is the algebraic expression 2(-n-3)-7(5+2n). This expression involves variables, constants, and parentheses, which require careful attention to the order of operations. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. In this case, we will first address the parentheses by applying the distributive property. The distributive property states that a(b+c) = ab + ac, which allows us to multiply a term outside the parentheses by each term inside the parentheses. This is a crucial step in simplifying expressions, as it eliminates the parentheses and allows us to combine like terms later on. Before we delve into the step-by-step simplification, it's important to recognize the structure of the expression. We have two terms, 2(-n-3) and -7(5+2n), each involving multiplication of a constant by a binomial expression within parentheses. The negative sign in front of the second term is crucial and must be carefully considered during the distribution process. A common mistake is to overlook the negative sign, which can lead to an incorrect solution. By meticulously applying the distributive property and the order of operations, we can systematically simplify this expression and arrive at the correct answer. Let's proceed to the next step, where we will apply the distributive property to both terms in the expression.
Applying the Distributive Property
The next key step in simplifying our expression, 2(-n-3)-7(5+2n), is to apply the distributive property. This property, a cornerstone of algebra, allows us to multiply a single term by multiple terms within parentheses. Remember, the distributive property states that a(b + c) = ab + ac. We'll apply this principle to both parts of our expression. First, let's focus on the term 2(-n-3). We need to distribute the 2 to both -n and -3. This means we multiply 2 by -n, resulting in -2n, and then multiply 2 by -3, resulting in -6. So, 2(-n-3) simplifies to -2n - 6. Now, let's move on to the second term, -7(5+2n). Here, we need to distribute the -7 to both 5 and 2n. It's crucial to pay attention to the negative sign. Multiplying -7 by 5 gives us -35, and multiplying -7 by 2n gives us -14n. Therefore, -7(5+2n) simplifies to -35 - 14n. By carefully applying the distributive property, we've successfully removed the parentheses from our expression. We've transformed 2(-n-3) into -2n - 6 and -7(5+2n) into -35 - 14n. Our expression now looks like this: -2n - 6 - 35 - 14n. This is a significant step forward, as we've eliminated the parentheses and are now ready to combine like terms. However, it is important to emphasize accuracy in this step, ensuring correct multiplications and the proper handling of the negative sign, as even a minor error can lead to an incorrect final result. The next step involves identifying and combining like terms, which will further simplify our expression.
Combining Like Terms
After successfully applying the distributive property, our expression now stands as -2n - 6 - 35 - 14n. The next crucial step in simplification is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have two types of terms: terms with the variable 'n' and constant terms (numbers without variables). The terms with 'n' are -2n and -14n. To combine them, we simply add their coefficients (the numbers in front of the variable). In this case, we add -2 and -14, which gives us -16. So, -2n - 14n simplifies to -16n. Next, we need to combine the constant terms, which are -6 and -35. Adding these two numbers together, we get -6 - 35 = -41. Therefore, the constant terms combine to -41. Now that we've combined the 'n' terms and the constant terms, we can rewrite the expression. Our simplified expression is -16n - 41. This process of combining like terms is a fundamental aspect of algebraic simplification. It allows us to reduce the number of terms in an expression, making it more concise and easier to work with. By carefully identifying like terms and adding their coefficients, we can streamline complex expressions into simpler forms. It's essential to double-check your work at this stage to ensure that you've correctly identified and combined all like terms. A common mistake is to overlook a term or to incorrectly add the coefficients. By paying close attention to detail, you can confidently move on to the final step, which is presenting the simplified expression.
The Simplified Expression: -16n - 41
Having meticulously applied the distributive property and combined like terms, we have successfully simplified the expression 2(-n-3)-7(5+2n). Our final result is -16n - 41. This simplified expression is equivalent to the original expression, meaning it will yield the same value for any given value of 'n'. However, the simplified form is much more concise and easier to understand. We started with a more complex expression involving parentheses and multiple terms, and through a step-by-step process, we have reduced it to a simple binomial expression with just two terms. This process of simplification is not just about finding the correct answer; it's also about developing a deeper understanding of algebraic manipulation. By mastering these techniques, you can confidently tackle a wide range of algebraic problems. It's important to remember that each step in the simplification process plays a crucial role. From applying the distributive property to combining like terms, every action contributes to the final result. Accuracy and attention to detail are paramount, as even a small error can propagate through the steps and lead to an incorrect answer. Therefore, it's always a good practice to double-check your work, especially when dealing with negative signs and multiple terms. In conclusion, the simplified form of the expression 2(-n-3)-7(5+2n) is -16n - 41. This result demonstrates the power of algebraic simplification in making complex expressions more manageable and understandable. With a solid grasp of the fundamental principles, you can confidently approach similar problems and achieve accurate solutions.
Simplify the expression to create an equivalent expression. Which of the following is the correct simplified form? (A) (B) (C) (D)
Simplify Algebraic Expressions: Step-by-Step Solution for 2(-n-3)-7(5+2n)
