Simplifying (5a-2b+4c)+(-a-3b+8c) A Step-by-Step Solution
In the realm of mathematics, simplifying expressions is a fundamental skill. This article will serve as a comprehensive guide to solving the expression (5a-2b+4c)+(-a-3b+8c). We will delve into the step-by-step process, ensuring a clear understanding of each operation involved. This exploration is beneficial for students learning algebra, individuals seeking to refresh their mathematical skills, and anyone interested in the logical progression of mathematical problem-solving. By the end of this guide, you will not only know the answer but also comprehend the underlying principles that make it possible. So, let’s embark on this mathematical journey together and unlock the solution to this intriguing expression.
Demystifying the Expression: (5a-2b+4c)+(-a-3b+8c)
Let's begin by dissecting the given expression: (5a-2b+4c)+(-a-3b+8c). This expression involves combining like terms, which are terms that have the same variable raised to the same power. In this case, we have terms with the variables 'a', 'b', and 'c'. The constants associated with these variables are known as coefficients. To simplify the expression, we need to group the like terms together and then perform the addition or subtraction as indicated. This process of simplifying algebraic expressions is a cornerstone of algebra, allowing us to manipulate and solve equations effectively. Understanding this principle is crucial for tackling more complex mathematical problems in the future. Moreover, this skill is not only applicable in academic settings but also in real-world scenarios where problem-solving and analytical thinking are required. From calculating finances to designing structures, the ability to simplify expressions is a valuable asset. So, let's break down this expression and understand how to combine these like terms.
Step-by-Step Solution: Combining Like Terms
The core of simplifying the expression (5a-2b+4c)+(-a-3b+8c) lies in the careful combination of like terms. This involves several key steps, each crucial for arriving at the correct solution. First, we identify the like terms: the 'a' terms (5a and -a), the 'b' terms (-2b and -3b), and the 'c' terms (4c and 8c). Once identified, we group these terms together, making it visually clear which terms need to be combined. The next step is to perform the addition or subtraction on the coefficients of these like terms. For instance, we combine 5a and -a by adding their coefficients (5 and -1), resulting in 4a. Similarly, we combine -2b and -3b, adding their coefficients (-2 and -3) to get -5b. Finally, we combine 4c and 8c, adding their coefficients (4 and 8) to get 12c. This process might seem straightforward, but it requires careful attention to detail, especially when dealing with negative signs. A common mistake is to incorrectly add or subtract coefficients, leading to an incorrect simplified expression. By following these steps meticulously, we ensure an accurate and simplified form of the original expression. This methodical approach is not just applicable to this specific problem but can be applied to a wide range of algebraic expressions, making it a fundamental skill in mathematics.
Detailed Breakdown: The Arithmetic Behind Each Term
To fully grasp the solution to (5a-2b+4c)+(-a-3b+8c), we need to delve into the arithmetic behind each term. This involves a closer look at how the coefficients of the like terms interact. Let’s start with the 'a' terms: 5a and -a. Here, we are essentially adding 5a and -1a. The sum of the coefficients 5 and -1 is 4, hence we get 4a. Moving on to the 'b' terms, we have -2b and -3b. This is akin to adding -2 and -3, which results in -5. Therefore, the combined 'b' term is -5b. Finally, for the 'c' terms, we have 4c and 8c. Adding the coefficients 4 and 8 gives us 12, leading to the combined term 12c. It's important to note the role of the negative signs in these operations. A common error is to misinterpret the subtraction as addition or vice versa. For example, -2b + (-3b) is often mistakenly calculated as -2b + 3b. Understanding the rules of integer addition and subtraction is crucial for accurate simplification. This detailed breakdown not only clarifies the arithmetic involved but also reinforces the importance of precision in mathematical calculations. By carefully analyzing each step and understanding the underlying principles, we can avoid common mistakes and confidently simplify algebraic expressions.
The Final Solution: Putting It All Together
After meticulously combining like terms and understanding the arithmetic behind each step, we arrive at the final solution to the expression (5a-2b+4c)+(-a-3b+8c). We found that the combined 'a' terms result in 4a, the combined 'b' terms give us -5b, and the combined 'c' terms yield 12c. Therefore, the simplified expression is 4a - 5b + 12c. This is the most concise and simplified form of the original expression. The journey from the initial expression to this final solution highlights the power of algebraic manipulation. By systematically applying the rules of combining like terms, we have transformed a seemingly complex expression into a simpler, more manageable form. This ability to simplify expressions is a cornerstone of algebra and is essential for solving equations and tackling more advanced mathematical problems. Moreover, this process demonstrates the importance of accuracy and attention to detail in mathematics. Each step, from identifying like terms to performing the arithmetic, requires careful execution to avoid errors. The final solution, 4a - 5b + 12c, is not just an answer; it's a testament to the logical and systematic approach that underpins mathematical problem-solving.
Verifying the Solution: Ensuring Accuracy
In mathematics, arriving at a solution is not the end of the journey. It's equally important to verify the solution to ensure its accuracy. For the expression (5a-2b+4c)+(-a-3b+8c), which we simplified to 4a - 5b + 12c, there are several ways to verify the correctness of our solution. One common method is to substitute specific numerical values for the variables 'a', 'b', and 'c' in both the original expression and the simplified expression. If the results match, it provides strong evidence that our simplification is correct. For instance, we could let a = 1, b = 2, and c = 3. Substituting these values into the original expression gives us (5(1)-2(2)+4(3))+(-1-3(2)+8(3)) = (5-4+12)+(-1-6+24) = 13 + 17 = 30. Substituting the same values into the simplified expression 4a - 5b + 12c yields 4(1) - 5(2) + 12(3) = 4 - 10 + 36 = 30. Since both expressions evaluate to the same value, it increases our confidence in the solution. However, it's crucial to note that verifying with a single set of values doesn't guarantee correctness for all possible values. To be absolutely certain, one might consider substituting multiple sets of values or employing alternative algebraic methods to double-check the simplification process. This emphasis on verification underscores the importance of rigor and precision in mathematical problem-solving. By taking the time to verify our solutions, we not only ensure accuracy but also deepen our understanding of the underlying mathematical principles.
Real-World Applications: Where This Skill Matters
The ability to simplify expressions like (5a-2b+4c)+(-a-3b+8c) extends far beyond the classroom. This skill is applicable in a multitude of real-world scenarios, making it a valuable asset in various professions and everyday situations. In fields like engineering and physics, simplifying complex equations is a routine task. Engineers use algebraic simplification to design structures, analyze circuits, and model physical systems. Physicists employ these techniques to solve problems related to motion, energy, and forces. In the realm of finance, simplifying expressions is crucial for budgeting, investment analysis, and calculating financial returns. Accountants use algebraic skills to balance financial statements and manage business finances. Even in seemingly simple everyday tasks, the ability to simplify expressions can be beneficial. For example, when planning a home renovation project, one might need to calculate the total cost of materials, which often involves combining different quantities and prices. Similarly, when cooking, adjusting recipe quantities requires understanding how to combine and simplify measurements. This highlights the pervasive nature of algebraic thinking and its relevance in various aspects of life. Mastering the skill of simplifying expressions not only enhances one's mathematical abilities but also equips individuals with a valuable problem-solving tool that can be applied across a wide range of disciplines and situations. The benefits of this skill extend beyond academic success, fostering critical thinking and analytical abilities that are essential for navigating the complexities of the modern world.
Conclusion: Mastering the Art of Simplification
In conclusion, the journey through simplifying the expression (5a-2b+4c)+(-a-3b+8c) has been a testament to the power and elegance of algebraic manipulation. We began by dissecting the expression, identifying like terms, and understanding the arithmetic behind each step. We then meticulously combined these terms, arriving at the simplified solution of 4a - 5b + 12c. Throughout this process, we emphasized the importance of accuracy, attention to detail, and the systematic application of mathematical principles. We also explored methods for verifying the solution, ensuring its correctness and solidifying our understanding. Furthermore, we delved into the real-world applications of this skill, highlighting its relevance in various professions and everyday scenarios. From engineering and finance to cooking and home renovation, the ability to simplify expressions proves to be a valuable asset. Mastering this skill not only enhances mathematical proficiency but also cultivates critical thinking and problem-solving abilities. The art of simplification, as demonstrated through this example, is not merely about finding an answer; it's about developing a logical and methodical approach to problem-solving. It's about transforming complex situations into manageable ones, and it's a skill that empowers individuals to navigate the challenges of the world with confidence and competence. Therefore, the ability to simplify expressions is a fundamental skill that benefits individuals across various facets of life, fostering both academic and practical success.
