Simplifying Algebraic Expressions A Step-by-Step Guide To (6x^4y^2 - 3xy^3 - 4xy) + (4x^4y^2 - Xy^3 + 4xy)

Introduction

In this article, we will delve into the process of simplifying algebraic expressions, specifically focusing on the given expression: $(6x^4y^2 - 3xy^3 - 4xy) + (4x^4y^2 - xy^3 + 4xy)$. Simplifying expressions is a fundamental skill in algebra, crucial for solving equations, understanding mathematical relationships, and making complex problems more manageable. We will break down the steps involved, explaining the underlying principles and techniques to ensure a clear understanding. This comprehensive guide aims to not only provide the solution but also to enhance your algebraic manipulation skills. We will cover the basics of combining like terms, paying close attention to the signs and coefficients, and provide detailed explanations to help you grasp each step. By the end of this article, you will be equipped with the knowledge and confidence to tackle similar algebraic simplification problems.

Understanding the Basics of Algebraic Expressions

Before we dive into the specifics of the given expression, it's crucial to establish a firm understanding of the foundational concepts of algebraic expressions. Algebraic expressions are combinations of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables, typically represented by letters like x and y, symbolize unknown values, while constants are fixed numerical values. The terms within an algebraic expression are the individual components separated by addition or subtraction. For example, in the expression $6x^4y^2 - 3xy^3 - 4xy$, the terms are $6x^4y^2$, $-3xy^3$, and $-4xy$. Understanding the structure of these expressions is the first step towards simplifying them effectively. Each term comprises a coefficient (the numerical factor) and a variable part (the variables and their exponents). In the term $6x^4y^2$, 6 is the coefficient, and $x^4y^2$ is the variable part. The exponent indicates the power to which the variable is raised, and it plays a critical role when combining like terms. We will see how these concepts come into play as we simplify the target expression, emphasizing the importance of recognizing and manipulating like terms to reach the simplified form. Mastering these basics not only helps in solving this particular problem but also lays a strong foundation for more advanced algebraic concepts.

Identifying Like Terms

The cornerstone of simplifying algebraic expressions lies in the ability to identify like terms. Like terms are terms that have the same variables raised to the same powers. In other words, they can only differ in their coefficients. For instance, $3x^2$ and $-5x^2$ are like terms because they both contain the variable x raised to the power of 2. On the other hand, $3x^2$ and $3x^3$ are not like terms because the exponents of x are different. Similarly, $2xy$ and $5yx$ are like terms due to the commutative property of multiplication, which allows us to rearrange the variables without changing the term's value. However, $2xy$ and $2x^2y$ are not like terms as the exponents of x are different. When dealing with expressions involving multiple variables, it is crucial to ensure that all variables and their respective exponents match for terms to be considered alike. This careful attention to detail is key to avoiding errors in simplification. Recognizing like terms allows us to combine them, which is the essence of simplifying an algebraic expression. We'll apply this concept to the given expression, systematically identifying and grouping like terms to pave the way for simplification. Accurate identification of like terms ensures that we are combining only those terms that can be combined, maintaining the integrity of the expression throughout the simplification process.

Step-by-Step Simplification Process

Now, let's embark on the step-by-step simplification process of the given expression: $(6x^4y^2 - 3xy^3 - 4xy) + (4x^4y^2 - xy^3 + 4xy)$. The first crucial step in simplifying any algebraic expression is to remove the parentheses. In this case, since we are adding the two expressions, we can simply rewrite the expression without the parentheses, preserving the signs of each term: $6x^4y^2 - 3xy^3 - 4xy + 4x^4y^2 - xy^3 + 4xy$. This step eliminates any potential confusion arising from the order of operations and sets the stage for combining like terms. Next, we need to identify and group the like terms. From the expression, we can see that $6x^4y^2$ and $4x^4y^2$ are like terms, $-3xy^3$ and $-xy^3$ are like terms, and $-4xy$ and $+4xy$ are like terms. Grouping these terms together helps visualize the combination process. After identifying the like terms, we combine them by adding or subtracting their coefficients while keeping the variable part the same. This is the heart of simplification, where we reduce the number of terms and make the expression more concise. We will meticulously combine the coefficients, paying close attention to the signs, to ensure accuracy. Each step in this process is designed to methodically reduce the complexity of the expression, ultimately leading to the simplest form.

Combining Like Terms in Detail

The pivotal step in simplifying the expression $(6x^4y^2 - 3xy^3 - 4xy) + (4x^4y^2 - xy^3 + 4xy)$ is the detailed combination of like terms. We've already identified the like terms, now let's combine them meticulously. First, we focus on the terms with $x^4y^2$: $6x^4y^2 + 4x^4y^2$. To combine these, we add their coefficients: $6 + 4 = 10$. Thus, the combined term is $10x^4y^2$. This process demonstrates the fundamental principle of combining like terms – adding or subtracting the coefficients while keeping the variable part unchanged. Next, we address the terms with $xy^3$: $-3xy^3 - xy^3$. Here, we are adding two negative coefficients: $-3 + (-1) = -4$. So, the combined term is $-4xy^3$. It's crucial to remember that when a term has no explicit coefficient, it is understood to be 1. This is a common point of oversight that can lead to errors. Finally, we combine the terms with $xy$: $-4xy + 4xy$. In this case, we have $-4 + 4 = 0$. This results in the term $0xy$, which is simply 0. When terms cancel each other out in this way, they disappear from the simplified expression. By systematically combining each set of like terms, we ensure that every part of the expression is simplified correctly. This step-by-step approach minimizes the risk of errors and highlights the importance of careful arithmetic within algebraic manipulation. The result of this detailed combination will give us the simplified form of the original expression.

The Final Simplified Expression

After meticulously combining like terms, we arrive at the final simplified expression. From the previous steps, we have: $6x^4y^2 + 4x^4y^2 = 10x^4y^2$, $-3xy^3 - xy^3 = -4xy^3$, and $-4xy + 4xy = 0$. Now, we gather these simplified terms to form the final expression. The simplified expression is $10x^4y^2 - 4xy^3$. This expression is the most concise form of the original expression, achieved by combining all like terms. It is essential to present the final answer in a clear and organized manner, which helps in both understanding and verification. The process of simplification has reduced the complexity of the original expression, making it easier to work with in subsequent mathematical operations or analyses. This final form highlights the power of algebraic manipulation in streamlining expressions and revealing their essential components. By successfully simplifying this expression, we have demonstrated a key skill in algebra that is fundamental to solving equations and tackling more complex mathematical problems. The simplified expression, $10x^4y^2 - 4xy^3$, is the culmination of our step-by-step approach, showcasing the efficiency and elegance of algebraic simplification.

Conclusion

In conclusion, we have successfully navigated the process of simplifying the algebraic expression $(6x^4y^2 - 3xy^3 - 4xy) + (4x^4y^2 - xy^3 + 4xy)$, arriving at the simplified form $10x^4y^2 - 4xy^3$. Throughout this journey, we've emphasized the importance of understanding the basics of algebraic expressions, identifying like terms, and meticulously combining them. This process not only simplifies the expression but also enhances our algebraic skills and problem-solving abilities. Simplifying algebraic expressions is a fundamental skill in mathematics, with applications ranging from solving equations to understanding complex mathematical models. The ability to manipulate expressions effectively is crucial for success in algebra and beyond. By breaking down the problem into manageable steps, we've demonstrated that even seemingly complex expressions can be simplified with a systematic approach. The key takeaways from this article include the importance of recognizing like terms, paying close attention to signs and coefficients, and organizing the simplification process. These skills are transferable to a wide range of algebraic problems, making this a valuable exercise in mathematical proficiency. As we conclude, remember that practice is key to mastering algebraic simplification. Continue to apply these techniques to various expressions, and you'll find your skills growing stronger with each problem you solve.