Simplifying Algebraic Expressions A Step-by-Step Guide

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In this article, we will delve into the simplification of the algebraic expression (4x³y - 5x²y + 6y) - (9x³y - 3x²y - 6y). Algebraic expressions form the foundation of many mathematical concepts, and mastering the techniques to simplify them is crucial for success in algebra and beyond. This process involves combining like terms, which are terms that have the same variables raised to the same powers. Simplifying expressions not only makes them easier to understand but also facilitates solving equations and performing further algebraic manipulations. Let's embark on this journey of simplification, breaking down each step to ensure clarity and comprehension. We will explore the underlying principles of combining like terms and applying the distributive property to remove parentheses. By the end of this discussion, you will have a solid understanding of how to simplify such expressions and be able to apply these skills to more complex problems.

Understanding the Basics of Algebraic Expressions

Before we dive into the simplification process, it’s essential to understand the fundamental components of algebraic expressions. An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. In the expression (4x³y - 5x²y + 6y) - (9x³y - 3x²y - 6y), we have variables (x and y), constants (the numerical coefficients), and operations (addition and subtraction). Terms are the individual components of an expression separated by addition or subtraction. For example, in the term 4x³y, 4 is the coefficient, x³ is the variable x raised to the power of 3, and y is the variable y raised to the power of 1. Understanding these basics is the first step toward mastering algebraic simplification. It allows us to identify like terms, which are crucial for simplifying expressions. Like terms have the same variables raised to the same powers, differing only in their coefficients. Recognizing like terms is like sorting similar items together, making it easier to combine them. This foundational knowledge will empower you to tackle more complex algebraic problems with confidence and clarity.

Step-by-Step Simplification Process

Now, let's break down the simplification process step by step. The expression we're working with is (4x³y - 5x²y + 6y) - (9x³y - 3x²y - 6y). The first step is to remove the parentheses. When we have a negative sign in front of a parenthesis, we need to distribute the negative sign to each term inside the parenthesis. This means multiplying each term inside the second set of parentheses by -1. So, -(9x³y - 3x²y - 6y) becomes -9x³y + 3x²y + 6y. Now our expression looks like this: 4x³y - 5x²y + 6y - 9x³y + 3x²y + 6y. The next step is to identify and combine like terms. Remember, like terms have the same variables raised to the same powers. In this expression, 4x³y and -9x³y are like terms, -5x²y and 3x²y are like terms, and 6y and 6y are like terms. Combining these terms, we add or subtract their coefficients while keeping the variables and their powers the same. This process is akin to adding apples to apples and oranges to oranges; you can only combine what is similar. This methodical approach ensures that we don't miss any terms and that the simplification is accurate.

Combining Like Terms

Let's delve deeper into the process of combining like terms. We've identified the like terms in our expression: 4x³y and -9x³y, -5x²y and 3x²y, and 6y and 6y. To combine 4x³y and -9x³y, we add their coefficients: 4 + (-9) = -5. So, 4x³y - 9x³y = -5x³y. Next, we combine -5x²y and 3x²y by adding their coefficients: -5 + 3 = -2. Thus, -5x²y + 3x²y = -2x²y. Finally, we combine 6y and 6y: 6 + 6 = 12. So, 6y + 6y = 12y. Now, we put these simplified terms together to get the final simplified expression. This step is crucial as it brings all the individual simplifications together into a coherent and concise form. The act of combining like terms is a fundamental skill in algebra, allowing us to reduce complex expressions into manageable forms. By practicing this skill, you'll become more adept at identifying and combining like terms in various algebraic scenarios.

The Simplified Expression

After combining all the like terms, we arrive at the simplified expression. We had -5x³y from combining 4x³y and -9x³y, -2x²y from combining -5x²y and 3x²y, and 12y from combining 6y and 6y. Putting these together, the simplified expression is -5x³y - 2x²y + 12y. This is the final simplified form of the original expression (4x³y - 5x²y + 6y) - (9x³y - 3x²y - 6y). This simplified expression is much easier to work with and understand. It clearly shows the relationship between the terms and makes it simpler to perform further operations, such as solving equations or substituting values for the variables. The process of simplification is not just about reducing the length of the expression; it's about making it more accessible and useful. Mastering this skill is a key step in building a strong foundation in algebra and mathematics in general. This final simplified form represents the most concise and understandable representation of the original expression.

Importance of Simplifying Algebraic Expressions

Simplifying algebraic expressions is not just a mathematical exercise; it's a crucial skill with practical applications in various fields. In mathematics, simplified expressions make it easier to solve equations, evaluate functions, and perform other algebraic manipulations. For example, when solving an equation, a simplified expression can lead to a more straightforward solution. In physics and engineering, simplifying expressions can help in modeling and analyzing complex systems. Often, real-world problems are represented by complex equations, and simplifying these equations is necessary to find solutions. In computer science, simplified expressions can optimize algorithms and improve the efficiency of computer programs. A more concise expression often translates to fewer computations, which can significantly impact performance. Beyond these specific fields, the ability to simplify expressions enhances problem-solving skills in general. It encourages logical thinking, attention to detail, and the ability to break down complex problems into smaller, manageable steps. Simplifying expressions is a fundamental skill that empowers you to tackle mathematical challenges with greater confidence and efficiency.

Practice Problems

To solidify your understanding of simplifying algebraic expressions, let's look at some practice problems. Try simplifying the following expressions:

  1. (3a²b + 2ab - 5) - (a²b - 4ab + 2)
  2. 2(x³ - 3x² + 4) + (5x³ + x² - 2)
  3. (7pq - 4p + 3q) - (2pq + 5p - q)

These problems will give you an opportunity to apply the steps we've discussed: removing parentheses, identifying like terms, and combining like terms. Remember to distribute the negative sign carefully when removing parentheses and to pay close attention to the variables and their powers when identifying like terms. Working through these problems will reinforce your understanding and build your confidence in simplifying algebraic expressions. Practice is the key to mastery in mathematics, and these exercises will help you develop the necessary skills to tackle more complex problems. Don't hesitate to review the steps we've covered if you encounter any difficulties, and remember that each problem you solve strengthens your understanding.

Conclusion

In conclusion, simplifying the expression (4x³y - 5x²y + 6y) - (9x³y - 3x²y - 6y) involves a methodical process of removing parentheses, identifying like terms, and combining them. By distributing the negative sign, we transformed the expression into 4x³y - 5x²y + 6y - 9x³y + 3x²y + 6y. Subsequently, we combined the like terms: 4x³y and -9x³y to get -5x³y, -5x²y and 3x²y to get -2x²y, and 6y and 6y to get 12y. This resulted in the simplified expression -5x³y - 2x²y + 12y. This process not only reduces the complexity of the expression but also enhances its clarity and utility. Simplifying algebraic expressions is a fundamental skill in mathematics with far-reaching applications. It's essential for solving equations, evaluating functions, and tackling real-world problems in various fields. By mastering this skill, you'll gain a valuable tool for mathematical problem-solving and analytical thinking. Remember, practice is key, so continue to work on simplifying expressions to solidify your understanding and build your confidence. The ability to simplify expressions is a cornerstone of algebraic proficiency, and with consistent effort, you can achieve mastery.