Simplifying Algebraic Expressions A Step-by-Step Guide

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In mathematics, simplifying expressions is a fundamental skill that allows us to make complex equations more manageable and easier to understand. This article will delve into the process of simplifying the algebraic expression (3x³y - 9xy² + 2y) - (7x³y - 2xy² - 2y). We will break down each step, explaining the underlying principles and techniques involved. Whether you are a student learning algebra or someone looking to brush up on your math skills, this comprehensive guide will provide you with the knowledge and confidence to tackle similar problems. Our main focus will be on combining like terms and applying the distributive property correctly to arrive at the simplest form of the given expression. Understanding these concepts is crucial not only for academic success but also for various real-world applications where algebraic manipulation is necessary.

Understanding the Basics: Terms and Like Terms

Before we dive into the simplification process, it's essential to understand the basic building blocks of algebraic expressions: terms and like terms. A term is a single mathematical expression that can be a number, a variable, or a product of numbers and variables. For example, in the expression 3x³y - 9xy² + 2y, each part separated by a plus or minus sign is a term: 3x³y, -9xy², and 2y are all individual terms. Like terms, on the other hand, are terms that have the same variables raised to the same powers. The numerical coefficients (the numbers in front of the variables) can be different, but the variable parts must be identical for terms to be considered "like". For instance, 3x³y and 7x³y are like terms because they both have the same variable part, x³y. However, 3x³y and -9xy² are not like terms because the powers of the variables are different (x is cubed in the first term but to the power of 1 in the second, and y is to the power of 1 in the first term but squared in the second).

Recognizing like terms is crucial for simplifying expressions because only like terms can be combined. When we combine like terms, we essentially add or subtract their coefficients while keeping the variable part the same. For example, if we have 5x² and -2x², these are like terms, and we can combine them as (5 - 2)x² = 3x². Understanding this concept forms the foundation for simplifying more complex expressions, as it allows us to reduce the number of terms and make the expression more concise and manageable. This is a key step in many mathematical problems, and mastering it will significantly enhance your ability to solve algebraic equations and tackle more advanced mathematical concepts. In the given expression, we will identify and combine like terms such as the terms involving x³y and the terms involving xy², which will be a crucial step in simplifying the overall expression.

Step-by-Step Simplification Process

Now, let's go through the step-by-step process of simplifying the expression (3x³y - 9xy² + 2y) - (7x³y - 2xy² - 2y). The first key step in simplifying this expression involves understanding and applying the distributive property. This property allows us to remove the parentheses by distributing the negative sign in front of the second set of parentheses to each term inside. In other words, we change the sign of each term within the second parentheses. Applying the distributive property, we rewrite the expression as follows:

(3x³y - 9xy² + 2y) - (7x³y - 2xy² - 2y) = 3x³y - 9xy² + 2y - 7x³y + 2xy² + 2y

Notice that the negative sign in front of the parentheses has changed the signs of each term inside: 7x³y became -7x³y, -2xy² became +2xy², and -2y became +2y. This step is crucial because it eliminates the parentheses, making it easier to combine like terms. Many errors in simplifying expressions occur at this stage, so it is important to be meticulous and double-check that the signs have been distributed correctly. By correctly applying the distributive property, we set the stage for the next phase of simplification, which involves grouping and combining like terms to reduce the expression to its simplest form. This initial step is a cornerstone of algebraic manipulation, paving the way for clearer and more concise mathematical statements.

Combining Like Terms: A Detailed Explanation

Once we've applied the distributive property and removed the parentheses, the next critical step is to combine like terms. As discussed earlier, like terms are terms that have the same variables raised to the same powers. In our expanded expression, 3x³y - 9xy² + 2y - 7x³y + 2xy² + 2y, we can identify several pairs of like terms. The terms 3x³y and -7x³y are like terms because they both have the variable part x³y. Similarly, -9xy² and 2xy² are like terms as they share the variable part xy². Lastly, 2y and 2y are like terms since they both have the variable y raised to the power of 1.

To combine like terms, we add or subtract their coefficients while keeping the variable part the same. For the terms 3x³y and -7x³y, we add their coefficients (3 and -7) to get 3 - 7 = -4. Therefore, combining these terms gives us -4x³y. For the terms -9xy² and 2xy², we add their coefficients (-9 and 2) to get -9 + 2 = -7. So, these terms combine to -7xy². Finally, for the terms 2y and 2y, we add their coefficients (2 and 2) to get 2 + 2 = 4. Thus, these terms combine to 4y. By systematically identifying and combining like terms, we reduce the expression to a simpler form, which makes it easier to work with and understand. This process is not just a mechanical step; it’s a way of revealing the underlying structure of the expression, making it more accessible for further analysis or use in other mathematical contexts.

Final Simplified Expression

After combining all like terms, we arrive at the final simplified expression. From the previous step, we had identified and combined the like terms as follows:

  • 3x³y and -7x³y combined to -4x³y
  • -9xy² and 2xy² combined to -7xy²
  • 2y and 2y combined to 4y

Now, we simply write these combined terms together to form the simplified expression. Therefore, the simplified form of the original expression (3x³y - 9xy² + 2y) - (7x³y - 2xy² - 2y) is:

-4x³y - 7xy² + 4y

This final expression is much cleaner and easier to interpret than the original. It contains only three terms, each representing a unique combination of variables and their respective powers. By systematically applying the distributive property and combining like terms, we have successfully simplified the complex expression into its most basic form. This process not only reduces the expression's complexity but also makes it easier to use in further calculations or analyses. The ability to simplify algebraic expressions is a cornerstone of mathematical proficiency, and mastering this skill opens the door to more advanced topics and problem-solving techniques. This simplified expression clearly showcases the power of algebraic manipulation in making mathematical statements more concise and understandable.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's easy to make mistakes if you're not careful. One common mistake is incorrectly applying the distributive property. As we discussed earlier, when there's a negative sign in front of parentheses, it's crucial to change the sign of every term inside the parentheses. For example, in the expression (3x³y - 9xy² + 2y) - (7x³y - 2xy² - 2y), you must distribute the negative sign to each term in the second parentheses, making it -7x³y + 2xy² + 2y. A frequent error is forgetting to change the sign of one or more terms, which will lead to an incorrect simplification.

Another common pitfall is combining unlike terms. Remember, only terms with the same variables raised to the same powers can be combined. For instance, 3x³y and -9xy² cannot be combined because they have different variable parts (x³y versus xy²). Confusing these can lead to significant errors in the simplified expression. To avoid this, always double-check that the variable parts are identical before combining terms. A systematic approach, where you first identify and group like terms before combining them, can help prevent this mistake.

Finally, errors can arise from arithmetic mistakes when adding or subtracting coefficients. Even if you correctly identify like terms and apply the distributive property, a simple arithmetic error can derail the simplification. For example, if you incorrectly calculate -9 + 2 as -11 instead of -7, your final expression will be wrong. To minimize such errors, take your time and double-check your calculations. Using a calculator or writing out the arithmetic steps can also help. By being mindful of these common pitfalls and taking steps to avoid them, you can increase your accuracy and confidence in simplifying algebraic expressions. Regular practice and careful attention to detail are the keys to mastering this fundamental skill.

Practice Problems

To solidify your understanding of simplifying algebraic expressions, let's work through a few practice problems. These examples will help you apply the concepts we've discussed and identify areas where you might need further practice. Remember, the key steps are to apply the distributive property correctly, identify like terms, and combine them accurately. By working through these problems, you'll build confidence and proficiency in simplifying expressions.

Problem 1: Simplify the expression (5a²b - 3ab² + 4b) - (2a²b + 5ab² - b)

Problem 2: Simplify the expression (4x³ - 2x² + 7x - 1) + (x³ + 5x² - 3x + 6)

Problem 3: Simplify the expression 2(3y² - 4y + 2) - (y² + 2y - 5)

These problems cover a range of scenarios you might encounter when simplifying expressions. Take your time to work through each one, paying close attention to the signs and the variable parts of the terms. Once you've solved them, you can check your answers to see if you've simplified them correctly. The more you practice, the more comfortable and skilled you'll become at this important algebraic technique. Simplifying expressions is a fundamental skill that underpins many areas of mathematics, so investing time in mastering it will pay dividends in your future studies.

Solutions to Practice Problems

Now, let's go through the solutions to the practice problems to check your work and ensure you've grasped the concepts correctly. Understanding the steps in each solution is just as important as getting the final answer right, as it reinforces the process of simplification.

Solution to Problem 1: (5a²b - 3ab² + 4b) - (2a²b + 5ab² - b)

First, apply the distributive property to remove the parentheses:

5a²b - 3ab² + 4b - 2a²b - 5ab² + b

Next, identify and combine like terms:

  • 5a²b and -2a²b combine to 3a²b
  • -3ab² and -5ab² combine to -8ab²
  • 4b and b combine to 5b

The simplified expression is:

3a²b - 8ab² + 5b

Solution to Problem 2: (4x³ - 2x² + 7x - 1) + (x³ + 5x² - 3x + 6)

Since we are adding the two expressions, we can simply remove the parentheses:

4x³ - 2x² + 7x - 1 + x³ + 5x² - 3x + 6

Now, identify and combine like terms:

  • 4x³ and x³ combine to 5x³
  • -2x² and 5x² combine to 3x²
  • 7x and -3x combine to 4x
  • -1 and 6 combine to 5

The simplified expression is:

5x³ + 3x² + 4x + 5

Solution to Problem 3: 2(3y² - 4y + 2) - (y² + 2y - 5)

First, apply the distributive property to both sets of parentheses:

6y² - 8y + 4 - y² - 2y + 5

Next, identify and combine like terms:

  • 6y² and -y² combine to 5y²
  • -8y and -2y combine to -10y
  • 4 and 5 combine to 9

The simplified expression is:

5y² - 10y + 9

By reviewing these solutions, you can identify any areas where you might have made a mistake and learn from them. Remember, practice is key to mastering algebraic simplification.

Conclusion

In this article, we have thoroughly explored the process of simplifying the algebraic expression (3x³y - 9xy² + 2y) - (7x³y - 2xy² - 2y). We began by defining the key concepts of terms and like terms, which form the foundation of algebraic simplification. We then walked through the step-by-step simplification process, emphasizing the importance of correctly applying the distributive property and meticulously combining like terms. Through detailed explanations and examples, we have demonstrated how to reduce complex expressions to their simplest forms.

We also highlighted common mistakes to avoid, such as incorrectly distributing the negative sign, combining unlike terms, and making arithmetic errors. By being aware of these pitfalls, you can increase your accuracy and confidence when simplifying algebraic expressions. The practice problems and their solutions provided an opportunity to apply the learned concepts and reinforce your understanding. The ability to simplify algebraic expressions is a fundamental skill in mathematics, essential for solving equations, tackling more advanced mathematical concepts, and even in various real-world applications.

By mastering this skill, you enhance your problem-solving abilities and gain a deeper appreciation for the elegance and power of algebraic manipulation. As you continue your mathematical journey, remember that consistent practice and a keen attention to detail are the keys to success. Simplifying expressions is not just about finding the right answer; it's about developing a systematic approach to problem-solving and building a solid foundation for future mathematical endeavors. With the knowledge and practice gained from this article, you are well-equipped to tackle a wide range of algebraic problems and excel in your mathematical studies.