Simplifying (5a - 7b) - (2a - 5b) A Step-by-Step Guide

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Introduction to Algebraic Simplification

In the realm of mathematics, simplifying algebraic expressions is a foundational skill. This process involves manipulating expressions to make them more concise and easier to understand. Algebraic simplification is crucial in solving equations, understanding formulas, and performing more complex mathematical operations. In this article, we will delve into the step-by-step process of simplifying the expression (5a - 7b) - (2a - 5b). This example serves as an excellent illustration of how to combine like terms and handle subtraction in algebraic expressions. Mastering these techniques is essential for anyone pursuing further studies in algebra and related fields.

When you first encounter an expression like (5a - 7b) - (2a - 5b), it might seem daunting. However, by breaking it down into smaller, manageable steps, the simplification becomes quite straightforward. The key lies in understanding the properties of algebraic operations, such as the distributive property and the commutative property. These properties allow us to rearrange and combine terms effectively. Additionally, being meticulous with signs, especially when dealing with subtraction, is crucial to avoid errors. The ability to simplify algebraic expressions not only enhances problem-solving skills but also builds a solid foundation for tackling more advanced mathematical concepts.

This guide aims to provide a comprehensive understanding of algebraic simplification, focusing on the specific expression (5a - 7b) - (2a - 5b). We will explore each step in detail, providing explanations and examples to ensure clarity. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this article will serve as a valuable resource. By the end of this guide, you will have a clear understanding of how to simplify this expression and similar ones, empowering you to confidently tackle algebraic challenges.

Breaking Down the Expression (5a - 7b) - (2a - 5b)

To effectively simplify the algebraic expression (5a - 7b) - (2a - 5b), the first step involves understanding its structure. This expression consists of two binomials, 5a - 7b and 2a - 5b, with a subtraction operation between them. The terms within each binomial are 5a, -7b, 2a, and -5b. These terms involve variables (a and b) and coefficients (the numerical values multiplying the variables). Recognizing these components is crucial for applying the correct simplification techniques. A common mistake is to overlook the negative sign preceding the second binomial, which requires careful distribution to ensure accurate simplification. Understanding the roles of variables, coefficients, and operations is fundamental to mastering algebraic simplification.

One of the primary techniques used in simplifying expressions is the distributive property. In this case, the subtraction sign outside the parentheses acts as a negative one that needs to be distributed to each term inside the second binomial. This means we multiply -1 by both 2a and -5b. This step is essential because it transforms the subtraction of the entire binomial into the addition of the negative of each term. For example, subtracting 2a becomes adding -2a, and subtracting -5b becomes adding 5b. Failing to properly distribute the negative sign is a common error that leads to incorrect results. Mastering the distributive property is crucial for handling expressions with parentheses and negative signs accurately.

After distributing the negative sign, the expression becomes 5a - 7b - 2a + 5b. The next step involves identifying like terms. Like terms are terms that have the same variable raised to the same power. In this expression, 5a and -2a are like terms because they both contain the variable a raised to the power of one. Similarly, -7b and 5b are like terms because they both contain the variable b raised to the power of one. Identifying like terms is a critical step in simplification because only like terms can be combined. Attempting to combine unlike terms, such as 5a and -7b, would be an algebraic error. The ability to correctly identify like terms ensures that the simplification process leads to an accurate and concise result.

Step-by-Step Simplification Process

1. Distributing the Negative Sign

The initial step in simplifying the expression (5a - 7b) - (2a - 5b) involves distributing the negative sign (which can be thought of as multiplying by -1) across the terms within the second parentheses. This means we need to multiply -1 by both 2a and -5b. When we multiply -1 by 2a, we get -2a. Similarly, when we multiply -1 by -5b, we get 5b because the product of two negative numbers is positive. This step transforms the original expression into 5a - 7b - 2a + 5b. Properly distributing the negative sign is crucial because it changes the signs of the terms within the parentheses, which directly affects the subsequent steps in the simplification process. Overlooking this step or incorrectly applying the distributive property is a common source of errors in algebraic simplification.

The distributive property is a fundamental concept in algebra, and its correct application is essential for simplifying expressions. It ensures that we account for the impact of the subtraction operation on each term within the parentheses. In essence, distributing the negative sign allows us to rewrite the expression as a sum of terms, making it easier to identify and combine like terms in the following steps. This step highlights the importance of meticulousness in algebraic manipulations, as even a small error in sign can lead to a completely different result. Mastering the distributive property is a key skill for anyone working with algebraic expressions.

2. Identifying Like Terms

After distributing the negative sign, the expression becomes 5a - 7b - 2a + 5b. The next crucial step is to identify like terms. Like terms are terms that contain the same variable raised to the same power. In this expression, we have two sets of like terms: terms involving the variable a and terms involving the variable b. Specifically, 5a and -2a are like terms because they both have the variable a raised to the power of one. Similarly, -7b and 5b are like terms because they both have the variable b raised to the power of one. It is essential to accurately identify like terms because only these terms can be combined to simplify the expression further. Misidentifying terms can lead to incorrect simplifications, emphasizing the need for careful attention to detail.

The ability to recognize like terms is a foundational skill in algebra. It stems from the understanding that algebraic terms can only be combined if they represent the same variable raised to the same power. For instance, 5a and -2a can be combined because they both represent multiples of the variable a. However, terms like 5a and -7b cannot be combined because they represent different variables. This principle extends to terms with exponents as well; for example, x² and x are not like terms because the variable x is raised to different powers. Properly identifying like terms sets the stage for the final step in simplification: combining these terms to arrive at the most concise form of the expression.

3. Combining Like Terms

The final step in simplifying the expression 5a - 7b - 2a + 5b is to combine the like terms that we identified in the previous step. To combine like terms, we add or subtract their coefficients while keeping the variable part the same. Let’s start with the a terms: we have 5a and -2a. To combine these, we subtract the coefficients: 5 - 2 = 3. So, 5a - 2a simplifies to 3a. Next, we combine the b terms: we have -7b and 5b. To combine these, we add the coefficients: -7 + 5 = -2. So, -7b + 5b simplifies to -2b. Combining the simplified a and b terms, we get the final simplified expression: 3a - 2b. This final step demonstrates how combining like terms results in a more concise and manageable form of the original expression.

The process of combining like terms is a direct application of the distributive property in reverse. For instance, 5a - 2a can be thought of as (5 - 2)a, which simplifies to 3a. Similarly, -7b + 5b can be thought of as (-7 + 5)b, which simplifies to -2b. This highlights the underlying mathematical principle that allows us to combine terms with the same variable. The result, 3a - 2b, is the simplest form of the original expression (5a - 7b) - (2a - 5b). There are no more like terms to combine, and the expression is now in its most reduced form. This skill is essential for solving equations and tackling more advanced algebraic problems.

Common Mistakes to Avoid

When simplifying algebraic expressions, several common mistakes can lead to incorrect answers. One frequent error is failing to correctly distribute the negative sign when dealing with expressions in parentheses. For example, in the expression (5a - 7b) - (2a - 5b), it's crucial to distribute the negative sign to both terms inside the second set of parentheses. This means changing (2a - 5b) to -2a + 5b. Many students mistakenly only apply the negative sign to the first term, resulting in an incorrect expression. To avoid this, always remember to multiply each term inside the parentheses by the negative sign (or -1). This careful distribution is essential for maintaining the accuracy of the simplification process.

Another common mistake is incorrectly combining unlike terms. Remember, like terms have the same variable raised to the same power. For instance, 5a and -2a are like terms and can be combined, but 5a and -7b are not like terms and cannot be combined. Similarly, x² and x are unlike terms because the variable x is raised to different powers. Students sometimes mistakenly add or subtract the coefficients of unlike terms, leading to an incorrect simplified expression. To avoid this, always double-check that terms have the same variable and exponent before attempting to combine them. This careful distinction between like and unlike terms is crucial for accurate simplification.

Sign errors are also a common pitfall in algebraic simplification. These errors can occur when adding, subtracting, or multiplying terms, especially when dealing with negative numbers. For example, when combining -7b and 5b, it's essential to correctly add the coefficients: -7 + 5 = -2. A sign error here could lead to an incorrect result, such as 2b instead of -2b. To minimize sign errors, take your time, write each step clearly, and double-check your work. Pay close attention to the rules for adding and subtracting negative numbers, and consider using a number line as a visual aid if needed. Accuracy in handling signs is fundamental to success in algebra.

Practice Problems and Solutions

To solidify your understanding of simplifying algebraic expressions, let’s work through a few practice problems. These examples will help you apply the techniques we’ve discussed and reinforce the importance of avoiding common mistakes. Each problem includes a step-by-step solution to guide you through the process.

Problem 1: Simplify the expression (3x + 2y) - (x - 4y).

Solution:

  1. Distribute the negative sign: First, distribute the negative sign across the terms inside the second parentheses: (3x + 2y) - (x - 4y) = 3x + 2y - x + 4y.
  2. Identify like terms: Next, identify the like terms. In this expression, 3x and -x are like terms, and 2y and 4y are like terms.
  3. Combine like terms: Now, combine the like terms: 3x - x = 2x and 2y + 4y = 6y. Therefore, the simplified expression is 2x + 6y.

Problem 2: Simplify the expression (4a - 5b) + (2a + 3b).

Solution:

  1. Distribute (if necessary): In this case, since we are adding the expressions, we don't need to distribute a negative sign. We can simply remove the parentheses: 4a - 5b + 2a + 3b.
  2. Identify like terms: Identify the like terms: 4a and 2a are like terms, and -5b and 3b are like terms.
  3. Combine like terms: Combine the like terms: 4a + 2a = 6a and -5b + 3b = -2b. Therefore, the simplified expression is 6a - 2b.

Problem 3: Simplify the expression 2(x - 3y) - (3x + y).

Solution:

  1. Distribute: First, distribute the 2 across the terms inside the first parentheses and the negative sign across the terms inside the second parentheses: 2(x - 3y) - (3x + y) = 2x - 6y - 3x - y.
  2. Identify like terms: Next, identify the like terms: 2x and -3x are like terms, and -6y and -y are like terms.
  3. Combine like terms: Now, combine the like terms: 2x - 3x = -x and -6y - y = -7y. Therefore, the simplified expression is -x - 7y.

By working through these practice problems, you can gain confidence in your ability to simplify algebraic expressions. Remember to focus on each step, avoid common mistakes, and double-check your work. Consistent practice is the key to mastering this essential skill.

Conclusion: Mastering Algebraic Simplification

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics, essential for solving equations and understanding more complex concepts. Throughout this guide, we have meticulously dissected the process of simplifying the expression (5a - 7b) - (2a - 5b). We began by understanding the structure of the expression, identifying terms, variables, and coefficients. Then, we broke down the simplification into three key steps: distributing the negative sign, identifying like terms, and combining those terms. Each step is crucial, and mastering them ensures accurate and efficient algebraic manipulation.

The distributive property plays a pivotal role in algebraic simplification, particularly when dealing with expressions in parentheses and negative signs. The correct application of this property is vital for changing the expression into a form where like terms can be easily identified. We emphasized the importance of distributing the negative sign across all terms within the parentheses, as this is a common source of errors. Furthermore, we highlighted the significance of accurately identifying like terms, which are terms with the same variable raised to the same power. Only like terms can be combined, and the ability to distinguish between like and unlike terms is critical for simplifying expressions correctly.

Finally, combining like terms involves adding or subtracting their coefficients while keeping the variable part the same. This step condenses the expression into its simplest form, making it easier to work with in subsequent mathematical operations. We also addressed common mistakes to avoid, such as failing to distribute the negative sign, incorrectly combining unlike terms, and sign errors. These pitfalls can lead to incorrect answers, so awareness and careful attention to detail are paramount. To reinforce your understanding, we provided several practice problems with step-by-step solutions. Consistent practice is the key to mastering algebraic simplification and building confidence in your mathematical abilities.

Algebraic simplification is not just a standalone skill; it is a building block for more advanced topics in mathematics, such as solving equations, working with polynomials, and calculus. By mastering this skill, you lay a solid foundation for future mathematical endeavors. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, the principles and techniques outlined in this guide will serve as a valuable resource. Embrace the challenge, practice diligently, and you will find that simplifying algebraic expressions becomes a straightforward and rewarding process.