Simplifying Algebraic Expressions A Step-by-Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill that allows us to make complex problems more manageable. Algebraic expressions, in particular, often appear daunting at first glance, but with a systematic approach, they can be reduced to their simplest forms. This article delves into the process of simplifying the algebraic expression (14.2a + 9.8b) - (13.1b - 0.2a) - (3.7a + 4.8b), providing a comprehensive step-by-step guide suitable for students and anyone looking to refresh their algebra skills. By understanding the underlying principles and applying them diligently, you'll be able to tackle similar problems with confidence and accuracy.

Understanding the Basics of Algebraic Expressions

Before we dive into the specifics of our expression, let's establish a firm understanding of what algebraic expressions are and the basic operations involved in simplifying them. An algebraic expression is a combination of variables (represented by letters like 'a' and 'b'), constants (numbers), and mathematical operations such as addition, subtraction, multiplication, and division. The goal of simplifying an algebraic expression is to rewrite it in a more concise and manageable form without changing its value. This often involves combining like terms, which are terms that have the same variable raised to the same power. For instance, 3a and 5a are like terms because they both contain the variable 'a' raised to the power of 1, while 3a and 5a² are not like terms because the variable 'a' is raised to different powers. The cornerstone of simplifying algebraic expressions lies in the application of the distributive property and the commutative and associative properties of addition and multiplication. The distributive property states that a(b + c) = ab + ac, which allows us to remove parentheses by multiplying the term outside the parentheses by each term inside. The commutative property (a + b = b + a and a * b = b * a) allows us to change the order of terms in an addition or multiplication without changing the result, while the associative property ((a + b) + c = a + (b + c) and (a * b) * c = a * (b * c)) allows us to regroup terms without changing the result. Mastering these fundamental concepts is crucial for successfully simplifying any algebraic expression.

Step-by-Step Simplification of (14.2a + 9.8b) - (13.1b - 0.2a) - (3.7a + 4.8b)

Now, let's embark on the journey of simplifying the expression (14.2a + 9.8b) - (13.1b - 0.2a) - (3.7a + 4.8b) step by step. This process will not only yield the simplified form but also provide a clear understanding of the techniques involved. First, we need to address the parentheses. The expression contains two sets of parentheses preceded by a minus sign, which indicates that we need to distribute the negative sign to each term within those parentheses. Remember that subtracting a quantity is the same as adding its negative. So, -(13.1b - 0.2a) becomes -13.1b + 0.2a, and -(3.7a + 4.8b) becomes -3.7a - 4.8b. After distributing the negative signs, our expression transforms into 14.2a + 9.8b - 13.1b + 0.2a - 3.7a - 4.8b. The next crucial step is to combine like terms. We identify terms that contain the same variable raised to the same power. In this case, we have terms with 'a' (14.2a, 0.2a, and -3.7a) and terms with 'b' (9.8b, -13.1b, and -4.8b). We then group these like terms together, which can be done mentally or by physically rearranging the terms (using the commutative property). Grouping the terms, we get (14.2a + 0.2a - 3.7a) + (9.8b - 13.1b - 4.8b). Now, we perform the arithmetic operations within each group. Adding the 'a' terms, 14.2a + 0.2a equals 14.4a, and then subtracting 3.7a gives us 10.7a. For the 'b' terms, 9.8b - 13.1b equals -3.3b, and then subtracting 4.8b gives us -8.1b. Therefore, the simplified expression is 10.7a - 8.1b. This is the simplest form of the original expression, as there are no more like terms to combine.

Detailed Breakdown of Each Step

To further solidify your understanding, let's dissect each step involved in simplifying the algebraic expression. This detailed breakdown will highlight the rationale behind each operation and provide clarity on potential areas of confusion. The initial expression we're working with is (14.2a + 9.8b) - (13.1b - 0.2a) - (3.7a + 4.8b). The first step involves the distribution of the negative sign. As mentioned earlier, subtracting a quantity is equivalent to adding its negative. This means we need to multiply each term inside the parentheses by -1. When we distribute the negative sign across (13.1b - 0.2a), we get -13.1b + 0.2a. Similarly, distributing the negative sign across (3.7a + 4.8b) yields -3.7a - 4.8b. It's crucial to remember to change the sign of every term within the parentheses. A common mistake is to only change the sign of the first term. After this distribution, the expression becomes 14.2a + 9.8b - 13.1b + 0.2a - 3.7a - 4.8b. The next step is identifying and grouping like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have two groups of like terms: terms with 'a' (14.2a, 0.2a, and -3.7a) and terms with 'b' (9.8b, -13.1b, and -4.8b). To make the process clearer, we can rearrange the terms to group the like terms together: (14.2a + 0.2a - 3.7a) + (9.8b - 13.1b - 4.8b). This rearrangement doesn't change the value of the expression due to the commutative property of addition. Finally, we combine the like terms by performing the arithmetic operations within each group. For the 'a' terms, we have 14.2a + 0.2a - 3.7a. Adding 14.2a and 0.2a gives us 14.4a, and then subtracting 3.7a results in 10.7a. For the 'b' terms, we have 9.8b - 13.1b - 4.8b. Subtracting 13.1b from 9.8b gives us -3.3b, and then subtracting 4.8b results in -8.1b. Therefore, the simplified expression is 10.7a - 8.1b.

Common Mistakes to Avoid

While simplifying algebraic expressions is a straightforward process when approached systematically, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate results. One of the most frequent errors is incorrectly distributing the negative sign. When an expression in parentheses is preceded by a minus sign, it's crucial to change the sign of every term inside the parentheses. For example, -(a - b) should be simplified as -a + b, not -a - b. Failing to distribute the negative sign properly can lead to significant errors in the final answer. Another common mistake is combining unlike terms. Remember that like terms must have the same variable raised to the same power. You cannot combine terms like 3a and 3a² or 2x and 2y. Only terms that have the exact same variable and exponent can be added or subtracted. Confusing these terms will lead to an incorrect simplification. Arithmetic errors are also a significant source of mistakes. When adding or subtracting coefficients, it's essential to pay close attention to the signs and perform the calculations accurately. A simple arithmetic mistake can throw off the entire solution. To avoid these errors, it's helpful to double-check your calculations and use a calculator if necessary. Forgetting to simplify completely is another oversight. Sometimes, after performing the initial steps of simplification, there may still be like terms that can be combined. Make sure you've reduced the expression to its simplest form by combining all possible like terms. Lastly, not following the order of operations can lead to errors, especially in more complex expressions. Remember to address parentheses first, then exponents, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right). By being mindful of these common mistakes and practicing diligently, you can improve your accuracy and confidence in simplifying algebraic expressions.

Practice Problems and Solutions

To truly master the art of simplifying algebraic expressions, practice is essential. Working through various examples will help you solidify your understanding of the concepts and techniques involved. Here are a few practice problems along with their solutions to get you started:

Problem 1: Simplify the expression 5x + 3y - 2x + 7y.

Solution: First, group the like terms: (5x - 2x) + (3y + 7y). Then, combine the like terms: 3x + 10y. Therefore, the simplified expression is 3x + 10y.

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

Solution: First, distribute the 2 and the negative sign: 2a + 6b - 4a + b. Then, group the like terms: (2a - 4a) + (6b + b). Combine the like terms: -2a + 7b. Therefore, the simplified expression is -2a + 7b.

Problem 3: Simplify the expression 3(x² - 2x + 1) + 2(x² + x - 3).

Solution: First, distribute the 3 and the 2: 3x² - 6x + 3 + 2x² + 2x - 6. Then, group the like terms: (3x² + 2x²) + (-6x + 2x) + (3 - 6). Combine the like terms: 5x² - 4x - 3. Therefore, the simplified expression is 5x² - 4x - 3.

Problem 4: Simplify the expression (7p - 4q) - (2p + 5q) + (3q - p).

Solution: First, distribute the negative sign: 7p - 4q - 2p - 5q + 3q - p. Then, group the like terms: (7p - 2p - p) + (-4q - 5q + 3q). Combine the like terms: 4p - 6q. Therefore, the simplified expression is 4p - 6q.

By working through these examples and attempting similar problems on your own, you'll develop a strong foundation in simplifying algebraic expressions. Remember to pay close attention to the signs, group like terms carefully, and double-check your calculations to ensure accuracy.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics that involves rewriting an expression in its most concise form while preserving its value. This process often involves distributing, combining like terms, and carefully applying the rules of arithmetic. Throughout this article, we have systematically simplified the expression (14.2a + 9.8b) - (13.1b - 0.2a) - (3.7a + 4.8b), demonstrating each step in detail. We also highlighted common mistakes to avoid and provided practice problems to reinforce your understanding. Mastering these techniques will not only help you succeed in algebra but also in more advanced mathematical fields. The simplified form of the expression (14.2a + 9.8b) - (13.1b - 0.2a) - (3.7a + 4.8b) is 10.7a - 8.1b. Remember to practice regularly and apply these methods to a variety of problems to solidify your skills. With consistent effort, you'll become proficient in simplifying algebraic expressions and tackling mathematical challenges with confidence.