Adding And Subtracting Algebraic Expressions A Step By Step Guide

Algebraic expressions are the foundation of algebra, and mastering the operations of addition and subtraction is crucial for success in more advanced mathematical concepts. This comprehensive guide will walk you through various examples, providing step-by-step explanations and insights to solidify your understanding. We will explore how to combine like terms, handle parentheses, and simplify expressions effectively. By the end of this article, you'll be well-equipped to tackle any algebraic expression addition or subtraction problem with confidence. This article aims to provide a clear and concise explanation of how to add and subtract algebraic expressions. We will break down the process into manageable steps and illustrate each step with examples. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this guide will help you grasp the fundamental principles and techniques involved.

1) (4am³ + 11m²) + (-8m⁵ - 2m²)

To begin our journey into algebraic manipulation, let's tackle the first expression: (4am³ + 11m²) + (-8m⁵ - 2m²). The key to adding algebraic expressions lies in identifying and combining like terms. Like terms are terms that have the same variables raised to the same powers. In this expression, we have several terms with different variables and powers. The first step is to remove the parentheses. Since we are adding the expressions, we can simply drop the parentheses without changing the signs of the terms inside. This gives us:

4am³ + 11m² - 8m⁵ - 2m²

Now, we identify the like terms. We have two terms with m²: 11m² and -2m². The term 4am³ has variables a and m, and the term -8m⁵ has the variable m raised to the power of 5. So, the like terms are 11m² and -2m². Next, we combine the like terms by adding their coefficients. The coefficient of 11m² is 11, and the coefficient of -2m² is -2. Adding these coefficients, we get 11 + (-2) = 9. Therefore, 11m² - 2m² = 9m². Now, let's rewrite the entire expression by combining the like terms and arranging the terms in descending order of the powers of the variable m. This gives us:

-8m⁵ + 4am³ + 9m²

This is the simplified form of the original expression. We have successfully combined the like terms and presented the result in a clear and organized manner. Remember, the order of terms can be changed as long as the sign of each term remains the same. However, it is a common practice to arrange the terms in descending order of powers for better readability and consistency.

2) (27a³ - 18b² + 23c) + (14d³ + 35b² - 18c)

Now, let's move on to the second expression: (27a³ - 18b² + 23c) + (14d³ + 35b² - 18c). Again, we start by removing the parentheses. Since we are adding the expressions, we can drop the parentheses without changing the signs. This gives us:

27a³ - 18b² + 23c + 14d³ + 35b² - 18c

Next, we identify the like terms. In this expression, we have terms with the variables a³, b², c, and d³. The like terms are -18b² and 35b², and 23c and -18c. The terms 27a³ and 14d³ do not have any like terms in the expression. Now, we combine the like terms. For the b² terms, we have -18b² + 35b². Adding the coefficients, we get -18 + 35 = 17. Therefore, -18b² + 35b² = 17b². For the c terms, we have 23c - 18c. Subtracting the coefficients, we get 23 - 18 = 5. Therefore, 23c - 18c = 5c. Now, let's rewrite the entire expression by combining the like terms. This gives us:

27a³ + 17b² + 5c + 14d³

This is the simplified form of the original expression. We have successfully identified and combined the like terms. Notice that the terms are arranged alphabetically by the variable. This is another common practice to maintain consistency and readability. Remember, the key to simplifying algebraic expressions is to carefully identify and combine like terms, paying attention to the signs and coefficients. By following these steps, you can simplify even complex expressions with ease.

3) (-9ab⁵ - 17) - (8a²b⁵ - 7)

Let's consider the third expression: (-9ab⁵ - 17) - (8a²b⁵ - 7). This expression involves subtraction, which requires an extra step compared to addition. When subtracting algebraic expressions, we need to distribute the negative sign to all the terms inside the parentheses that are being subtracted. First, we rewrite the expression by distributing the negative sign:

-9ab⁵ - 17 - 8a²b⁵ + 7

Notice that the sign of each term inside the second parentheses has been changed. The term 8a²b⁵ became -8a²b⁵, and the term -7 became +7. Now, we identify the like terms. In this expression, we have two constant terms: -17 and +7. The terms -9ab⁵ and -8a²b⁵ do not have any like terms. Next, we combine the like terms. Adding the constants, we get -17 + 7 = -10. Now, let's rewrite the entire expression by combining the like terms and arranging the terms in a logical order. This gives us:

-8a²b⁵ - 9ab⁵ - 10

This is the simplified form of the original expression. We have successfully distributed the negative sign, identified and combined the like terms, and presented the result in a clear and organized manner. When dealing with subtraction, always remember to distribute the negative sign carefully to ensure the correct signs for each term. This step is crucial for simplifying the expression accurately.

4) (6a² + 14a - 47) - (a² - a - 4)

Moving on to the fourth expression: (6a² + 14a - 47) - (a² - a - 4). Similar to the previous example, this expression involves subtraction. Therefore, we need to distribute the negative sign to the terms inside the second parentheses. First, we rewrite the expression by distributing the negative sign:

6a² + 14a - 47 - a² + a + 4

Notice that the sign of each term inside the second parentheses has been changed. The term a² became -a², the term -a became +a, and the term -4 became +4. Now, we identify the like terms. In this expression, we have three sets of like terms: 6a² and -a², 14a and a, and -47 and +4. Next, we combine the like terms. For the a² terms, we have 6a² - a². Subtracting the coefficients, we get 6 - 1 = 5. Therefore, 6a² - a² = 5a². For the a terms, we have 14a + a. Adding the coefficients, we get 14 + 1 = 15. Therefore, 14a + a = 15a. For the constant terms, we have -47 + 4. Adding these numbers, we get -47 + 4 = -43. Now, let's rewrite the entire expression by combining the like terms. This gives us:

5a² + 15a - 43

This is the simplified form of the original expression. We have successfully distributed the negative sign, identified and combined the like terms, and presented the result in a clear and organized manner. This example reinforces the importance of careful distribution and accurate arithmetic when simplifying algebraic expressions.

5) (35m⁵ + 27m² - 13m) + (-67m² - 21m)

Finally, let's consider the fifth expression: (35m⁵ + 27m² - 13m) + (-67m² - 21m). This expression involves addition. We can start by removing the parentheses. Since we are adding the expressions, we can simply drop the parentheses without changing the signs. This gives us:

35m⁵ + 27m² - 13m - 67m² - 21m

Now, we identify the like terms. In this expression, we have two sets of like terms: 27m² and -67m², and -13m and -21m. The term 35m⁵ does not have any like terms. Next, we combine the like terms. For the m² terms, we have 27m² - 67m². Subtracting the coefficients, we get 27 - 67 = -40. Therefore, 27m² - 67m² = -40m². For the m terms, we have -13m - 21m. Adding the coefficients, we get -13 + (-21) = -34. Therefore, -13m - 21m = -34m. Now, let's rewrite the entire expression by combining the like terms and arranging the terms in descending order of the powers of the variable m. This gives us:

35m⁵ - 40m² - 34m

This is the simplified form of the original expression. We have successfully identified and combined the like terms, and presented the result in a clear and organized manner. This final example demonstrates the process of simplifying algebraic expressions when dealing with both positive and negative coefficients.

Conclusion

In this article, we have explored the fundamental principles and techniques for adding and subtracting algebraic expressions. We have seen how to identify and combine like terms, how to handle parentheses, and how to distribute negative signs when subtracting expressions. By working through various examples, we have reinforced the importance of careful attention to detail and accurate arithmetic. Mastering these skills is essential for success in algebra and beyond. Remember, practice is key. The more you work with algebraic expressions, the more confident and proficient you will become. So, keep practicing, and you'll be well on your way to mastering the art of algebraic manipulation. The ability to simplify algebraic expressions is a valuable skill that will serve you well in various mathematical and scientific contexts. Whether you are solving equations, graphing functions, or working with more advanced mathematical concepts, a solid understanding of algebraic manipulation is essential. By mastering the techniques discussed in this article, you will be well-prepared to tackle a wide range of algebraic problems with confidence and accuracy. Keep up the great work, and happy simplifying!