Adding And Subtracting Algebraic Expressions A Step By Step Guide

In the realm of mathematics, algebraic expressions form the bedrock of numerous concepts and problem-solving techniques. Understanding how to manipulate these expressions, particularly through addition and subtraction, is crucial for students and professionals alike. This comprehensive guide will delve into the process of adding and subtracting algebraic expressions, providing clear explanations and step-by-step solutions to the following problems:

1. (6xy + 2y + 16x) + (4xy - x)
2. (2x² + 5xy + 4y²) + (-xy - 6x² + 2y²)
3. (3p² - 2p + 3) + (2p² - 7p + 7)
4. (x²y²) - (-x² + y²)
5. (9xy + y - 2x) - (6xy - 2x)

By mastering these examples, you will gain a solid foundation in algebraic manipulation, empowering you to tackle more complex mathematical challenges with confidence. So, let's embark on this journey of algebraic exploration and unravel the intricacies of adding and subtracting expressions.

1. Adding (6xy + 2y + 16x) and (4xy - x)

To effectively add algebraic expressions, the core principle lies in identifying and combining like terms. Like terms are those that share the same variables raised to the same powers. In the expression (6xy + 2y + 16x) + (4xy - x), we have several terms to consider. The terms '6xy' and '4xy' are like terms because they both contain the variables 'x' and 'y' multiplied together. Similarly, '16x' and '-x' are like terms as they both involve the variable 'x' raised to the power of 1. The term '2y' stands alone as it doesn't have any other like terms in the expression.

Now, let's proceed with the addition by grouping the like terms together. This can be visually represented as (6xy + 4xy) + 2y + (16x - x). By rearranging the terms, we make it easier to perform the addition. The next step involves combining the coefficients of the like terms. The coefficients are the numerical values that multiply the variables. For the 'xy' terms, we have 6 + 4 = 10, resulting in 10xy. For the 'x' terms, we have 16 - 1 = 15, resulting in 15x. The term '2y' remains unchanged as it has no like terms to combine with. Therefore, the simplified expression after adding (6xy + 2y + 16x) and (4xy - x) is 10xy + 2y + 15x. This resulting expression represents the sum of the original two expressions, combining like terms to present the answer in its simplest form. Understanding the process of identifying and combining like terms is fundamental to algebraic manipulation and forms the basis for more complex operations.

2. Adding (2x² + 5xy + 4y²) and (-xy - 6x² + 2y²)

The key to adding these algebraic expressions lies in the precise identification and grouping of like terms. In the given expressions, (2x² + 5xy + 4y²) and (-xy - 6x² + 2y²), we need to carefully examine each term and its variable components. Like terms are those that possess the same variables raised to the same powers. For instance, '2x²' and '-6x²' are like terms because they both contain the variable 'x' raised to the power of 2. Similarly, '5xy' and '-xy' are like terms, both having the variables 'x' and 'y' multiplied together. Lastly, '4y²' and '2y²' are like terms, sharing the variable 'y' raised to the power of 2.

To add these expressions effectively, we first group the like terms together. This can be represented as (2x² - 6x²) + (5xy - xy) + (4y² + 2y²). This rearrangement helps in visually organizing the terms that can be combined. The next step involves adding the coefficients of the like terms. The coefficients are the numerical values preceding the variables. For the 'x²' terms, we have 2 - 6 = -4, resulting in -4x². For the 'xy' terms, we have 5 - 1 = 4, resulting in 4xy. Lastly, for the 'y²' terms, we have 4 + 2 = 6, resulting in 6y². Combining these results, the simplified expression after adding (2x² + 5xy + 4y²) and (-xy - 6x² + 2y²) is -4x² + 4xy + 6y². This final expression represents the sum of the original two expressions, with all like terms combined to present the answer in its most concise form. This process underscores the importance of meticulous term identification and coefficient manipulation in algebraic addition.

3. Adding (3p² - 2p + 3) and (2p² - 7p + 7)

The core concept in adding these algebraic expressions remains the same: identifying and combining like terms. The expressions (3p² - 2p + 3) and (2p² - 7p + 7) consist of terms involving the variable 'p' raised to different powers, as well as constant terms. Like terms, as we've established, are terms that share the same variable raised to the same power. In this case, '3p²' and '2p²' are like terms because they both contain the variable 'p' raised to the power of 2. Similarly, '-2p' and '-7p' are like terms, both involving the variable 'p' raised to the power of 1. Lastly, the constants '3' and '7' are also like terms, as they are both numerical values without any variable component.

To proceed with the addition, we group the like terms together. This can be written as (3p² + 2p²) + (-2p - 7p) + (3 + 7). This rearrangement allows for a clearer visualization of the terms that can be combined. The next step involves adding the coefficients of the like terms. For the 'p²' terms, we have 3 + 2 = 5, resulting in 5p². For the 'p' terms, we have -2 - 7 = -9, resulting in -9p. For the constant terms, we have 3 + 7 = 10. Therefore, the simplified expression after adding (3p² - 2p + 3) and (2p² - 7p + 7) is 5p² - 9p + 10. This resulting expression represents the sum of the original two expressions, with all like terms combined to present the answer in its simplest form. This example reinforces the fundamental principle of combining like terms in algebraic addition, applicable across various expressions.

4. Subtracting (-x² + y²) from (x²y²)

Subtracting algebraic expressions introduces a slight variation to the process of addition, but the core principle of identifying and combining like terms remains crucial. When we subtract one expression from another, we are essentially adding the negative of the expression being subtracted. In the problem (x²y²) - (-x² + y²), we are subtracting the expression (-x² + y²) from (x²y²). To handle this subtraction, we can rewrite the problem as (x²y²) + (-1)(-x² + y²). This step involves distributing the negative sign (or -1) across the terms within the parentheses of the second expression.

Distributing the -1, we get (x²y²) + (x² - y²). Now, we have a standard addition problem. We need to identify like terms. In this case, we have 'x²y²', 'x²', and '-y²'. However, there are no like terms among them. 'x²y²' is a term with both 'x' and 'y' squared and multiplied, while 'x²' has only 'x' squared, and '-y²' has only 'y' squared. Since there are no like terms to combine, the expression remains as it is. Therefore, the simplified expression after subtracting (-x² + y²) from (x²y²) is x²y² + x² - y². This example highlights that not all subtraction problems will result in simplification through combining like terms. Sometimes, the expression remains as is after distributing the negative sign, emphasizing the importance of careful observation and application of algebraic rules.

5. Subtracting (6xy - 2x) from (9xy + y - 2x)

Subtracting algebraic expressions requires careful attention to the signs and the proper distribution of the negative sign. In the problem (9xy + y - 2x) - (6xy - 2x), we are subtracting the expression (6xy - 2x) from (9xy + y - 2x). To execute this subtraction, we can rewrite the problem as an addition problem by distributing the negative sign across the terms within the parentheses of the expression being subtracted. This means we change the subtraction to addition and multiply each term inside the second set of parentheses by -1.

So, (9xy + y - 2x) - (6xy - 2x) becomes (9xy + y - 2x) + (-1)(6xy - 2x). Now, we distribute the -1 to both terms inside the second parentheses: -1 * 6xy = -6xy and -1 * -2x = +2x. The expression then becomes (9xy + y - 2x) + (-6xy + 2x). Now that we've transformed the subtraction into addition, we can proceed by identifying and combining like terms. Like terms are terms that have the same variables raised to the same powers. In this expression, '9xy' and '-6xy' are like terms, as are '-2x' and '+2x'. The term 'y' does not have any like terms in the expression.

Next, we combine the coefficients of the like terms. For the 'xy' terms, we have 9xy - 6xy = 3xy. For the 'x' terms, we have -2x + 2x = 0. The 'x' terms cancel each other out. The term 'y' remains as it is. Therefore, the simplified expression after subtracting (6xy - 2x) from (9xy + y - 2x) is 3xy + y. This example illustrates the critical role of distributing the negative sign correctly and then combining like terms to arrive at the simplified expression. Mastering this technique is essential for accurate algebraic manipulation.

In conclusion, the ability to add and subtract algebraic expressions is a fundamental skill in mathematics. By understanding the principles of identifying and combining like terms, and by paying close attention to the distribution of negative signs during subtraction, you can effectively simplify complex expressions. The examples provided in this guide offer a step-by-step approach to solving such problems, reinforcing the importance of meticulousness and accuracy in algebraic manipulation. As you continue your mathematical journey, these skills will serve as a solid foundation for tackling more advanced concepts and problem-solving scenarios. Practice and application are key to mastering these techniques, empowering you to confidently navigate the world of algebra and beyond.