Subtracting Algebraic Expressions A Comprehensive Guide

When dealing with algebraic expressions, subtraction is a fundamental operation that requires careful attention to signs and like terms. In this section, we will delve into the process of subtracting the expression (2p + 3q) from (3p - 2q). This exercise not only reinforces basic algebraic principles but also lays the groundwork for more complex manipulations in algebra. Understanding how to subtract algebraic expressions is crucial for solving equations, simplifying formulas, and tackling various mathematical problems.

To subtract (2p + 3q) from (3p - 2q), we need to distribute the negative sign across the terms of the expression being subtracted. This means that the expression (2p + 3q) will become (-2p - 3q). The next step involves combining like terms. Like terms are those that have the same variable raised to the same power. In this case, the like terms are those involving p and those involving q. By carefully rearranging and combining these terms, we can simplify the expression and arrive at the correct result. This process highlights the importance of paying close attention to the signs of the terms and ensuring that like terms are accurately combined to achieve the simplified form of the expression. Mastering this skill is essential for success in algebra and related mathematical fields.

So, let's proceed step-by-step to ensure clarity and precision. First, rewrite the subtraction as an addition of the negative:

(3p - 2q) - (2p + 3q) = (3p - 2q) + (-2p - 3q)

Now, combine the like terms:

= (3p - 2p) + (-2q - 3q)

This simplifies to:

= p - 5q

Therefore, the result of subtracting (2p + 3q) from (3p - 2q) is p - 5q. This demonstrates a clear application of algebraic principles, particularly the distribution of the negative sign and the combination of like terms. The ability to perform such operations accurately and efficiently is fundamental to more advanced mathematical concepts and problem-solving techniques. This example serves as a solid foundation for further exploration of algebraic manipulations and their practical applications.

In this section, we address the task of subtracting the algebraic expression (5x² - 3xy + y²) from (7x² + xy - 4y²). This problem introduces expressions with multiple variables and terms, making it a crucial step in mastering algebraic subtraction. The process involves distributing the negative sign, identifying like terms, and combining them to simplify the expression. This exercise reinforces the understanding of how to handle different terms and variables, which is essential for solving more complex equations and problems in algebra. By carefully following each step, we can arrive at the correct solution and strengthen our algebraic skills.

To begin, subtraction in algebra is equivalent to adding the negative of the expression being subtracted. This means we need to distribute the negative sign across all terms of the expression (5x² - 3xy + y²). This distribution is a critical step, as it ensures that the signs of all terms are correctly reversed before combining like terms. Once the negative sign is properly distributed, we can then proceed to identify and group like terms. Like terms are terms that have the same variables raised to the same powers. In this case, we will group the terms with x², the terms with xy, and the terms with y². This grouping allows us to combine the coefficients of the like terms, simplifying the expression and leading us to the final result.

The problem at hand requires a meticulous approach to ensure accuracy. Let’s rewrite the problem by distributing the negative sign:

(7x² + xy - 4y²) - (5x² - 3xy + y²) = (7x² + xy - 4y²) + (-5x² + 3xy - y²)

Next, we combine like terms:

= (7x² - 5x²) + (xy + 3xy) + (-4y² - y²)

Simplifying each group of like terms gives us:

= 2x² + 4xy - 5y²

Therefore, the result of subtracting (5x² - 3xy + y²) from (7x² + xy - 4y²) is 2x² + 4xy - 5y². This exercise underscores the importance of paying close attention to detail when working with algebraic expressions, especially when subtraction and multiple terms are involved. The ability to accurately subtract such expressions is a fundamental skill in algebra, paving the way for solving more intricate problems in mathematics and related fields.

In this section, we will tackle subtracting the algebraic expression (5a + 4b + 7c) from (8a - 3b + 11c). This exercise is designed to reinforce the principles of algebraic subtraction with expressions involving multiple variables. The key to solving this problem lies in the correct distribution of the negative sign and the accurate combination of like terms. Mastering this process is essential for simplifying complex algebraic equations and solving problems in various mathematical contexts. By working through this example step-by-step, we can enhance our understanding and proficiency in algebraic manipulations.

The initial step in this subtraction problem is to rewrite the subtraction as an addition of the negative. This involves distributing the negative sign across the terms of the expression being subtracted, which is (5a + 4b + 7c) in this case. Distributing the negative sign is crucial because it changes the sign of each term within the parentheses, ensuring that we subtract each term correctly. Once the negative sign has been properly distributed, we can proceed to the next step, which is identifying and combining like terms. Like terms are those that contain the same variables raised to the same powers. In this expression, the like terms are those involving a, those involving b, and those involving c. By grouping these like terms together, we can simplify the expression and arrive at the final answer. The ability to accurately perform these steps is fundamental to algebraic problem-solving.

To solve this, let’s start by rewriting the subtraction as addition of the negative:

(8a - 3b + 11c) - (5a + 4b + 7c) = (8a - 3b + 11c) + (-5a - 4b - 7c)

Now, we combine the like terms:

= (8a - 5a) + (-3b - 4b) + (11c - 7c)

Simplifying each group of like terms gives us:

= 3a - 7b + 4c

Therefore, the result of subtracting (5a + 4b + 7c) from (8a - 3b + 11c) is 3a - 7b + 4c. This example demonstrates the importance of meticulous attention to detail when performing algebraic operations, especially when dealing with multiple variables and terms. The ability to accurately subtract algebraic expressions is a fundamental skill that underpins more advanced mathematical concepts and problem-solving techniques. By mastering these basic operations, we can build a strong foundation for tackling more complex mathematical challenges.

By mastering these algebraic subtraction techniques, you will be well-equipped to handle a wide range of mathematical problems. The ability to accurately subtract expressions is a cornerstone of algebra, paving the way for success in higher-level mathematics and related fields.