Simplifying Algebraic Expressions Using The Distributive Property A Step-by-Step Guide

Algebraic expressions can often appear complex, but by understanding the fundamental properties of real numbers, we can systematically simplify them. In this article, we will walk through the simplification of the expression $-3+2(2+x+3)$, highlighting the properties used at each step. We'll particularly focus on the distributive property, which plays a crucial role in this process. Simplifying expressions is a core skill in algebra, and mastering it is essential for solving more complex equations and problems. This article aims to provide a clear, step-by-step explanation, making it easier for anyone to understand and apply these concepts.

Step 1: $= -3 + 2(x+2+3)$

The initial step in simplifying the expression $-3+2(2+x+3)$ involves rearranging the terms within the parentheses. This might seem like a small change, but it sets the stage for combining like terms, which is a fundamental aspect of simplifying algebraic expressions. The key here is to recognize that the order of addition doesn't affect the sum, a principle known as the commutative property of addition. By rearranging the terms to group the constants together, we make it easier to see the next simplification step. Rearranging terms in this way allows us to visually organize the expression, making it less cluttered and more manageable. The commutative property ensures that this rearrangement does not alter the value of the expression, maintaining mathematical accuracy throughout the process. This first step, while seemingly simple, is crucial for creating a clear path toward the final simplified form. Understanding and applying this principle of rearrangement is a cornerstone of algebraic manipulation. This meticulous approach ensures that each subsequent step builds upon a solid foundation, ultimately leading to an accurate and simplified expression. Furthermore, this step underscores the importance of recognizing and utilizing basic algebraic properties to streamline complex expressions, making them easier to work with. The careful attention to detail in this initial phase lays the groundwork for more complex operations and ensures that the final result is both correct and concise. In essence, this step is a testament to the power of organization and strategic manipulation in simplifying algebraic expressions. It's a reminder that even small adjustments can have a significant impact on the overall clarity and solvability of a problem.

Step 2: $= -3 + 2(x+5)$

In the second step, we further simplify the expression by combining the constant terms within the parentheses. Specifically, we add $2$ and $3$ together, resulting in $5$. This step relies on the basic arithmetic operation of addition and the principle of combining like terms, which is a fundamental concept in algebra. By combining these constants, we reduce the complexity of the expression inside the parentheses, making it easier to handle in subsequent steps. This simplification is crucial because it reduces the number of terms and prepares the expression for the application of the distributive property. Combining like terms is a strategy that allows us to condense and organize expressions, making them more manageable and less prone to errors. This step demonstrates the importance of recognizing and grouping similar terms to streamline algebraic manipulations. By simplifying the expression within the parentheses, we are essentially making the expression more concise and easier to work with. This careful attention to detail ensures that we are maintaining accuracy while reducing the complexity of the expression. The act of simplifying through combining like terms not only streamlines the immediate step but also lays the groundwork for future operations, making the entire simplification process more efficient and less prone to errors. This meticulous approach underscores the value of clarity and precision in algebraic manipulations, ensuring that each step contributes to the overall goal of arriving at the simplest form of the expression. Moreover, the combination of constants serves as a practical application of basic arithmetic within a broader algebraic context, reinforcing the interconnectedness of mathematical concepts and their utility in problem-solving.

Step 3: $= -3 + 2x + 10$

This step is the heart of the simplification process, where the distributive property is applied. The distributive property states that $a(b+c) = ab + ac$. In our case, we are distributing the $2$ across the terms inside the parentheses, $x$ and $5$. This means we multiply $2$ by $x$ to get $2x$, and $2$ by $5$ to get $10$. The application of the distributive property is a crucial step in expanding and simplifying algebraic expressions. Understanding the distributive property is fundamental to manipulating expressions and solving equations. This step effectively removes the parentheses, transforming the expression into a more manageable form. By distributing the $2$, we are essentially breaking down the multiplication into smaller, more easily handled components. This process not only simplifies the expression but also reveals the individual terms that will be combined in subsequent steps. The correct application of the distributive property is essential for maintaining the equivalence of the expression throughout the simplification process. Any error in this step can lead to an incorrect final result, underscoring the importance of precision and accuracy in algebraic manipulations. Furthermore, this step highlights the power of the distributive property in transforming expressions, allowing us to rewrite them in different forms that are often more convenient for further simplification or problem-solving. The ability to apply the distributive property effectively is a hallmark of algebraic proficiency and is essential for success in more advanced mathematical topics. In essence, this step is a cornerstone of the simplification process, enabling us to move from a compact form with parentheses to an expanded form that is ready for further manipulation.

The property of real numbers used to transition from step 2 to step 3 is the distributive property.

Step 4: $= 2x + (-3) + 10$

In step four, we rearrange the terms in the expression, placing the term with the variable ($2x$) first, followed by the constant terms $(-3)$ and $10$. This rearrangement is done to group like terms together, which is a standard practice in simplifying algebraic expressions. The property that allows us to change the order of terms in addition without changing the result is the commutative property of addition. Rearranging terms in this way helps to visually organize the expression and makes it easier to combine the constant terms in the next step. This step might seem minor, but it contributes to the overall clarity and organization of the simplification process. By placing the variable term first, we are adhering to a common convention in algebraic notation, which often makes expressions easier to read and understand. This convention is particularly helpful when dealing with more complex expressions, where the organization of terms can significantly impact clarity. Furthermore, the rearrangement highlights the like terms that can be combined, making the next step more intuitive. The commutative property of addition ensures that this rearrangement does not alter the value of the expression, maintaining mathematical integrity throughout the process. This step underscores the importance of careful organization and attention to detail in algebraic manipulations, even in seemingly simple operations. By strategically rearranging terms, we can often simplify the simplification process and reduce the likelihood of errors. In essence, this step is a testament to the power of visual organization and strategic manipulation in making algebraic expressions more accessible and easier to work with.

Step 5: $= 2x + 7$

The final step in simplifying the expression involves combining the constant terms. We add $-3$ and $10$ together, which results in $7$. This step utilizes the basic arithmetic operation of addition and represents the culmination of the simplification process. By combining these constants, we arrive at the simplest form of the expression, $2x + 7$. Combining the constant terms is a crucial step in reducing the expression to its most concise form. This final simplification provides a clear and unambiguous representation of the original expression, making it easier to understand and use in further calculations or problem-solving. The result, $2x + 7$, is a simplified algebraic expression that cannot be further reduced without additional information or context. This step highlights the importance of following a systematic approach to simplification, ensuring that all like terms are combined and the expression is presented in its most efficient form. The process of adding $-3$ and $10$ demonstrates the practical application of integer arithmetic within an algebraic context. This reinforces the interconnectedness of different mathematical concepts and their utility in solving problems. Furthermore, the final simplified form provides a clear and concise answer, making it easier to communicate the result and use it in subsequent steps of a larger problem. In essence, this step represents the culmination of the simplification process, demonstrating the power of algebraic manipulation in transforming complex expressions into simpler, more manageable forms. The final result serves as a testament to the effectiveness of the step-by-step approach and the importance of precision in algebraic operations.

Conclusion

In summary, we have successfully simplified the expression $-3+2(2+x+3)$ to $2x + 7$ by systematically applying the properties of real numbers. The key steps involved rearranging terms, combining like terms, and, most importantly, applying the distributive property. Understanding and applying these properties is essential for simplifying algebraic expressions and solving more complex mathematical problems. This step-by-step guide demonstrates the importance of clarity, precision, and organization in algebraic manipulations. By mastering these fundamental skills, you can confidently tackle a wide range of algebraic challenges. The ability to simplify expressions efficiently and accurately is a cornerstone of mathematical proficiency and is essential for success in more advanced mathematical topics. This article serves as a reminder of the power of basic algebraic principles and their utility in transforming complex expressions into simpler, more manageable forms. The process of simplification not only reduces the complexity of an expression but also provides valuable insights into its underlying structure and relationships. By carefully following the steps outlined in this guide, anyone can learn to simplify algebraic expressions with confidence and accuracy. This skill is not only valuable in mathematics but also in various other fields that require analytical thinking and problem-solving. In conclusion, the ability to simplify expressions is a fundamental skill that empowers individuals to tackle a wide range of mathematical and real-world challenges.