Solving Algebraic Expressions A Step-by-Step Guide To Simplifying 3.6(-3.7r + 3.8a - 3.2)

Introduction: Delving into Algebraic Expressions

In the realm of mathematics, algebraic expressions form the bedrock of more complex equations and formulas. Algebraic expressions are mathematical phrases that combine variables (letters representing unknown values), constants (fixed numerical values), and mathematical operations such as addition, subtraction, multiplication, and division. Understanding how to manipulate and simplify these expressions is a crucial skill in mathematics. This article focuses on dissecting and solving the algebraic expression $3.6(-3.7r + 3.8a - 3.2)$, providing a step-by-step guide to its simplification and offering insights into the underlying mathematical principles. This exploration not only aids in solving this specific problem but also equips you with the tools necessary to tackle a wide array of algebraic challenges. Mastering these fundamental concepts is essential for progress in algebra and beyond, laying the groundwork for success in higher-level mathematics, science, and engineering disciplines. So, let’s embark on this journey of mathematical discovery, unraveling the intricacies of this expression and gaining a deeper understanding of algebraic simplification.

Breaking Down the Expression: $3.6(-3.7r + 3.8a - 3.2)$

To effectively tackle the algebraic expression $3.6(-3.7r + 3.8a - 3.2)$, we need to break it down into its constituent parts. This expression involves the multiplication of a constant (3.6) with a trinomial expression enclosed in parentheses (-3.7r + 3.8a - 3.2). The trinomial itself is composed of three terms: -3.7r, which represents a variable term where 'r' is a variable multiplied by the coefficient -3.7; 3.8a, another variable term where 'a' is a variable multiplied by the coefficient 3.8; and -3.2, a constant term. The structure of the expression suggests that we need to apply the distributive property of multiplication over addition and subtraction to simplify it. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. By applying this principle, we can multiply the constant 3.6 with each term inside the parentheses, thereby expanding the expression. This process is crucial for removing the parentheses and combining like terms, which is the next step in simplifying the expression. Understanding the individual components and the overall structure of the expression is the first step towards solving it effectively.

Applying the Distributive Property: A Step-by-Step Approach

The distributive property is the cornerstone of simplifying the expression $3.6(-3.7r + 3.8a - 3.2)$. This property allows us to multiply a single term by each term within a set of parentheses. In this case, we need to distribute the 3.6 across each of the three terms inside the parentheses: -3.7r, 3.8a, and -3.2. Let's break down the application of this property step-by-step:

1. Multiply 3.6 by -3.7r:

   • This results in (3.6 * -3.7r) = -13.32r.

2. Multiply 3.6 by 3.8a:

   • This results in (3.6 * 3.8a) = 13.68a.

3. Multiply 3.6 by -3.2:

   • This results in (3.6 * -3.2) = -11.52.

By performing these multiplications, we've successfully distributed the 3.6 across the trinomial. The expression now transforms from a compact form to an expanded form, which makes it easier to identify and combine like terms, if any. This step is crucial because it removes the parentheses, which is often a necessary precursor to further simplification. The expanded expression now reads as -13.32r + 13.68a - 11.52, setting the stage for the final simplification steps.

Simplifying the Expanded Expression: Combining Like Terms

After applying the distributive property, our expression stands as -13.32r + 13.68a - 11.52. The next step in simplifying this expression involves identifying and combining like terms. Like terms are terms that have the same variable raised to the same power. In the given expression, we have three terms: -13.32r, 13.68a, and -11.52. Upon closer inspection, we can see that there are no like terms present. This is because each term contains a different variable ('r' and 'a') or is a constant. The term -13.32r contains the variable 'r', the term 13.68a contains the variable 'a', and -11.52 is a constant term without any variable. Since there are no like terms, we cannot combine any of them. This means that the expression -13.32r + 13.68a - 11.52 is already in its simplest form. The absence of like terms often occurs in algebraic expressions, and recognizing this is crucial to avoid unnecessary steps. In such cases, the expanded form obtained after distribution becomes the final simplified form. This outcome highlights the importance of understanding the definition of like terms and applying it correctly in the simplification process.

Final Simplified Form: -13.32r + 13.68a - 11.52

Having applied the distributive property and assessed the possibility of combining like terms, we arrive at the final simplified form of the expression $3.6(-3.7r + 3.8a - 3.2)$. The simplified form is -13.32r + 13.68a - 11.52. This expression represents the most concise way to express the original algebraic phrase. It's important to note that this form is equivalent to the initial expression but is easier to interpret and use in further calculations or problem-solving scenarios. The process of simplification has allowed us to transform the expression from a product of a constant and a trinomial into a sum of individual terms. This transformation not only makes the expression more readable but also highlights the relationships between the variables and constants involved. The absence of like terms in this expression means that no further reduction is possible, and the expression is in its most elementary state. This final form is the culmination of our step-by-step simplification process, demonstrating the power of algebraic manipulation in streamlining mathematical expressions.

Conclusion: Mastering Algebraic Simplification

In conclusion, simplifying algebraic expressions like $3.6(-3.7r + 3.8a - 3.2)$ is a fundamental skill in mathematics. This process involves understanding the structure of the expression, applying the distributive property, identifying and combining like terms, and arriving at the simplest possible form. Through our step-by-step analysis, we transformed the initial expression into its simplified form: -13.32r + 13.68a - 11.52. This journey highlights the importance of each step in the simplification process, from recognizing the components of the expression to applying the correct algebraic rules. Mastering these techniques not only enables us to solve specific problems but also builds a strong foundation for more advanced mathematical concepts. The ability to simplify algebraic expressions is crucial in various fields, including science, engineering, and economics, where mathematical models are used to represent real-world phenomena. By understanding and practicing these principles, students and professionals alike can enhance their problem-solving skills and confidently tackle complex mathematical challenges. The journey through algebraic simplification is a testament to the power and elegance of mathematical principles in making complex problems more manageable and understandable.