Simplifying Algebraic Expressions A Step-by-Step Solution For -(1/5)(5x + 20) - (1/4)(4x - 28)
Introduction: Mastering Algebraic Simplification
In the realm of mathematics, algebraic expressions often appear complex and daunting. However, with a systematic approach and a clear understanding of fundamental principles, these expressions can be simplified to reveal their underlying simplicity. This article delves into the process of simplifying the expression $\bf{-\frac{1}{5}(5x + 20) - \frac{1}{4}(4x - 28)}$, providing a comprehensive step-by-step guide to arrive at the correct solution. This is a crucial skill for students learning algebra, as simplifying expressions is the foundation for solving more complex equations and problems. This article will not only provide the solution but also break down each step, ensuring a complete understanding of the process. We will explore the distributive property, combining like terms, and other essential algebraic techniques. By mastering these concepts, you will be well-equipped to tackle a wide range of algebraic challenges. So, let's embark on this journey of algebraic simplification together!
Problem Statement: Deconstructing the Expression
The problem at hand involves simplifying the expression $\bf{-\frac{1}{5}(5x + 20) - \frac{1}{4}(4x - 28)}$. This expression may appear intimidating at first glance, but it can be systematically simplified by applying the distributive property and combining like terms. The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms within parentheses. By understanding how to properly apply this property, we can begin to break down the complex expression into smaller, more manageable parts. Additionally, the ability to identify and combine like terms is essential for simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. By combining these terms, we can reduce the expression to its simplest form. The given expression includes fractions, which adds another layer of complexity. However, by carefully applying the rules of arithmetic and algebra, we can navigate these fractions and arrive at the correct solution. The goal is to transform the given expression into a simpler, equivalent form, which will make it easier to understand and work with in future calculations. So, let's delve into the steps required to achieve this simplification.
Step-by-Step Solution: Unveiling the Simplified Form
Step 1: Applying the Distributive Property
The first step in simplifying the expression is to apply the distributive property. This property states that $a(b + c) = ab + ac$. We need to distribute the fractions outside the parentheses to the terms inside. Let's apply this to our expression:
Distribute $-\frac{1}{5}$ to both terms inside the first parentheses:
Next, distribute $-\frac{1}{4}$ to both terms inside the second parentheses:
Now, combine the results:
Step 2: Combining Like Terms
The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, the like terms are the terms with $x$ and the constant terms. Let's rewrite the expression:
Combine the $x$ terms: $-x - x = -2x$
Combine the constant terms: $-4 + 7 = 3$
Step 3: Presenting the Simplified Expression
Now, combine the simplified terms to obtain the final expression:
Therefore, the simplified expression for $\bf{-\frac{1}{5}(5x + 20) - \frac{1}{4}(4x - 28)}$ is $\bf{-2x + 3}$. This process showcases how breaking down a complex expression into smaller, manageable steps can lead to a clear and concise solution. Understanding the distributive property and the concept of like terms are essential for mastering algebraic simplification.
Answer: The Final Simplified Form
Based on the step-by-step simplification process outlined above, the correct answer is:
This result confirms that by applying the distributive property and combining like terms, we can successfully simplify complex algebraic expressions. The journey from the initial expression to the final simplified form demonstrates the power of algebraic manipulation and the importance of understanding fundamental mathematical principles. It also highlights the importance of accuracy and attention to detail when working with algebraic expressions. A single error in distribution or combining like terms can lead to an incorrect result. Therefore, it is crucial to carefully review each step and ensure that all calculations are performed correctly. The ability to simplify algebraic expressions is a valuable skill in mathematics and is essential for success in more advanced topics. By mastering this skill, you will be well-prepared to tackle a wide range of mathematical challenges and problems.
Conclusion: Reinforcing Algebraic Simplification Skills
In conclusion, simplifying the expression $\bf{-\frac{1}{5}(5x + 20) - \frac{1}{4}(4x - 28)}$ to $\bf{-2x + 3}$ demonstrates the power of applying fundamental algebraic principles. The distributive property and the process of combining like terms are essential tools in simplifying complex expressions. This exercise reinforces the importance of a systematic approach to problem-solving in mathematics. By breaking down a complex problem into smaller, more manageable steps, we can arrive at the correct solution with greater confidence. Moreover, this example serves as a reminder of the significance of accuracy and attention to detail in algebraic manipulations. A single error can propagate through the entire solution, leading to an incorrect result. Therefore, it is crucial to carefully review each step and ensure that all calculations are performed correctly. The ability to simplify algebraic expressions is a fundamental skill in mathematics and is essential for success in more advanced topics. By mastering this skill, you will be well-prepared to tackle a wide range of mathematical challenges and problems. Continued practice and application of these principles will further solidify your understanding and enhance your problem-solving abilities in algebra. This understanding will serve as a strong foundation for future mathematical endeavors and will empower you to approach complex problems with confidence and clarity.
