Simplifying Algebraic Expressions A Step-by-Step Guide

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This article provides a detailed walkthrough on simplifying the algebraic expression $\bf{-\frac{1}{5}(5x + 20) - \frac{1}{4}(4x - 28)}$ . Algebraic expressions are a fundamental part of mathematics, and mastering the techniques to simplify them is crucial for success in algebra and beyond. In this guide, we'll break down the process step-by-step, ensuring a clear understanding of each operation involved. Whether you're a student looking to improve your algebra skills or someone brushing up on their math, this article will provide you with the tools and knowledge you need. Understanding how to simplify such expressions not only enhances your problem-solving abilities but also lays a solid foundation for more advanced mathematical concepts. So, let's dive in and unravel this expression together!

Understanding the Basics of Algebraic Expressions

Before we tackle the main problem, let's briefly discuss what algebraic expressions are and why simplifying them is important. Algebraic expressions are combinations of variables (like $x$), constants (like 5, 20, $-\frac{1}{5}$, and $\frac{1}{4}$), and mathematical operations (addition, subtraction, multiplication, division). Simplifying an algebraic expression means rewriting it in a more concise and manageable form, without changing its value. This often involves combining like terms and applying the distributive property. Why is simplification important? Simplified expressions are easier to work with. They make it easier to solve equations, graph functions, and perform other mathematical operations. Think of it as decluttering a workspace – a clean and organized expression makes the rest of the problem-solving process much smoother. In this particular expression, we have fractions, parentheses, and multiple terms, making simplification essential for finding the most straightforward form.

The Importance of Order of Operations

When simplifying expressions, it's vital to follow the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order ensures that we perform the operations in the correct sequence, leading to the correct answer. Ignoring the order of operations can result in significant errors. For example, in our expression, we need to address the parentheses first by applying the distributive property before we can combine any like terms. Understanding and applying PEMDAS is a fundamental skill in algebra and is crucial for accurately simplifying expressions. This ensures that each step we take is mathematically sound and that we arrive at the correct simplified form. So, before we dive into the specific steps for our problem, keep PEMDAS in mind as our guiding principle.

Step-by-Step Simplification of -(1/5)(5x + 20) - (1/4)(4x - 28)

Now, let's break down the simplification of the expression $\bf{-\frac{1}{5}(5x + 20) - \frac{1}{4}(4x - 28)}$ into manageable steps. We'll focus on clarity and accuracy, ensuring you understand each operation as we go. Our primary goal is to eliminate the parentheses by applying the distributive property, then combine like terms to achieve the simplified form.

Step 1: Applying the Distributive Property

The first step in simplifying this expression is to apply the distributive property. The distributive property states that $a(b + c) = ab + ac$. We'll apply this to both terms in our expression. For the first term, $\bf{-\frac{1}{5}(5x + 20)}$, we distribute the $-\frac{1}{5}$ across both $5x$ and $20$. This means we multiply $-\frac{1}{5}$ by $5x$ and then by $20$. Similarly, for the second term, $\bf{-\frac{1}{4}(4x - 28)}$, we distribute the $-\frac{1}{4}$ across both $4x$ and $-28$. This involves multiplying $-\frac{1}{4}$ by $4x$ and then by $-28$. It's crucial to pay close attention to the signs during this process, as incorrect sign handling is a common source of errors. By carefully applying the distributive property, we expand the expression, setting the stage for combining like terms in the subsequent steps. This step is the foundation for the rest of the simplification process, so let's perform it meticulously.

Step 2: Performing the Multiplication

Following the distribution, our next step is to perform the multiplications. Let's break this down:

  • For the first term, we have $\bf{-\frac{1}{5} \times 5x}$. This simplifies to $\bf{-x}$, because the 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with $-1$ multiplied by $x$, which is $-x$.
  • Next, we multiply $\bf{-\frac{1}{5}}$ by $20$. This gives us $\bf{-\frac{20}{5}}$, which simplifies to $\bf{-4}$. So, the first part of our expression now looks like $\bf{-x - 4}$.
  • Moving on to the second term, we have $\bf{-\frac{1}{4} \times 4x}$. Similar to the first term, the 4s cancel out, leaving us with $\bf{-x}$.
  • Finally, we multiply $\bf{-\frac{1}{4}}$ by $\bf{-28}$. A negative times a negative is a positive, so we have $\bf{\frac{28}{4}}$, which simplifies to $\bf{7}$. The second part of our expression is now $\bf{-x + 7}$.

Performing these multiplications carefully and accurately is essential. Each multiplication is a building block, and any mistake here will carry through the rest of the simplification. After this step, we'll have a clearer expression with no parentheses, making it easier to combine like terms.

Step 3: Combining Like Terms

Now that we've applied the distributive property and performed the multiplications, our expression looks like this: $\bf{-x - 4 - x + 7}$. The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have two types of like terms: terms with the variable $x$ and constant terms (numbers without a variable).

  • Let's start by combining the terms with $x$. We have $\bf{-x}$ and $\bf{-x}$. Adding these together gives us $\bf{-2x}$.
  • Next, we combine the constant terms: $\bf{-4}$ and $\bf{7}$. Adding these together gives us $\bf{3}$.

Combining like terms simplifies our expression significantly. It reduces the number of terms and makes the expression easier to understand and work with. This step is a crucial part of simplifying algebraic expressions and is a skill that will be used extensively in algebra and beyond. After combining like terms, we're very close to our final simplified expression.

Step 4: Writing the Simplified Expression

After combining like terms, we have $\bf{-2x + 3}$. This is the simplified form of the original expression, $\bf{-\frac{1}{5}(5x + 20) - \frac{1}{4}(4x - 28)}$. We've successfully eliminated the parentheses, performed the necessary multiplications, and combined like terms to arrive at this concise form. This final expression is much easier to work with and provides a clear representation of the original expression's value for any given $x$.

Conclusion: The Simplified Form and Its Significance

In conclusion, the simplified expression for $\bf{-\frac{1}{5}(5x + 20) - \frac{1}{4}(4x - 28)}$ is $\bf{-2x + 3}$. This result corresponds to option A in the original question. The process we followed involved applying the distributive property, performing multiplications, and combining like terms. Each step was crucial in transforming the complex original expression into a simpler, more manageable form. Simplifying algebraic expressions is a fundamental skill in mathematics. It allows us to rewrite expressions in a way that is easier to understand and work with. Simplified expressions are essential for solving equations, graphing functions, and performing other mathematical operations. The ability to simplify expressions efficiently and accurately is a valuable asset in algebra and beyond. By mastering these techniques, you'll be well-equipped to tackle more complex mathematical problems. Remember to always follow the order of operations (PEMDAS) and pay close attention to signs to ensure accurate results. With practice, simplifying expressions will become second nature, making your mathematical journey smoother and more successful.

Practice Problems

To solidify your understanding of simplifying algebraic expressions, try working through these practice problems. Each problem provides an opportunity to apply the steps we've discussed in this article. Remember to follow the order of operations, pay attention to signs, and combine like terms carefully. The more you practice, the more confident you'll become in your ability to simplify expressions. Practice is key to mastering any mathematical skill, and these problems will help you reinforce what you've learned.

Further Resources

If you're looking to deepen your understanding of algebraic expressions and simplification techniques, there are many excellent resources available. Online platforms like Khan Academy offer comprehensive lessons and practice exercises. Textbooks and workbooks are also valuable tools, providing structured explanations and a wide range of problems to solve. Don't hesitate to seek help from teachers, tutors, or classmates if you encounter challenges. Collaboration and discussion can often clarify concepts and provide new perspectives. Continuous learning and exploration are key to building a strong foundation in mathematics.