Simplifying Algebraic Expressions A Step-by-Step Guide With Practice Problems

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. This article delves into the process of simplifying a given algebraic expression, providing a step-by-step solution and a detailed explanation to enhance understanding. We will dissect the expression ${\frac{1}{5}(150x - 80y + 50 - 50x - 25y + 20)}$ and simplify it to its most basic form, while also exploring the underlying mathematical principles. Understanding how to simplify algebraic expressions is crucial for success in algebra and beyond. Algebraic expressions are the building blocks of more complex equations and formulas, making proficiency in this area essential for students and professionals alike. This article aims to provide a clear and concise explanation of the simplification process, empowering readers to tackle similar problems with confidence.

Understanding the Expression

Let's begin by examining the given expression: ${\frac{1}{5}(150x - 80y + 50 - 50x - 25y + 20)}$. This expression involves variables ${x}$ and ${y}$, as well as constants. The presence of parentheses indicates that we need to apply the distributive property after simplifying the terms inside the parentheses. Simplifying algebraic expressions involves combining like terms and then applying the distributive property. Before diving into the solution, it's crucial to understand the order of operations (PEMDAS/BODMAS), which dictates that we should simplify expressions within parentheses first. This means we need to combine the ${x}$ terms, the ${y}$ terms, and the constant terms separately. This foundational step sets the stage for applying the distributive property and arriving at the simplified expression. By carefully following this process, we can ensure accuracy and a clear understanding of the underlying mathematical principles. This section emphasizes the importance of understanding the expression's structure before attempting to simplify it, highlighting the significance of identifying like terms and the role of the distributive property.

Step-by-Step Solution

To simplify the expression, we'll follow these steps:

1. Combine Like Terms Inside the Parentheses

First, identify and combine the like terms within the parentheses:

• ${x}$ terms: ${150x - 50x = 100x}$
• ${y}$ terms: ${-80y - 25y = -105y}$
• Constants: ${50 + 20 = 70}$

So, the expression inside the parentheses becomes: ${100x - 105y + 70}$.

2. Apply the Distributive Property

Next, we distribute the ${\frac{1}{5}}$ across the simplified expression inside the parentheses:

${\frac{1}{5}(100x - 105y + 70) = \frac{1}{5}(100x) - \frac{1}{5}(105y) + \frac{1}{5}(70)}$

3. Perform the Multiplication

Now, we perform the multiplication for each term:

• ${\frac{1}{5}(100x) = 20x}$
• ${\frac{1}{5}(105y) = 21y}$
• ${\frac{1}{5}(70) = 14}$

4. Write the Simplified Expression

Finally, we combine the results to get the simplified expression:

${20x - 21y + 14}$

This step-by-step solution breaks down the simplification process into manageable parts. Each step is clearly explained, ensuring that the reader can follow along and understand the logic behind each operation. The emphasis on combining like terms first is crucial for accurate simplification, and the distributive property is applied methodically to avoid errors. By providing a detailed breakdown, this section empowers readers to approach similar problems with confidence and a clear understanding of the necessary steps. The solution highlights the importance of precision and attention to detail in algebraic simplification.

Identifying the Equivalent Expression

Comparing our simplified expression, ${20x - 21y + 14}$, with the given options, we can see that it matches option A.

• A. ${20x - 21y + 14}$
• B. ${20x + 11y + 14}$
• C. ${20x - 11y + 6}$
• D. ${20x + 21y + 6}$

Therefore, the equivalent expression is option A. This section emphasizes the importance of careful comparison and attention to detail. After performing the simplification, it's crucial to compare the result with the given options to identify the correct answer. This step reinforces the accuracy of the simplification process and ensures that the correct equivalent expression is selected. By highlighting the matching process, this section completes the solution and provides a clear conclusion. It underscores the importance of double-checking the answer to avoid errors and ensure confidence in the final result. The ability to accurately identify equivalent expressions is a key skill in algebra, and this section reinforces that skill by demonstrating the final step in the problem-solving process.

Common Mistakes to Avoid When Simplifying Algebraic Expressions

When simplifying algebraic expressions, several common mistakes can lead to incorrect answers. Being aware of these pitfalls is crucial for accuracy and success in algebra. Let's explore some of the most frequent errors and how to avoid them.

1. Incorrectly Combining Like Terms

One of the most common mistakes is combining terms that are not alike. Remember, like terms have the same variables raised to the same powers. For example, ${3x}$ and ${5x}$ are like terms, but ${3x}$ and ${5x^2}$ are not. To avoid this, carefully examine each term and ensure that you are only combining those with identical variable parts. Double-checking your work can help catch these errors. This mistake often arises from a misunderstanding of the definition of like terms, so a solid grasp of this concept is essential. Emphasizing the importance of meticulous examination and double-checking can significantly reduce the occurrence of this error.

2. Sign Errors

Sign errors are another frequent source of mistakes. When distributing a negative sign or combining terms with different signs, it's easy to make a mistake. For instance, when subtracting a negative number, remember to add its positive counterpart. Similarly, when distributing a negative sign, ensure that you change the sign of every term inside the parentheses. To avoid sign errors, take your time and write out each step clearly. Using parentheses to keep track of signs can also be helpful. This type of error often stems from carelessness, so promoting a methodical and deliberate approach is key to preventing it. The use of parentheses as a visual aid can also serve as a helpful strategy for students.

3. Incorrect Application of the Distributive Property

The distributive property, which states that ${a(b + c) = ab + ac}$, is a fundamental concept in algebra. However, it's often misapplied, particularly when dealing with more complex expressions. A common mistake is forgetting to distribute the term to every term inside the parentheses. For example, in the expression ${2(x + y + z)}$, you must multiply the 2 by ${x}$, ${y}$, and ${z}$. To avoid this, make sure you multiply the term outside the parentheses by each term inside. Writing out the distribution step-by-step can help ensure accuracy. This error often arises from a lack of attention to detail, so emphasizing the importance of thoroughness is crucial. Providing clear examples and practice problems can help students master the correct application of the distributive property.

4. Order of Operations Mistakes

The order of operations (PEMDAS/BODMAS) dictates the sequence in which mathematical operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Failing to follow this order can lead to incorrect simplification. For example, if an expression contains both multiplication and addition, you must perform the multiplication before the addition. To avoid order of operations mistakes, always follow the PEMDAS/BODMAS rule. Writing out each step in the correct order can help you stay on track. This mistake often stems from a lack of understanding or a lapse in memory, so reinforcing the order of operations through practice is essential. Using mnemonic devices like PEMDAS/BODMAS can also help students remember the correct sequence.

5. Forgetting to Simplify Completely

A final common mistake is not simplifying the expression completely. After performing the initial steps, such as combining like terms and applying the distributive property, it's essential to check if there are any further simplifications that can be made. For example, you might need to combine like terms again or factor out a common factor. To avoid this mistake, always double-check your work and ensure that the expression is in its simplest form. Practice and familiarity with different types of expressions can help you recognize when further simplification is possible. This mistake often arises from a premature conclusion, so encouraging students to always double-check their work is crucial. Highlighting the concept of the simplest form and providing examples can help students understand when an expression is fully simplified.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in simplifying algebraic expressions. Each mistake highlights the importance of specific skills and concepts, emphasizing the need for a solid foundation in algebra. Regular practice and attention to detail are key to mastering the art of simplifying expressions.

Practice Problems

To solidify your understanding of simplifying algebraic expressions, let's work through a few practice problems.

Practice Problem 1

Simplify the expression: ${3(2x - 5y) + 4(x + 2y)}$

Solution:

1. Apply the distributive property:
   • ${3(2x - 5y) = 6x - 15y}$
   • ${4(x + 2y) = 4x + 8y}$
2. Combine the results: ${6x - 15y + 4x + 8y}$
3. Combine like terms:
   • ${x}$ terms: ${6x + 4x = 10x}$
   • ${y}$ terms: ${-15y + 8y = -7y}$
4. Write the simplified expression: ${10x - 7y}$

Practice Problem 2

Simplify the expression: ${\frac{1}{2}(8a + 6b) - \frac{1}{3}(9a - 12b)}$

Solution:

1. Apply the distributive property:
   • ${\frac{1}{2}(8a + 6b) = 4a + 3b}$
   • ${-\frac{1}{3}(9a - 12b) = -3a + 4b}$
2. Combine the results: ${4a + 3b - 3a + 4b}$
3. Combine like terms:
   • ${a}$ terms: ${4a - 3a = a}$
   • ${y}$ terms: ${3b + 4b = 7b}$
4. Write the simplified expression: ${a + 7b}$

Practice Problem 3

Simplify the expression: ${-2(3m - 4n + 5) + 5(2m + n - 1)}$

Solution:

1. Apply the distributive property:
   • ${-2(3m - 4n + 5) = -6m + 8n - 10}$
   • ${5(2m + n - 1) = 10m + 5n - 5}$
2. Combine the results: ${-6m + 8n - 10 + 10m + 5n - 5}$
3. Combine like terms:
   • ${m}$ terms: ${-6m + 10m = 4m}$
   • ${n}$ terms: ${8n + 5n = 13n}$
   • Constants: ${-10 - 5 = -15}$
4. Write the simplified expression: ${4m + 13n - 15}$

These practice problems provide an opportunity to apply the concepts and techniques discussed in this article. Each problem is accompanied by a detailed solution, allowing readers to check their work and identify areas for improvement. By working through these examples, readers can reinforce their understanding of algebraic simplification and build confidence in their problem-solving abilities. The variety of problems ensures a comprehensive understanding of the topic.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics. By following the steps outlined in this article—combining like terms, applying the distributive property, and performing the necessary operations—you can simplify complex expressions with confidence. Remember to avoid common mistakes, such as incorrectly combining like terms, making sign errors, or misapplying the distributive property. Practice regularly and double-check your work to ensure accuracy. Mastering this skill will not only help you in algebra but also in various other areas of mathematics and beyond. This article has provided a comprehensive guide to simplifying algebraic expressions, empowering readers to tackle these problems effectively. From understanding the basic principles to working through practice problems, this resource aims to build a solid foundation in algebraic simplification. With consistent effort and attention to detail, anyone can master this essential mathematical skill.