Identify The Errors In Gatien's Work And Provide A Correct Solution For Simplifying The Expression A = 3(2x + 7) − 4.

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Introduction: Decoding the Algebraic Expression

In the realm of mathematics, precision is paramount. Algebraic expressions, with their intricate dance of variables and constants, demand meticulous attention to detail. In this article, we delve into a common pitfall encountered by students: errors in simplifying algebraic expressions. We will dissect Gatien's attempt to simplify the expression A = 3(2x + 7) − 4, pinpointing the missteps and offering a corrected solution. This exploration will not only illuminate the specific errors but also underscore the fundamental principles that govern algebraic manipulation. Understanding these principles is crucial for building a strong foundation in mathematics, empowering students to tackle complex problems with confidence and accuracy. We will dissect each step of Gatien's work, highlighting the specific errors and providing a clear explanation of the correct procedure. This detailed analysis will serve as a valuable learning tool for students, enabling them to avoid similar mistakes in the future and develop a deeper understanding of algebraic principles. This article serves as a comprehensive guide to identifying and correcting errors in algebraic simplification. By examining Gatien's work, we can gain valuable insights into common pitfalls and develop a stronger understanding of the underlying mathematical principles. Let's embark on this journey of mathematical discovery, transforming mistakes into opportunities for learning and growth.

Unveiling Gatien's Attempt: A Step-by-Step Analysis

Gatien's attempt to simplify the expression A = 3(2x + 7) − 4 reveals a few critical errors in the application of the order of operations and the distributive property. Let's meticulously examine each step of his solution to identify these missteps.

Gatien's initial step correctly states the expression:

A = 3(2x + 7) − 4

This demonstrates a proper understanding of the starting point. However, the subsequent steps unveil the areas where Gatien's understanding falters. The first error appears in the application of the distributive property. The distributive property states that a(b + c) = ab + ac. Gatien correctly multiplies 3 by 2x to get 6x, and 3 by 7 to get 21. However, in the next step, an error emerges in the simplification process.

Gatien's second step: = 6x + 21 − 12

This is where the first error occurs. Gatien incorrectly subtracts 12 from 21. There is no 12 in the original equation to subtract. The original equation should remain = 6x + 21 - 4. The next step should be to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, 21 and -4 are like terms because they are both constants. The error in this step highlights the importance of carefully applying the distributive property and maintaining accuracy in arithmetic operations. A seemingly small mistake can cascade through the rest of the solution, leading to an incorrect final answer.

Gatien's third step: = 6x + 9

In this step, Gatien correctly performs the subtraction 21 - 12 to arrive at 9. This part of the solution is correct, following from the erroneous second step. However, because the second step was incorrect, this step, while arithmetically sound in isolation, contributes to the overall incorrect solution. This illustrates how crucial it is to ensure each step in a mathematical process is accurate, as errors can propagate and distort the final result. The correct step here, following from the corrected second step (6x + 21 - 4), should be 6x + 17.

Gatien's fourth step: = 15x

This final step reveals a fundamental misunderstanding of algebraic terms. Gatien incorrectly adds the constant term 9 to the coefficient of the x term, 6. This is a classic error, confusing the rules for combining like terms. Only terms with the same variable and exponent can be added together. 6x is a term with the variable x, while 9 is a constant term. They cannot be combined in this way. This error underscores the importance of understanding the definition of like terms and the rules for combining them in algebraic expressions. To correctly simplify the expression, the terms 6x and 9 should remain separate, as they are not like terms.

Correcting the Course: A Step-by-Step Solution

To rectify Gatien's errors and arrive at the correct simplification of the expression A = 3(2x + 7) − 4, we must meticulously apply the order of operations and the distributive property. Let's embark on a step-by-step journey to the accurate solution.

Step 1: Applying the Distributive Property

The distributive property is the cornerstone of simplifying expressions involving parentheses. It dictates that we multiply the term outside the parentheses by each term inside the parentheses. In our case, we need to multiply 3 by both 2x and 7.

A = 3(2x + 7) − 4 = (3 * 2x) + (3 * 7) − 4

This yields:

A = 6x + 21 − 4

Here, we've successfully distributed the 3, removing the parentheses and setting the stage for the next step.

Step 2: Combining Like Terms

The next crucial step involves identifying and combining like terms. Like terms are those that share the same variable raised to the same power. In our expression, 6x is a term with the variable 'x', while 21 and -4 are constant terms. We can only combine the constant terms.

A = 6x + (21 − 4)

Performing the subtraction, we get:

A = 6x + 17

This step showcases the importance of recognizing like terms and applying the correct arithmetic operations to them.

Step 3: The Final Simplified Expression

We've now arrived at the final simplified form of the expression. The terms 6x and 17 are not like terms, so they cannot be combined further. Therefore, the simplified expression is:

A = 6x + 17

This is the correct simplification of the original expression. It demonstrates the power of the distributive property and the importance of accurately combining like terms. This step-by-step solution serves as a clear guide, highlighting the correct application of algebraic principles.

Key Takeaways: Mastering Algebraic Simplification

Gatien's attempt to simplify the expression A = 3(2x + 7) − 4 provides a valuable learning opportunity. By dissecting his errors, we can glean crucial insights into the fundamental principles of algebraic simplification. Let's encapsulate the key takeaways from this exercise.

  • The Distributive Property is Paramount: The distributive property, a(b + c) = ab + ac, is a cornerstone of simplifying expressions with parentheses. It allows us to multiply a term outside the parentheses by each term inside, effectively removing the parentheses and paving the way for further simplification. Gatien's error in not correctly applying the distributive property led to an incorrect initial step, setting off a chain reaction of errors.
  • Combining Like Terms with Precision: Like terms, those with the same variable raised to the same power, can be combined through addition or subtraction. However, it's crucial to recognize and combine only like terms. Gatien's mistake of adding a constant term to a term with a variable highlights a common misunderstanding. A firm grasp of the definition of like terms is essential for accurate simplification.
  • Order of Operations: The Unwavering Guide: The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), provides a roadmap for simplifying expressions. Adhering to this order ensures consistent and accurate results. While Gatien's errors weren't directly related to the order of operations, a strong understanding of PEMDAS is crucial for navigating more complex expressions.
  • Accuracy in Arithmetic: The Bedrock of Algebra: Even with a solid understanding of algebraic principles, arithmetic errors can derail the simplification process. Gatien's arithmetic mistake in the initial steps underscores the importance of careful calculation and attention to detail. Accuracy in basic arithmetic operations is the bedrock upon which algebraic skills are built.
  • Step-by-Step Approach: Clarity and Control: Simplifying complex expressions can be daunting, but breaking the process down into manageable steps promotes clarity and control. By meticulously applying each step, we minimize the risk of errors and gain a deeper understanding of the underlying process. A step-by-step approach transforms a seemingly complex problem into a series of simpler, more manageable tasks.

Conclusion: From Mistakes to Mastery

Gatien's journey through algebraic simplification, though marked by errors, ultimately serves as a powerful lesson. By dissecting his missteps and providing a corrected solution, we've highlighted the critical principles that govern algebraic manipulation. The distributive property, the combination of like terms, the order of operations, and arithmetic accuracy are the cornerstones of successful simplification. This exploration underscores the importance of a meticulous, step-by-step approach, transforming complex problems into manageable tasks. Remember, mistakes are not roadblocks but rather stepping stones on the path to mastery. By learning from errors, we deepen our understanding and fortify our mathematical foundation. Embrace the challenge, persevere through the complexities, and unlock the power of algebra!