Lorena's Equation Solving Analysis Of Algebraic Steps And Error Identification

Let's dive into Lorena's attempt to solve the equation $5k - 3(2k - \frac{2}{3}) - 9 = 0$. In this detailed analysis, we'll meticulously examine each step of her work to pinpoint any errors and understand the underlying mathematical principles involved. By dissecting Lorena's approach, we aim to not only identify mistakes but also reinforce the correct procedures for solving algebraic equations. This exploration is crucial for anyone looking to enhance their algebra skills and develop a deeper understanding of equation solving. Join us as we break down the equation step-by-step, ensuring clarity and accuracy in our mathematical journey. Our goal is to provide a comprehensive analysis that benefits students and math enthusiasts alike.

Step 1 Unveiling the Distribution

In Step 1, Lorena expands the expression by distributing the -3 across the terms inside the parentheses. This is a crucial step in simplifying the equation. The original equation is $5k - 3(2k - \frac{2}{3}) - 9 = 0$. Lorena's first step transforms this into $5k - 6k + 2 - 9 = 0$. To properly analyze this step, we need to meticulously check the distribution. When -3 is multiplied by 2k, the result is indeed -6k. However, the critical part lies in the multiplication of -3 by $-\frac{2}{3}$. The product of these two terms should be positive since a negative times a negative is a positive. Specifically, $-3 * -\frac{2}{3}$ equals 2. Lorena has correctly executed this part of the distribution. Therefore, the expansion up to this point, $5k - 6k + 2$, is accurate. The final term, -9, remains unchanged as it is not part of the distribution. Consequently, the entire expression $5k - 6k + 2 - 9 = 0$ in Step 1 is mathematically sound and correctly derived from the original equation. This step demonstrates a good understanding of the distributive property, which is fundamental in algebraic manipulations. The ability to correctly distribute terms is essential for simplifying and solving equations, making it a key skill in algebra. This detailed examination confirms that Lorena's initial step is a solid foundation for solving the equation.

Step 2 Combining Like Terms A Critical Evaluation

Moving onto Step 2, Lorena combines like terms to further simplify the equation. From Step 1, the equation stands at $5k - 6k + 2 - 9 = 0$. The like terms here are the terms containing 'k' (5k and -6k) and the constant terms (2 and -9). Combining 5k and -6k yields -k, which is a straightforward application of adding coefficients. Now, let's focus on the constant terms. Adding 2 and -9 results in -7. So far, Lorena's work seems accurate. She combines these terms to form the equation $-k - 7 = 0$. This step is crucial because combining like terms simplifies the equation, making it easier to isolate the variable 'k'. Accuracy in this step is paramount; any error here would propagate through the remaining steps, leading to an incorrect solution. The process of combining like terms is a fundamental algebraic technique, and Lorena's application of it in this step appears to be flawless. This simplification is a key step towards solving the equation, and Lorena has executed it with precision. Therefore, Step 2, where the equation is simplified to $-k - 7 = 0$, is a correct progression from the previous step, demonstrating a strong grasp of algebraic simplification techniques. This careful combination of like terms sets the stage for the next phase of solving the equation.

Step 3 Isolating the Variable A Key Algebraic Manipulation

In Step 3, Lorena aims to isolate the term containing the variable 'k'. Starting from the equation $-k - 7 = 0$, she adds 7 to both sides of the equation. This is a fundamental algebraic manipulation based on the principle that adding the same value to both sides maintains the equation's balance. By adding 7, the equation transforms to $-k - 7 + 7 = 0 + 7$, which simplifies to $-k = 7$. This step is crucial because it brings us closer to solving for 'k' by isolating the term with 'k' on one side of the equation. The operation is mathematically sound and follows the rules of algebraic manipulation. Isolating the variable is a key strategy in solving equations, and Lorena's execution in this step is accurate and efficient. The importance of this step cannot be overstated, as it sets the stage for the final determination of the value of 'k'. The result, $-k = 7$, is a direct and correct consequence of the previous step, showcasing a clear understanding of how to manipulate equations to solve for unknowns. Thus, Step 3 represents a valid and necessary step in Lorena's solution process, reflecting a solid grasp of algebraic principles.

Step 4 The Final Solution Unveiling the Error

Finally, in Step 4, Lorena attempts to solve for 'k'. She starts with the equation $-k = 7$ from the previous step. To find 'k', she needs to eliminate the negative sign. The correct procedure here is to multiply both sides of the equation by -1. This would give $(-1) * -k = (-1) * 7$, which simplifies to $k = -7$. However, Lorena's result is $k = \frac{1}{7}$, which is incorrect. The error likely stems from a misunderstanding of how to handle the negative sign or potentially dividing instead of multiplying by -1. This is a critical mistake that leads to a wrong answer. While the preceding steps were executed flawlessly, this final misstep invalidates the entire solution. Solving for the variable correctly is the ultimate goal in equation solving, and this is where Lorena's solution falters. The correct value of 'k' is -7, not $\frac{1}{7}$. This error underscores the importance of careful attention to detail in the final steps of solving an equation. Therefore, Step 4 is where Lorena's solution goes astray, highlighting a misunderstanding of basic algebraic principles in the final stage of problem-solving.

Correct Answer

The correct answer is k = -7, not 1/7.

Conclusion A Comprehensive Review

In conclusion, while Lorena demonstrated a strong understanding of algebraic principles in the initial steps of solving the equation $5k - 3(2k - \frac{2}{3}) - 9 = 0$, her solution ultimately falters in Step 4. Steps 1, 2, and 3 were executed correctly, showcasing her grasp of the distributive property, combining like terms, and isolating the variable. However, the final step of solving for 'k' contained a critical error, leading to an incorrect answer. The detailed analysis reveals that Lorena's mistake was in handling the negative sign when solving $-k = 7$. The correct operation is to multiply both sides by -1, resulting in $k = -7$, but Lorena arrived at $k = \frac{1}{7}$ instead. This error highlights the importance of precision and careful attention to detail, especially in the final stages of problem-solving. It serves as a valuable lesson in the necessity of double-checking each step to ensure accuracy. Lorena's journey through this equation provides a clear illustration of how a single mistake can derail an otherwise correct solution process. This analysis is beneficial for students and math enthusiasts, emphasizing the significance of mastering basic algebraic manipulations and avoiding common pitfalls. The correct solution, $k = -7$, stands in stark contrast to Lorena's answer, underscoring the impact of that final misstep. This comprehensive review not only identifies the error but also reinforces the correct methods for solving similar equations, contributing to a deeper understanding of algebraic problem-solving strategies.