Analyzing Lorena's Equation Solving Steps A Detailed Breakdown

In the intricate world of mathematics, solving equations is a fundamental skill. It requires a meticulous approach, a keen eye for detail, and a solid understanding of algebraic principles. In this article, we embark on a journey to analyze Lorena's attempt to solve the equation $5k - 3(2k - \frac{2}{3}) - 9 = 0$. By carefully examining each step of her solution, we aim to pinpoint any errors, highlight the correct steps, and ultimately, deepen our understanding of equation-solving strategies. Mathematics is not just about finding the right answer; it's about the process, the logic, and the ability to identify and correct mistakes. So, let's delve into Lorena's work and dissect her mathematical journey, step by step. Our analysis will not only reveal the correctness of each step but also provide valuable insights into common pitfalls and effective problem-solving techniques in algebra. This exploration is crucial for students and math enthusiasts alike, as it emphasizes the importance of precision and critical thinking in mathematical problem-solving. Understanding where errors occur is as important as understanding how to solve the problem correctly, as it builds a stronger foundation for future mathematical endeavors. Through this detailed examination, we aim to provide a comprehensive understanding of the equation-solving process, ensuring that learners can approach similar problems with confidence and accuracy.

Problem Statement: The Equation at Hand

The equation Lorena attempted to solve is:

$5k - 3(2k - \frac{2}{3}) - 9 = 0$

This equation involves a variable, k, and requires the application of the order of operations, distribution, combining like terms, and isolating the variable to find the solution. Before we dive into Lorena's steps, it's crucial to recognize the structure of the equation. The presence of parentheses indicates the need for distribution, while the combination of constant terms and variable terms suggests the necessity of simplification. Solving this equation correctly demands a systematic approach, where each step is carefully executed to maintain equality. Any error in a single step can lead to an incorrect final answer. Therefore, a thorough understanding of algebraic principles is essential. Moreover, this equation provides an excellent opportunity to reinforce the importance of precision in mathematical calculations. By breaking down the equation into smaller, manageable parts, we can identify potential areas for error and develop strategies to minimize mistakes. This approach not only aids in solving the current equation but also enhances overall problem-solving skills in algebra. Remember, the goal is not just to find the value of k, but to understand the process and reasoning behind each step.

Step-by-Step Analysis of Lorena's Solution

Step 1: $5k - 6k + 2 - 9 = 0$

In this crucial first step, Lorena distributes the -3 across the terms inside the parentheses. This is a critical operation where accuracy is paramount. Let's dissect this step meticulously. Lorena begins by multiplying -3 with 2k, which correctly yields -6k. This demonstrates a good understanding of the distributive property. However, the next part of the distribution requires careful attention. Lorena multiplies -3 by -2/3. A negative times a negative results in a positive, which Lorena correctly identifies. The multiplication itself, -3 * (-2/3), simplifies to +2. This part of the step is also executed flawlessly. The remaining term, -9, is simply carried down, as it is not affected by the distribution. Thus, the entire first step, transforming the original equation to $5k - 6k + 2 - 9 = 0$, is perfectly executed. This sets a strong foundation for the subsequent steps. This step highlights the importance of paying close attention to signs and numerical coefficients during distribution. A minor error here can propagate through the rest of the solution, leading to a wrong answer. Lorena's success in this step demonstrates a solid grasp of the distributive property and attention to detail. Therefore, this step is not only correct but also showcases Lorena's proficiency in algebraic manipulation.

Step 2: $-k - 7 = 0$

Step 2 involves combining like terms from the previous step. Here, Lorena groups the terms with the variable k (5k and -6k) and the constant terms (+2 and -9). This is a standard algebraic technique to simplify equations. Let's examine her execution. Combining 5k and -6k should result in -1k, which is commonly written as -k. Lorena correctly performs this operation. Next, she combines the constants, +2 and -9. Adding these numbers, 2 + (-9), gives -7. Lorena accurately calculates this as well. Therefore, the resulting equation, $-k - 7 = 0$, is a correct simplification of the previous step. This step is crucial as it reduces the complexity of the equation, making it easier to isolate the variable k. Lorena's accurate combination of like terms showcases her understanding of algebraic simplification. The ability to correctly combine like terms is fundamental to solving equations efficiently and accurately. Any mistake in this step would lead to an incorrect equation, affecting the final solution. However, Lorena's precise execution ensures that the equation remains balanced and the solution path remains viable. This step serves as a testament to her algebraic skills and her ability to maintain accuracy throughout the problem-solving process.

Step 3: $-k = 7$

In Step 3, Lorena aims to isolate the term containing the variable, -k. To do this, she adds 7 to both sides of the equation from Step 2 ($-k - 7 = 0$). This is a valid algebraic manipulation based on the principle of maintaining equality: whatever operation is performed on one side of the equation must also be performed on the other side. Adding 7 to both sides of $-k - 7 = 0$ results in $-k - 7 + 7 = 0 + 7$. Simplifying this, we get $-k = 7$. Lorena has executed this step correctly. The goal of isolating the variable term is crucial for eventually solving for the variable itself. By adding 7 to both sides, she effectively cancels out the -7 on the left side, leaving -k isolated. This step demonstrates a clear understanding of inverse operations and their role in equation solving. The accuracy in this step is vital, as it sets the stage for the final step where the value of k will be determined. A mistake here would propagate through the solution, leading to an incorrect final answer. Lorena's precise application of the addition property of equality underscores her proficiency in algebraic manipulations. This step not only showcases her mathematical skills but also reinforces the importance of maintaining balance in equations throughout the solving process.

Step 4: $k = \frac{1}{7}$

This final step is where Lorena attempts to solve for k. From Step 3, she has the equation $-k = 7$. To isolate k, she needs to eliminate the negative sign. This is typically done by multiplying or dividing both sides of the equation by -1. Correctly performing this operation on $-k = 7$ yields $(-1) * (-k) = (-1) * 7$, which simplifies to $k = -7$. However, Lorena states that $k = \frac{1}{7}$. This is incorrect. She appears to have made a mistake in this final step. It's possible she confused multiplying by -1 with taking the reciprocal, or perhaps made a simple arithmetic error. This step highlights the importance of double-checking the final steps of a solution, as even a small mistake can lead to a wrong answer. Despite the correct execution of the previous steps, the error in this final step invalidates the solution. The correct value of k should be -7, not 1/7. This discrepancy underscores the need for careful attention to detail and accuracy in mathematical calculations, particularly in the concluding stages of problem-solving. Therefore, while Lorena demonstrated a strong understanding of algebraic principles in the initial steps, the error in Step 4 led to an incorrect final answer.

Correct Solution

To reiterate, let's solve the equation $5k - 3(2k - \frac{2}{3}) - 9 = 0$ step by step:

1. Distribute the -3: $5k - 6k + 2 - 9 = 0$
2. Combine like terms: $-k - 7 = 0$
3. Add 7 to both sides: $-k = 7$
4. Multiply both sides by -1: $k = -7$

Therefore, the correct solution is $k = -7$.

Conclusion: Key Takeaways from Lorena's Attempt

In conclusion, while Lorena demonstrated a strong grasp of algebraic principles and equation-solving techniques in the initial steps of her attempt to solve the equation $5k - 3(2k - \frac{2}{3}) - 9 = 0$, she made a critical error in the final step. Her work showcased a correct understanding of distribution, combining like terms, and isolating the variable term. However, the mistake in Step 4, where she incorrectly stated $k = \frac{1}{7}$ instead of $k = -7$, highlights the importance of meticulousness and double-checking in mathematical problem-solving. This analysis provides valuable insights into the equation-solving process. It underscores the significance of accuracy in each step, as errors can propagate and lead to incorrect final answers. Lorena's journey serves as a learning opportunity, emphasizing the need for careful attention to detail and the importance of verifying solutions. The correct solution, as we demonstrated, is $k = -7$. By dissecting Lorena's work, we not only identified the error but also reinforced the correct approach to solving such equations. This exercise is crucial for students and math enthusiasts alike, as it fosters a deeper understanding of algebraic manipulations and the critical role of accuracy in mathematical endeavors. The key takeaway is that even with a strong foundation in algebraic principles, vigilance and careful execution are essential for achieving correct solutions.