Analyzing Lorena's Equation Solving Steps Finding The Correct Solution

In the realm of mathematics, solving equations is a fundamental skill. This article delves into Lorena's attempt to solve the equation 5k - 3(2k - 2/3) - 9 = 0. We will meticulously analyze each step of her solution to identify any potential errors and pinpoint the correct steps. By examining her work, we aim to enhance our understanding of algebraic manipulations and equation-solving techniques. This exploration will not only benefit those learning algebra but also serve as a refresher for anyone seeking to sharpen their mathematical prowess. Solving equations accurately is a cornerstone of mathematics, crucial for various applications in science, engineering, and everyday problem-solving. Therefore, a detailed analysis like this is invaluable for anyone looking to master this essential skill.

Step-by-Step Analysis of Lorena's Solution

Step 1 Evaluating Lorena's Initial Distribution

In Lorena's first step, she expanded the expression by distributing the -3 across the terms inside the parentheses: 5k - 3(2k - 2/3) - 9 = 0 becomes 5k - 6k + 2 - 9 = 0. Let's dissect this step to ensure its accuracy. The distribution involves multiplying -3 by both 2k and -2/3. When -3 is multiplied by 2k, the result is indeed -6k. Now, let's examine the second part of the distribution: -3 multiplied by -2/3. A negative times a negative yields a positive, and 3(2/3) equals 2. Thus, the term should be +2, which Lorena correctly obtained. Therefore, the expansion 5k - 6k + 2 - 9 = 0 accurately reflects the distribution of -3. This initial step is crucial because any error here would propagate through the rest of the solution, leading to an incorrect final answer. Accuracy in distribution is paramount in algebraic manipulations, and Lorena's execution in this step appears flawless. It sets a solid foundation for the subsequent steps in solving the equation.

Step 2 Combining Like Terms Simplifying the Equation

Moving on to Step 2, Lorena combined like terms to simplify the equation 5k - 6k + 2 - 9 = 0. This step involves grouping the 'k' terms together and the constant terms together. Let's break it down: Combining the 'k' terms, 5k and -6k, results in -k. This is because 5 minus 6 equals -1, so we have -1k, which is simply written as -k. Next, we combine the constant terms, which are +2 and -9. Adding these together, 2 minus 9 equals -7. So, the constant term is -7. Putting these together, the simplified equation becomes -k - 7 = 0. Lorena has accurately combined the like terms, transitioning the equation from 5k - 6k + 2 - 9 = 0 to -k - 7 = 0. This simplification is a key step in solving equations, making the equation easier to manipulate and solve for the variable. Correctly combining like terms reduces the complexity of the equation, paving the way for the isolation of the variable. Lorena's meticulous execution of this step further solidifies the integrity of her solution process.

Step 3 Isolating the Variable Examining Lorena's Algebraic Manipulation

Step 3 in Lorena's solution involves isolating the variable -k in the equation -k - 7 = 0. To isolate -k, we need to eliminate the constant term, which is -7. This is achieved by adding 7 to both sides of the equation, maintaining the balance. When we add 7 to both sides, the equation transforms as follows: -k - 7 + 7 = 0 + 7. On the left side, -7 + 7 cancels out, leaving us with just -k. On the right side, 0 + 7 equals 7. Therefore, the equation simplifies to -k = 7. Lorena has correctly isolated -k by adding 7 to both sides, effectively moving the constant term to the other side of the equation. This is a fundamental algebraic manipulation technique used to separate the variable term from the constants. Isolating the variable is a crucial step in the equation-solving process, bringing us closer to finding the value of the variable. Lorena's accurate application of this technique demonstrates a solid understanding of algebraic principles.

Step 4 Solving for k Identifying the Error in Lorena's Final Step

In the final step, Step 4, Lorena attempts to solve for k in the equation -k = 7. This is where we need to exercise careful scrutiny. The equation -k = 7 implies that the negative of k is equal to 7. To find k, we need to eliminate the negative sign. This can be done by multiplying both sides of the equation by -1. When we multiply -k by -1, we get k. When we multiply 7 by -1, we get -7. Therefore, the correct solution for k should be -7. However, Lorena's solution states that k = 1/7, which is incorrect. The error lies in the final step of solving for k. Instead of correctly multiplying both sides by -1, Lorena seems to have taken the reciprocal of 7, which is an incorrect operation in this context. This error highlights the importance of carefully applying algebraic rules, especially when dealing with negative signs. The correct solution, k = -7, is significantly different from Lorena's answer, emphasizing the impact of a single mistake in the equation-solving process. This analysis underscores the need for meticulous attention to detail in mathematical manipulations.

Identifying the Correct Statements

After thoroughly analyzing each step of Lorena's solution, we can now identify the correct statements about her work. Steps 1, 2, and 3 were executed flawlessly, demonstrating a strong understanding of algebraic principles. However, Step 4 contains a critical error in the final step of solving for k. This error leads to an incorrect solution. Therefore, the statements corresponding to Steps 1, 2, and 3 being correct are the ones that apply. The statement pointing out the error in Step 4 is also correct, as Lorena failed to correctly isolate k by not multiplying both sides of the equation by -1. This detailed analysis allows us to pinpoint not only the correct steps but also the exact location and nature of the mistake, providing valuable insights into the problem-solving process.

Correcting Lorena's Error and Finding the Accurate Solution

To rectify Lorena's error, let's revisit Step 4. The equation we arrived at was -k = 7. The mistake was in not correctly eliminating the negative sign on k. To do this, we need to multiply both sides of the equation by -1. This gives us: (-1) * (-k) = (-1) * 7. On the left side, (-1) * (-k) simplifies to k, because a negative times a negative is a positive. On the right side, (-1) * 7 equals -7. Thus, the correct solution is k = -7. This contrasts sharply with Lorena's incorrect answer of k = 1/7. By accurately applying the rules of algebra, we've identified and corrected the mistake, arriving at the true solution. This exercise underscores the importance of verifying each step in the equation-solving process to avoid such errors. The correct solution, k = -7, satisfies the original equation, further validating our correction.

Conclusion The Importance of Precision in Equation Solving

In conclusion, our comprehensive analysis of Lorena's equation-solving journey reveals the critical importance of precision in each step. While Lorena demonstrated a strong grasp of initial algebraic manipulations in Steps 1, 2, and 3, a crucial error in Step 4 led to an incorrect final answer. This detailed examination highlights that even a single mistake in the latter stages of problem-solving can invalidate the entire solution. The error in Step 4, where Lorena incorrectly solved for k, underscores the necessity of meticulously applying algebraic rules, especially when dealing with negative signs and variable isolation. By identifying and correcting this error, we not only arrived at the accurate solution, k = -7, but also reinforced the significance of careful verification at every stage of the equation-solving process. This exercise serves as a valuable lesson in the importance of precision and attention to detail in mathematics, applicable to both students learning algebra and anyone seeking to enhance their problem-solving skills. The ability to solve equations accurately is a fundamental skill that underpins various disciplines, making this analysis a worthwhile endeavor for mathematical proficiency.