Is K=6 A Solution To (1/3)k=3 A Step-by-Step Explanation

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Introduction

In the realm of mathematics, solving equations is a fundamental skill. Equations represent relationships between variables and constants, and finding the solution involves determining the value(s) of the variable(s) that make the equation true. This article delves into the process of verifying whether a given value is a solution to an equation, using the specific example of the equation (1/3)k = 3 and the potential solution k = 6. We will explore the step-by-step method for substituting the value of the variable into the equation and checking if the equality holds. Additionally, we will discuss the importance of understanding the properties of equality and how they are applied in solving equations. This exploration is crucial for anyone seeking to enhance their algebraic skills and gain a deeper understanding of mathematical problem-solving.

The question at hand is whether k = 6 is a solution to the equation (1/3)k = 3. To answer this, we need to understand what it means for a value to be a solution to an equation. A solution is a value that, when substituted for the variable, makes the equation a true statement. In simpler terms, if we replace k with 6 in the equation, will the left side of the equation equal the right side? This process involves basic algebraic manipulation and a solid understanding of arithmetic operations, making it a core concept in introductory algebra. The ability to verify solutions is not just about finding the right answer; it's about developing a critical approach to problem-solving and ensuring accuracy in mathematical calculations. This article will guide you through the process, highlighting the key steps and potential pitfalls to avoid. By the end, you'll have a clear understanding of how to determine whether a given value satisfies an equation and, more broadly, how to approach solution verification in algebra.

Verifying the Solution: Step-by-Step

To determine if k = 6 is a solution to the equation (1/3)k = 3, we will follow a systematic approach. The core idea is to substitute the given value of k into the equation and then simplify both sides to see if they are equal. This process is a cornerstone of algebra, allowing us to check the validity of potential solutions and build confidence in our problem-solving abilities. Let's break down the steps:

  1. Substitute the value of k: Replace k with 6 in the equation (1/3)k = 3. This gives us (1/3)(6) = 3. This step is crucial as it transforms the equation from a general statement about the variable k to a specific arithmetic expression. The accuracy of the substitution is paramount; a simple mistake here can lead to an incorrect conclusion. It's a good practice to double-check this step before proceeding further.

  2. Simplify the left side of the equation: Multiply (1/3) by 6. This can be thought of as dividing 6 by 3, which results in 2. So, the left side simplifies to 2. This step involves basic arithmetic, but it's essential to perform it correctly. Understanding fraction multiplication is key here. Remember, multiplying a fraction by a whole number can be visualized as dividing the whole number by the denominator of the fraction. In this case, 6 divided by 3 equals 2.

  3. Compare both sides: Now we have 2 = 3. This statement is clearly false. The left side of the equation (2) does not equal the right side (3). This comparison is the crux of the verification process. If the two sides were equal, then k = 6 would be a solution. However, since they are not equal, we can conclude that k = 6 is not a solution to the equation.

This step-by-step verification process is not just applicable to this specific equation; it's a general method that can be used to check the solutions of various algebraic equations. By substituting the potential solution and simplifying, we can determine whether the equation holds true. This technique reinforces the fundamental concept of equality in mathematics and builds a solid foundation for more advanced problem-solving.

Alternative Solutions and Why k=6 Fails

Since k = 6 is not a solution to the equation (1/3)k = 3, it's important to understand why and to explore the correct solution. This not only clarifies the specific problem but also reinforces the underlying principles of solving equations. Let's delve into the process of finding the actual solution and understand the error that leads to k = 6 being incorrect.

To find the correct solution, we need to isolate the variable k in the equation (1/3)k = 3. This involves using the properties of equality, which allow us to perform the same operation on both sides of the equation without changing its balance. The key here is to undo the operation that is being applied to k. In this case, k is being multiplied by 1/3. To undo this, we can multiply both sides of the equation by the reciprocal of 1/3, which is 3. This is a fundamental technique in algebra, and mastering it is crucial for solving a wide range of equations. By applying this method, we ensure that we maintain the equality while simplifying the equation towards the solution.

Let's perform the steps:

  1. Multiply both sides by 3: Starting with (1/3)k = 3, we multiply both sides by 3. This gives us 3 * (1/3)k = 3 * 3. This step is the core of the solution process. Multiplying by the reciprocal isolates the variable k on the left side.

  2. Simplify: On the left side, 3 * (1/3)k simplifies to k because 3 times 1/3 is 1. On the right side, 3 * 3 simplifies to 9. So, we have k = 9. This simplification step is crucial for arriving at the solution. It demonstrates the power of inverse operations in solving equations.

Therefore, the correct solution to the equation (1/3)k = 3 is k = 9. This highlights why k = 6 fails to satisfy the equation. When we substitute k = 6 into the equation, we get (1/3)(6) = 2, which is not equal to 3. This discrepancy shows that k = 6 does not maintain the equality of the equation. Understanding this process of finding the correct solution and recognizing why a particular value fails is essential for developing strong algebraic skills. It emphasizes the importance of accurate calculations and the correct application of algebraic principles.

The Importance of Verification in Mathematics

Verifying solutions in mathematics, especially in algebra, is a critical step that goes beyond simply finding an answer. It's a process that ensures accuracy, builds confidence, and deepens understanding of mathematical concepts. This practice is not just a formality; it's an integral part of the problem-solving process, allowing us to catch errors and reinforce our grasp of the underlying principles. Let's explore why verification is so important.

One of the primary reasons for verifying solutions is to ensure accuracy. Mathematical calculations, especially in algebra, can involve multiple steps, increasing the chances of making a mistake. A simple arithmetic error, a sign mistake, or an incorrect application of a formula can lead to a wrong answer. By verifying the solution, we have a safeguard against these errors. This is particularly important in exams or real-world applications where accuracy is paramount. The verification process acts as a quality control check, helping us to identify and correct mistakes before they have significant consequences. It's a way of double-checking our work and ensuring that the solution we've arrived at is indeed correct.

Furthermore, verification builds confidence in our problem-solving abilities. When we verify a solution and find that it satisfies the equation or problem conditions, we gain assurance that our approach was correct and our calculations were accurate. This confidence is invaluable, especially when tackling more complex problems. It reduces anxiety and encourages us to persevere when faced with challenges. Knowing that we have a reliable method for checking our work can make the problem-solving process less daunting and more enjoyable. This confidence translates into better performance in mathematics and a more positive attitude towards the subject.

Moreover, verifying solutions deepens understanding of mathematical concepts. The verification process often requires us to revisit the original problem and retrace our steps. This review helps us to solidify our understanding of the underlying principles and relationships. It's not just about getting the right answer; it's about understanding why the answer is correct. By verifying, we reinforce the connection between the solution and the problem's conditions, gaining a more holistic view of the mathematical concepts involved. This deeper understanding is crucial for long-term retention and the ability to apply these concepts in different contexts.

In conclusion, verification is not just a final step in problem-solving; it's a vital part of the entire process. It ensures accuracy, builds confidence, and deepens understanding. By making verification a regular practice, we become more proficient and confident mathematicians.

Conclusion

In summary, we've explored the process of determining whether k = 6 is a solution to the equation (1/3)k = 3. Through step-by-step substitution and simplification, we've shown that k = 6 does not satisfy the equation. The correct solution, as we demonstrated, is k = 9. This exercise underscores the importance of solution verification in mathematics. Verifying solutions is not merely a formality; it's a crucial step that ensures accuracy, builds confidence, and deepens understanding of mathematical concepts. By making verification a regular part of our problem-solving routine, we can strengthen our algebraic skills and approach mathematical challenges with greater assurance. This detailed analysis provides a clear illustration of how to verify solutions effectively and highlights the significance of this practice in the broader context of mathematical problem-solving.