Solving For W In The Equation -1 + √(w-6) = 3 A Step-by-Step Guide

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In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of solving for the variable w in the equation -1 + √(w-6) = 3. We will break down each step, providing a clear and concise explanation to ensure a thorough understanding of the solution.

Understanding the Problem

Our primary goal is to isolate w on one side of the equation. The given equation involves a square root, which necessitates a strategic approach to eliminate the radical and ultimately determine the value(s) of w that satisfy the equation. In this comprehensive guide on solving for w, we embark on a step-by-step journey to dissect the equation -1 + √(w-6) = 3 and unveil the real number solutions. We begin by understanding the problem, which involves isolating w amidst the presence of a square root. The strategic approach we undertake is crucial in eliminating the radical and determining the values of w that hold the equation true. This task requires a blend of algebraic manipulation and a keen eye for detail, ensuring we navigate the complexities of the equation with precision. The significance of isolating w cannot be overstated, as it forms the cornerstone of solving for variables in mathematical equations. The presence of the square root introduces an additional layer of complexity, demanding a methodic approach to undo its effect and bring w into the spotlight. Our journey begins with a clear goal: to unveil the values of w that make the equation a harmonious balance. To achieve this, we must tread carefully, ensuring each step we take aligns with the golden rules of algebra and mathematical precision. The presence of a radical invites the need for isolation, squaring, and verification—a trifecta of actions aimed at revealing the hidden values of w. Our commitment is to demystify the process, providing explanations that illuminate the path to understanding and solution.

Step-by-Step Solution

  1. Isolate the square root: To begin, we need to isolate the square root term. We can achieve this by adding 1 to both sides of the equation: -1 + √(w-6) + 1 = 3 + 1 √(w-6) = 4

  2. Square both sides: To eliminate the square root, we square both sides of the equation: (√(w-6))^2 = 4^2 w - 6 = 16

  3. Solve for w: Now, we can solve for w by adding 6 to both sides: w - 6 + 6 = 16 + 6 w = 22

  4. Verify the solution: It is crucial to verify our solution by substituting w = 22 back into the original equation: -1 + √(22-6) = 3 -1 + √16 = 3 -1 + 4 = 3 3 = 3 The solution holds true.

Our meticulous journey through the equation's labyrinth has led us to a pivotal moment: the step-by-step solution. This is where we apply the principles of algebra to methodically dismantle the equation and isolate w. The isolation of the square root is our first conquest, achieved by adding 1 to both sides, a maneuver that sets the stage for the next act: squaring both sides. This is where the radical is confronted head-on, its power nullified as we elevate both sides of the equation to the second power. The result, w - 6 = 16, marks a significant milestone, a moment where the complexity wanes, and the path to solving for w becomes clearer. Next, we add 6 to both sides, an action that brings us to the threshold of revelation: w = 22. However, our journey is not yet complete. We embrace the discipline of verification, a cornerstone of mathematical problem-solving. Substituting w = 22 back into the original equation is not just a formality; it is a crucial test of our solution's integrity. Through this process, we ensure that our algebraic maneuvers have held the equation's balance, that the value we've found for w truly satisfies the conditions laid out in the initial problem. This step solidifies our understanding, turning a potential answer into a confirmed solution. The journey through this step-by-step process is a testament to the power of methodical problem-solving, where each action is deliberate, and each verification is a validation of our efforts.

The Solution

Therefore, the solution to the equation -1 + √(w-6) = 3 is w = 22.

Having navigated the intricate steps of solving the equation, we arrive at the solution: w = 22. This value is not just a number; it's the culmination of our algebraic endeavors, the answer we've diligently sought. However, in the realm of mathematics, the journey doesn't end with finding a solution. We must reflect on the process, understanding not just the 'what' but the 'why' behind our actions. Our solution, w = 22, stands as a testament to the power of algebraic manipulation and the importance of strategic problem-solving. It's a beacon of clarity, shining through the initial complexity of the equation. This number, 22, is more than just a figure; it's a point of equilibrium, the value that restores balance to the equation. It's the result of isolating, squaring, adding, and verifying—a sequence of actions that transformed an initial puzzle into a solved equation. Our solution, w = 22, is a mathematical truth, a statement that holds firm under the scrutiny of verification. It's the end of our calculation, but the start of deeper understanding. This number, 22, encapsulates the essence of our mathematical quest, a quest marked by precision, strategy, and the unwavering pursuit of truth.

Conclusion

In conclusion, solving for w in the equation -1 + √(w-6) = 3 involves isolating the square root, squaring both sides, solving the resulting equation, and verifying the solution. The solution is w = 22. This exercise underscores the importance of careful algebraic manipulation and verification in mathematical problem-solving.

In our concluding thoughts, the journey of solving for w in the equation -1 + √(w-6) = 3 serves as a microcosm of the broader mathematical process. It's a journey that emphasizes the power of strategic problem-solving, where each step is deliberate and each action is guided by mathematical principles. Isolating the square root was not just a maneuver; it was a strategic decision that paved the way for simplifying the equation. Squaring both sides was more than an operation; it was a technique to dismantle the radical, bringing us closer to unveiling the value of w. Solving the resulting equation was not just a calculation; it was the heart of our quest, where algebraic manipulation revealed the potential solution. And, crucially, verifying the solution was not merely a formality; it was the cornerstone of ensuring accuracy, confirming that w = 22 indeed holds true. This exercise is a testament to the methodical approach that underpins effective mathematical problem-solving. It highlights the significance of each step, from the initial assessment of the problem to the final confirmation of the solution. The journey to finding w = 22 is a reminder that mathematical solutions are not just about numbers; they're about the process, the understanding, and the rigorous verification that transforms a potential answer into a confirmed truth.