Solving For K: A Step-by-Step Guide To Rearranging Formulas

Have you ever stumbled upon a formula and thought, "How do I isolate this variable?" It's a common challenge in mathematics and other fields. In this article, we'll tackle the task of making '$k$' the subject of the formula $\frac{1}{n} = \frac{\sqrt{k^2 + a^2}}{hg}$. We'll break down each step, making it easy to follow along and understand the underlying principles. So, let's dive in and learn how to rearrange formulas like a pro!

Understanding the Formula and the Goal

Before we jump into the algebraic manipulations, let's take a moment to understand the formula we're working with:

$\frac{1}{n} = \frac{\sqrt{k^2 + a^2}}{hg}$

Here, '$k$' is nestled inside a square root, which adds a layer of complexity. Our goal is to isolate '$k$' on one side of the equation. This means we need to undo the operations that are currently acting on it. We'll do this step-by-step, ensuring we maintain the equality throughout the process. Remember, whatever we do to one side of the equation, we must do to the other. This principle is the cornerstone of algebraic manipulation. Let’s identify the operations affecting '$k$'. We see that '$k$' is squared, then added to '$a^2$', the square root is taken of the sum, and finally, the result is divided by '$hg$'. To isolate '$k$', we'll reverse these operations in the opposite order.

Why is understanding the formula crucial? It's not just about mechanically applying steps. Understanding the relationships between the variables and the operations involved gives you a deeper insight and makes it easier to solve similar problems in the future. Moreover, it helps in checking the validity of your answer. Once you've solved for '$k$', you can plug it back into the original equation to see if it holds true. This is a valuable technique for verifying your solution and ensuring accuracy. So, let's keep this understanding in mind as we proceed with the steps.

Step 1: Eliminate the Fraction on the Right-Hand Side

Our first task is to get rid of the fraction on the right-hand side of the equation. To do this, we'll multiply both sides of the equation by '$hg$'. This will cancel out the denominator on the right side, bringing us closer to isolating '$k$'.

$\frac{1}{n} \cdot hg = \frac{\sqrt{k^2 + a^2}}{hg} \cdot hg$

This simplifies to:

$\frac{hg}{n} = \sqrt{k^2 + a^2}$

Notice how multiplying by '$hg$' effectively moves it from the denominator on the right to the numerator on the left. This is a common technique in algebraic manipulation – using inverse operations to simplify the equation. By performing the same operation on both sides, we maintain the balance of the equation, ensuring that the equality remains valid. This step has cleared the way for us to deal with the square root, which is our next hurdle. We are making steady progress towards isolating '$k$'. This step demonstrates the power of multiplication and division as inverse operations in simplifying equations. Remember, our goal is to undo the operations acting on '$k$', and this step is a significant move in that direction.

Step 2: Eliminate the Square Root

Now, we need to deal with the square root. The inverse operation of taking a square root is squaring. So, we'll square both sides of the equation. This will eliminate the square root on the right-hand side, further simplifying the expression and bringing us closer to isolating '$k$'.

$(\frac{hg}{n})^2 = (\sqrt{k^2 + a^2})^2$

This simplifies to:

$\frac{h^2g^2}{n^2} = k^2 + a^2$

Squaring both sides not only eliminates the square root but also squares each term within the fraction on the left. It's crucial to remember to apply the squaring operation to the entire fraction, not just the numerator or denominator. This step is a critical turning point in solving for '$k$'. We've successfully removed the square root, which was a major obstacle. Now, we're left with a simpler equation involving '$k^2$', which is easier to manipulate. Squaring both sides is a common technique used to eliminate square roots in equations. It's important to perform this operation carefully and accurately to avoid errors. We are now in a much better position to isolate '$k$', and the subsequent steps will build upon this progress.

Step 3: Isolate $k^2$

Our next goal is to isolate the term containing '$k$', which is '$k^2$'. To do this, we need to get rid of the '$a^2$' term on the right-hand side. We can achieve this by subtracting '$a^2$' from both sides of the equation. This is another application of the principle of performing the same operation on both sides to maintain equality.

$\frac{h^2g^2}{n^2} - a^2 = k^2 + a^2 - a^2$

This simplifies to:

$\frac{h^2g^2}{n^2} - a^2 = k^2$

By subtracting '$a^2$' from both sides, we've successfully isolated '$k^2$' on the right-hand side. This is a significant step towards our final goal of isolating '$k$'. The equation is now in a form where we can easily see the next operation needed to solve for '$k$'. This step highlights the importance of using inverse operations to isolate variables. Subtraction is the inverse operation of addition, and we've used it here to undo the addition of '$a^2$' to '$k^2$'. The left-hand side of the equation might look a bit complex, but don't be intimidated. We're focused on isolating '$k$', and we've made excellent progress in that direction. We are now just one step away from solving for '$k$', which involves taking the square root of both sides.

Step 4: Solve for $k$

Finally, to solve for '$k$', we need to undo the square. The inverse operation of squaring is taking the square root. So, we'll take the square root of both sides of the equation. Remember that when we take the square root, we need to consider both the positive and negative solutions, as both positive and negative values, when squared, will result in a positive value.

$\pm \sqrt{\frac{h^2g^2}{n^2} - a^2} = \sqrt{k^2}$

This simplifies to:

$k = \pm \sqrt{\frac{h^2g^2}{n^2} - a^2}$

We have now successfully isolated '$k$'! The solution shows that '$k$' can be either the positive or negative square root of the expression. This is an important consideration in many mathematical and physical contexts. Taking the square root introduces the possibility of two solutions, and it's crucial to acknowledge this in our final answer. This step completes the process of making '$k$' the subject of the formula. We've systematically reversed the operations acting on '$k$' to isolate it. The final solution expresses '$k$' in terms of the other variables in the formula. This result can now be used to calculate the value of '$k$' if we know the values of '$h$', '$g$', '$n$', and '$a$'. Our journey of rearranging the formula has come to a successful conclusion!

Conclusion

In this article, we walked through the process of making '$k$' the subject of the formula $\frac{1}{n} = \frac{\sqrt{k^2 + a^2}}{hg}$. We broke down the problem into manageable steps, using inverse operations to isolate '$k$'. This process involved multiplying, squaring, subtracting, and finally, taking the square root. Remember, the key to rearranging formulas is understanding the order of operations and applying the inverse operations in reverse order. By following these steps, you can confidently tackle similar problems and master the art of formula manipulation.

Further your understanding of algebraic manipulations and equation solving by exploring resources like Khan Academy's algebra section: Khan Academy Algebra. This can help reinforce the concepts and techniques discussed in this article.