Problem 97 - Precalculus Student Solutions Manual 5th

Question

Solve each equation for the indicated variable. Assume all denominators are nonzero. $$d=k \sqrt{h}, \quad \text { for } h$$

Step-by-Step Solution

Answer

h = \left(\frac{d}{k}\right)^2

Step 1: Isolate the Square Root

To solve for the variable \(h\), the first step is to isolate \(\root h\) on one side of the equation. Starting from the given equation, \(d = k \sqrt{h}\), divide both sides by \(k\): \[ \frac{d}{k} = \sqrt{h} \]

Step 2: Square Both Sides

To solve for \(h\), square both sides of the equation to eliminate the square root: \[ \left(\frac{d}{k}\right)^2 = h \]

Step 3: Simplify

Simplify the equation to find the explicit form of \(h\): \[ h = \left(\frac{d}{k}\right)^2 \]

Key Concepts

Isolating the Variable

When solving equations, one common goal is to isolate the variable you're solving for. This means getting the variable alone on one side of the equation. For the equation given, our target variable is h. To isolate it, we first need to move everything else to the opposite side. We begin with:
\[ d = k \, \sqrt{h} \]
Since our goal is to solve for \( h \), we need to get \( h \) by itself. To start, divide both sides of the equation by the coefficient of the square root of h, which is k:
\[ \frac{d}{k} = \sqrt{h} \]
This step is crucial for isolating the variable because it separates the variable term from other numbers or expressions in the equation.

Square Roots

Square roots are a unique mathematical operation, and dealing with them properly is key to solving certain types of equations. In the context of our problem, once we've isolated the square root of h by dividing both sides by k, we have:
\[ \frac{d}{k} = \sqrt{h} \]
To remove the square root and solve for h directly, we need to perform the inverse operation, which is squaring:
\[ \left( \frac{d}{k} \right)^2 = h \]
This step eliminates the square root and leaves us with an expression that directly solves for h. Always remember, squaring is the inverse operation to taking the square root, which is why this step works perfectly.

Algebraic Manipulation

Algebraic manipulation involves various techniques to simplify and solve equations. In our example, we've already used a key technique: dividing both sides of the equation. This falls under the broader category of algebraic manipulation. Next, as we squared both sides of the isolated equation, we ensured we were systematically handling each term correctly. It's important to:

• Identify the operations needed to isolate the variable
• Apply inverse operations carefully
• Maintain the balance of the equation

Remember, algebraic manipulation is essential for ensuring you correctly and cleanly solve for the intended variable without errors.

Solving for a Variable

Solving for a variable means finding the value of the variable that makes the equation true. It's essentially the climax of our manipulations. After isolating the square root of h and squaring both sides of the equation, we've arrived at:
\[ h = \left( \frac{d}{k} \right)^2 \]
This final step provides a clear, simplified expression for h. By following a systematic approach involving isolating the variable, understanding and using square roots, and performing clean algebraic manipulations, we've easily solved for h. Whether you're dealing with simple or complex equations, the goal remains the same—finding a methodical way to isolate and solve for your desired variable.