Problem 14 - Beginning and Intermediate Algebra

Question

Solve each formula for the specified variable. (Leave \(\pm\) in the answers as needed.) See Examples I and 2. \(L=\frac{k d^{4}}{h^{2}}\) for \(h\)

Step-by-Step Solution

Answer

h = \pm \sqrt{\frac{k d^4}{L}}

Step 1: Isolate the Fraction

Start with the equation given: \[ L = \frac{k d^4}{h^2} \] To isolate the fraction, multiply both sides of the equation by h^2 to get rid of the denominator:

Step 2: Multiply Both Sides by h^2

\[ L h^2 = k d^4 \] This step gives us the equation without the fraction.

Step 3: Solve for h^2

To solve for \(h^2\), divide both sides of the equation by L: \[ h^2 = \frac{k d^4}{L} \]

Step 4: Solve for h

Now, take the square root of both sides to solve for h: \[ h = \pm \sqrt{\frac{k d^4}{L}} \]

Key Concepts

Isolating the Variable

When solving an equation for a specific variable, 'isolating the variable' is often the first step. This means getting the variable by itself on one side of the equation. We isolate the variable to make it easier to solve the equation step-by-step.

In our example, we start with the equation: \( L = \frac{k d^4}{h^2} \). Here, we want to solve for the variable 'h'.
To do this, we need to eliminate or move everything else that is combined with 'h' on the same side of the equation.

Initially, 'h' is in the denominator of the fraction \( \frac{k d^4}{h^2} \). So, we need to find a way to shift it. If a variable is part of a fraction, one common technique is to multiply both sides by the denominator, which leads us to the next step.

Multiplying Both Sides of an Equation

Multiplying both sides of an equation is a key algebraic step used to isolate a variable. This technique ensures that the balance of the equation remains consistent.

In our example: \( L = \frac{k d^4}{h^2} \), 'h' is part of the fraction's denominator.
To remove 'h^2' from the denominator, we multiply both sides by \( h^2 \): \[ L \times h^2 = \frac{k d^4}{h^2} \times h^2 \] This results in: \( L h^2 = k d^4 \).

Now, 'h' is no longer in the denominator, simplifying the equation to a form where the variable is easier to isolate.
From this step, we are ready to move to further simplify and solve the equation.

Taking the Square Root

Taking the square root of both sides of an equation helps to solve for a variable that is squared. This step can be particularly useful when isolating a variable that has been squared, such as 'h^2'.

Let's continue with our example: \( L h^2 = k d^4 \). To isolate 'h', we first divide both sides by 'L':
\[ h^2 = \frac{k d^4}{L} \]
Next, to solve for 'h', we take the square root of both sides:
\[ h = \pm \sqrt{\frac{k d^4}{L}} \]
The \( \pm \) symbol indicates that there are two possible solutions, one positive and one negative, for 'h'.

This final step of taking the square root allows us to reach the solution: 'h' by itself, making the equation easier to work with.