Problem 61 - Elementary Algebra

Question

For the following problems, solve each literal equation for the designated letter. \(V=\frac{G M m}{D}\) for \(D\).

Step-by-Step Solution

Answer

Question: Solve the literal equation \(V = \frac{G M m}{D}\) for \(D\). Answer: \(D = \frac{G M m}{V}\)

Step 1: Write the given equation

The given equation is \(V = \frac{G M m}{D}\).

Step 2: Isolate D by multiplying both sides by D

To do this, multiply both sides of the equation by \(D\) to cancel it out from the fraction on the right side: \(VD = G M m\)

Step 3: Solve for D

Now, we need to isolate \(D\) by dividing both sides by \(V\): \(D = \frac{G M m}{V}\)

Step 4: Write the final answer

The final answer is \(D = \frac{G M m}{V}\).

Key Concepts

Isolate Variable

When solving literal equations, such as the given equation for \(V = \frac{G M m}{D}\), it's often necessary to isolate a particular variable. Isolating a variable means rearranging the equation so that the variable you want to solve for is on one side, typically by itself. This makes it the subject of the equation.Here's a simple approach:

• Identify the variable you need to solve for. In this case, it's \(D\).
• Apply operations to move all other elements to the opposite side.
• Perform these operations step by step, so you maintain balance and correctness in the equation.

In our example, this starts with multiplying both sides by \(D\) to remove \(D\) from the denominator.

Algebraic Manipulation

Algebraic manipulation involves changing the form of an equation through mathematical operations without altering its equality. This is crucial in solving for a specific variable, as it allows us to simplify and rearrange terms.

To solve a literal equation through algebraic manipulation:

• Use operations like addition, subtraction, multiplication, or division.
• Ensure each operation is applied to both sides of the equation.
• Keep the equation balanced at every step.

In the given equation, after isolating \(D\) with multiplication, we divided by \(V\) to further isolate \(D\) and solve the equation by rewriting it as \(D = \frac{G M m}{V}\). This manipulation highlights the core of algebraic flexibility: maintaining equality while changing forms.

Fraction Cancellation

Fraction cancellation is a useful technique in algebra, particularly with literal equations involving division. When a variable is within a fraction, our goal is often to 'cancel out' terms to simplify the equation.

In the equation \(V = \frac{G M m}{D}\):

• We want to eliminate \(D\) from the denominator. This is done by multiplying both sides by \(D\), transforming the equation into \(VD = G M m\).
• Cancelling fractions involves applying operations that result in a cleaner, simpler equation while maintaining equality.

This technique is essential for moving terms around and is particularly handy in making equations easier to handle and solve. By effectively managing fractions, we simplify the steps needed to arrive at the problem's solution.