Problem 38 - Precalculus Mathematics for Calculus

Question

Solve the equation for the indicated variable. $$F=G \frac{m M}{r^{2}} ; \quad \text { for } r$$

Step-by-Step Solution

Answer

\( r = \sqrt{\frac{GmM}{F}} \)

Step 1: Write the Original Equation

The original equation given is: \[ F = G \frac{mM}{r^2} \] where we are to solve for the variable \( r \).

Step 2: Isolate the Denominator

To isolate \( r^2 \), let's multiply both sides of the equation by \( r^2 \): \[ Fr^2 = GmM \] This step removes the fraction by ensuring \( r^2 \) is alone on one side.

Step 3: Solve for r^2

Next, divide both sides by \( F \) to completely isolate \( r^2 \) on one side: \[ r^2 = \frac{GmM}{F} \] Now \( r^2 \) is on one side and the rest of the terms are on the other.

Step 4: Take the Square Root

To solve for \( r \), take the square root of both sides of the equation: \[ r = \sqrt{\frac{GmM}{F}} \] This step gives us the expression for \( r \) in terms of the other variables.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving equations. It involves reorganizing and transforming equations to facilitate easier computation or to achieve a specific goal, such as solving for a variable. The process often requires a combination of operations such as addition, subtraction, multiplication, and division.

When dealing with the original exercise—solving for a variable within a formula—algebraic manipulation helps break down complex formulas into manageable steps. This allows us to focus on the relationship between terms.

Key practices in algebraic manipulation include:

• Identifying like terms and combining them to simplify expressions.
• Using distributive and associative properties to rearrange terms as needed.
• Eliminating fractions by multiplying through by the denominator, as was done by multiplying both sides by \( r^2 \) in the given equation.

Applying these strategies correctly paves the way for isolating the desired variable.

Isolating Variables

Isolating a variable means manipulating the equation to have the variable by itself on one side. This is crucial for solving equations, as it directly provides the value or expression for the variable of interest.

In the exercise provided, the purpose was to isolate \( r \). By first multiplying both sides by \( r^2 \), we ensured that \( r^2 \) was no longer part of a fraction. Subsequently, dividing by \( F \) allowed us to position \( r^2 \) alone on one side of the equation.

To effectively isolate a variable, consider these steps:

• Observe and understand the arrangement of the equation. Identify which operations are being performed on the variable.
• Perform inverse operations to "untangle" the variable from other terms.
• Recheck your steps, ensuring the variable is fully isolated and the equation remains balanced.

Through isolating variables, we reach expressions that explicitly define the unknown in terms of known quantities.

Inverse Operations

Inverse operations are operations that undo each other, such as addition and subtraction, or multiplication and division. These operations are essential in the process of manipulating and solving equations, as they allow us to reverse actions and isolate variables.

In the given problem, using inverse operations was key to both isolating and solving for \( r \). Multiplying by \( r^2 \) and then dividing by \( F \) involved applying inverse operations to remove these factors from the variable we sought.

The main aspects of using inverse operations include:

• Recognizing pairs of inverse operations, such as square and square root. In our solution, taking the square root was the final inverse operation used to solve for \( r \).
• Understanding the sequence of operations, as performing them in the wrong order can result in incorrect solutions.
• Maintaining equation balance by applying operations to both sides at every step.

By mastering inverse operations, we enhance our ability to solve even the most intricate equations with confidence.