Problem 67 - Algebra and Trigonometry with Analytic Geometry

Question

The formula occurs in the indicated application. Solve for the specified variable. \(F=g \frac{m M}{d^{2}}\) for \(m\)

Step-by-Step Solution

Answer

The solution is \( m = \frac{F d^2}{g M} \).

Step 1: Understand the Problem

We need to solve the formula \( F=g \frac{m M}{d^{2}} \) for the variable \( m \). This means isolating \( m \) on one side of the equation.

Step 2: Reorganize the Formula

Rewrite the formula to make it easier to solve for \( m \). The given formula is \( F=g \frac{m M}{d^{2}} \). We can multiply both sides by \( d^2 \) to remove the fraction: \( F d^2 = g m M \).

Step 3: Isolate Variable m

To isolate \( m \), divide both sides of the equation by \( g M \): \( \frac{F d^2}{g M} = m \). This gives us \( m = \frac{F d^2}{g M} \).

Step 4: Verify the Solution

Double-check to ensure \( m = \frac{F d^2}{g M} \) is correctly isolated. By substituting back into the original equation, it satisfies \( F=g \frac{m M}{d^{2}} \), thus confirming its correctness.

Key Concepts

Solving for a Variable

Imagine being an explorer on a mission to find a hidden treasure, where the treasure is the variable you need to solve for in an equation. In our case, it is the variable \( m \) we want to isolate. When we solve for a variable, our goal is to rearrange the equation so that this variable sits alone on one side. This often involves:

• Using inverse operations, like transforming multiplication into division.
• Reorganizing the equation, which means changing the structure without altering the balance.
• Performing the same operation on both sides of the equation to maintain equality.

The mission is successful when your variable holds a solitary spot on one side of the equal sign \( = \). Consider the operation as a balance – whatever you do on one side, you have to do on the other. This ensures the equation remains true and balanced.

Algebraic Equations

At the heart of mathematics lies algebra, and algebraic equations are the framework of many mathematical concepts. An algebraic equation is like a mathematical statement, where two expressions are equal. This exercise provided the algebraic equation \( F = g \frac{m M}{d^2} \).These equations can include:

• Variables, which are symbols representing numbers (like \( m \) in our equation).
• Constants, which are fixed numbers that don't change (such as \( g \)).
• Coefficients, which are numbers multiplying a variable \( (M, g) \).
• Operators that indicate the mathematical operation (such as \( \times \) and \( \div \)).

Understanding the components of an algebraic equation is crucial for decoding and solving them, much like understanding the parts of speech helps in constructing a sentence. Each part has a role, and knowing their function helps us in manipulating and solving these equations.

Problem-Solving Steps

Problem-solving in algebra often involves a structured approach, as outlined in the step-by-step solution earlier. This systematic journey ensures that you don’t just guess the solution but arrive at it logically. Here’s how you can approach similar tasks:

• Identify what's given: Start by clearly understanding the variables and constants involved.
• Determine what you need to find: Specify the variable to solve for (like \( m \) in this case).
• Manipulate the formula: Rearrange the formula using mathematical operations to isolate the needed variable efficiently.
• Verify your solution: Plug it back into the original equation to check if it holds true.

Each of these steps is like a move in a dance – in the right sequence, they lead to a smooth solution. By systematically following these processes, you'll find problem-solving in algebra becomes much more manageable and even enjoyable!