Problem 14 - Algebra A Combined Function

Question

Solve each formula for the specified variable. \(T=m n r\) for \(n\)

Step-by-Step Solution

Answer

The solution is \( n = \frac{T}{m r} \).

Step 1: Identify the formula

We start with the formula given in the exercise, which is:\[ T = m n r \]Our task is to solve this formula for the variable \( n \).

Step 2: Isolate the variable

To solve for \( n \), we need to isolate it on one side of the equation. The formula can be rearranged by dividing both sides by \( m \) and \( r \). This gives:\[ n = \frac{T}{m r} \]

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. They involve variables, which are symbols that represent unknown or changeable values, and constants, which are fixed, known quantities. The primary goal in working with algebraic equations is to find the value of the variable that makes the equation true. For example, in the equation \( T = m n r \), \( n \) is the variable for which we are solving, while \( T \), \( m \), and \( r \) are considered constants in this context.

Algebraic equations can be linear or non-linear, depending on the degree of the variable(s). A linear algebraic equation, like the one in the exercise, involves variables raised only to the first power and typically represents a straight line when graphically depicted. Solving for a variable often requires manipulating the equation using inverse operations to maintain the equality and simplify the equation.

Variable Isolation

Variable isolation is a fundamental technique in solving algebraic equations, aimed at expressing the variable of interest alone on one side of the equation. This method involves performing a series of algebraic manipulations to "undo" the operations on the variable, thus revealing its value.

In the equation \( T = m n r \), our aim is to isolate \( n \). The strategy to achieve this involves dividing both sides of the equation by the product \( m r \), cancelling out these terms from the side with \( n \). What remains is \( n = \frac{T}{m r} \).

• Identify the operations affecting the variable.
• Perform inverse operations in reverse order to undo these actions.
• Re-evaluate the equation after each step to ensure the variable is progressively isolated.

Variable isolation shines when dealing with more complex expressions, as it systematically helps in tracing the variable back to its roots without disrupting the balance of the equation.

Rearranging Equations

Rearranging equations is the process of altering their format to better suit given conditions or needs, such as solving for a specific variable. This involves shifting terms within the equation without affecting its underlying equality.\( \)
The equation \( T = m n r \) can be rewritten to solve for \( n \) by rearranging, achieved through division in this case. \( n = \frac{T}{m r} \) results from moving \( m \) and \( r \) to the opposite side of the equation.

• Apply balance: Perform identical operations on both sides of the equation.
• Monitor each alteration: Ensure changes logically lead towards achieving the intended form.
• Simplify where possible: Simplification can reveal more intuitive insights about the equation's behavior.

Rearranging is crucial when an equation is initially set up in a form that doesn't directly provide the desired information, such as equations in physics or economics where one needs to solve for different variables conveniently.