Problem 48 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ T=\frac{3 R}{M-n} \text { for } R $$

Step-by-Step Solution

Answer

\( R = \frac{T(M-n)}{3} \)

Step 1: Understand the equation

The equation given is \( T = \frac{3R}{M-n} \). We are asked to solve for \( R \), which means we need to express \( R \) in terms of the other variables \( T \), \( M \), and \( n \).

Step 2: Isolate the term containing R

Since \( R \) is in a fraction, multiply both sides by \( M-n \) to eliminate the denominator. This gives us:\[T(M-n) = 3R\]

Step 3: Solve for R

Now that we have \( T(M-n) = 3R \), divide both sides by 3 to isolate \( R \):\[R = \frac{T(M-n)}{3}\]

Step 4: Write the final equation

The expression for \( R \) is now simplified and solved in terms of \( T \), \( M \), and \( n \):\[R = \frac{T(M-n)}{3}\]

Key Concepts

Solving Equations

When we approach solving equations, the goal is to find the value of an unknown variable. To solve the equation given in the exercise, we need to rearrange it so we can express the variable we're interested in—here, it's \( R \). The equation provided is \( T = \frac{3R}{M-n} \). To solve this equation, we want to do the opposite operations that were initially applied to the variable. Think of it as peeling away layers until \( R \) is by itself. For example:

• If the variable is multiplied by a number, we divide both sides of the equation by that number.
• If it's divided by a number, we multiply both sides.
• And if something is added, we subtract or, if subtracted, we add.

These opposite operations help us "undo" what's been done to the variable so we can isolate it.

Isolating Variables

To isolate a variable means to get it alone on one side of the equation. It helps us see the variable's relationship to the other quantities clearly. In this exercise, the goal was to isolate \( R \) from the equationFirst, notice that \( R \) is in the denominator, so the term \( M-n \) is dividing it. To get rid of this fraction, we multiply both sides by \( M-n \) to transfer \( M-n \) to the other side. This step is crucial because it turns a complex fraction into a simpler multiplication problem:\[ T(M-n) = 3R \]Now, \( R \) is almost isolated. The equation says \( 3 \) times \( R \). To fully isolate \( R \), the other side must be divided by 3:\[ R = \frac{T(M-n)}{3} \]This process of isolating \( R \) ensures it's alone by performing inverse operations, granting us the final expression of \( R \) in terms of the other variables.

Formula Manipulation

Formula manipulation is the art of rearranging algebraic expressions to solve for a different variable or to simplify relationships between variables. When manipulating formulas, it's essential to apply mathematical operations equally to both sides. This maintains the balance of the equation. In our exercise, we began with a formula that connected several variables: \( T = \frac{3R}{M-n} \). Our goal was to manipulate this formula so that \( R \) is isolated. Here's the step-by-step breakdown of our manipulation:

• Removing fractions: We multiply both sides by \( M-n \) to get rid of the denominator: \( T(M-n) = 3R \).
• Simplification: The equation becomes a straightforward multiplication problem: \( 3R \) equals some quantity, which is \( T(M-n) \). It's more straightforward now.
• Final division: We divide both sides by 3 to finally isolate \( R \): \( R = \frac{T(M-n)}{3} \).

Successful formula manipulation involves these strategies, emphasizing balance and simplicity. This skill is invaluable in solving complex problems efficiently.