Problem 37 - Algebra for College Students

Question

For Problems \(31-44\), solve each equation for the indicated variable. $$ \frac{R}{S}=\frac{T}{S+T} \quad \text { for } R $$

Step-by-Step Solution

Answer

\( R = \frac{T \times S}{S + T} \)

Step 1: Eliminate the Fraction

Multiply both sides of the equation by \( (S + T) \) to eliminate the fraction. This gives us the equation \( R imes (S + T) = T imes S \).

Step 2: Expand the Equation

Expand the left side of the equation by distributing \( R \). That results in \( R imes S + R imes T = T imes S \).

Step 3: Solve for R

Rearrange the equation to isolate terms involving \( R \). Subtract \( R imes T \) from both sides to get \( R imes S = T imes S - R imes T \).

Step 4: Factor Out R

Factor \( R \) from the left side of the equation: \( R (S + T) = T imes S \).

Step 5: Divide to Solve for R

To isolate \( R \), divide both sides by \( S + T \): \( R = \frac{T imes S}{S + T} \).

Key Concepts

Solving Equations

Solving equations involves finding the value of a variable that makes the equation true. In our exercise, we aim to solve for \( R \) in the equation \( \frac{R}{S} = \frac{T}{S+T} \).

• First, recognize the structure of the equation. It is a fractional equation requiring manipulation to isolate \( R \).
• Our goal is to manipulate this equation to express \( R \) explicitly in terms of other variables.

Understanding how to solve equations involves understanding operations such as expanding expressions, factoring, and simplifying fractions. Each of these steps helps in systematically rearranging the equation until the desired variable is isolated.

Fraction Elimination

Fraction elimination is crucial when dealing with equations involving fractions. This is achieved by multiplying each side of the equation by the denominator to 'get rid of' the fraction.
In our exercise, we have the equation \( \frac{R}{S} = \frac{T}{S+T} \). To eliminate fractions, multiply both sides by \( (S + T) \), the denominator of the fraction on the right.
This operation gives us:

• \( R \times (S + T) = T \times S \)

By removing fractions, the equation becomes easier to handle, allowing for further manipulation and eventually simplifying the process of solving for \( R \).

Variable Isolation

Variable isolation is the process of manipulating an equation to get the variable of interest, \( R \) in this case, by itself on one side of the equation.
After eliminating fractions and multiplying out the terms, we have:

• \( R \times S + R \times T = T \times S \)

The goal is to make \( R \) the subject of the formula. You start by subtracting \( R \times T \) from both sides, leading to:

• \( R \times S = T \times S - R \times T \)

Next, factoring \( R \) out of the terms allows you to isolate it further, preparing the equation for the final solving step.

Equation Manipulation

Equation manipulation involves using algebraic techniques to simplify or rearrange equations. In this scenario, the equation is manipulated to solve for \( R \).
From the equation \( R \times S = T \times S - R \times T \), notice that combining terms that contain \( R \) through factoring can simplify things further.

• Factor out \( R \) from the left side: \( R (S + T) = T \times S \).
• Then divide both sides by \( S + T \) to make \( R \) the subject.
• This gives: \( R = \frac{T \times S}{S + T} \).

This clear division and simplification achieve the isolation of \( R \), showcasing effective equation manipulation to solve the given equation.