Problem 38 - Algebra for College Students

Question

For Problems \(31-44\), solve each equation for the indicated variable. $$ \frac{1}{R}=\frac{1}{S}+\frac{1}{T} \text { for } R $$

Step-by-Step Solution

Answer

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

Step 1: Identify the target variable

We need to solve the equation \( \frac{1}{R}=\frac{1}{S}+\frac{1}{T} \) for the variable \( R \). This means we need to rearrange the equation to express \( R \) explicitly in terms of \( S \) and \( T \).

Step 2: Combine the fractions on the right side of the equation

The right-hand side of the equation consists of the sum \( \frac{1}{S} + \frac{1}{T} \). To combine these fractions, find a common denominator, which in this case is \( ST \). Thus, we can rewrite the equation as: \[ \frac{1}{R} = \frac{T}{ST} + \frac{S}{ST} \]

Step 3: Simplify the expression

Now, rewrite the combined fractions with the common denominator:\[ \frac{1}{R} = \frac{T + S}{ST} \].

Step 4: Invert the equation to solve for \( R \)

To isolate \( R \), take the reciprocal of both sides of the equation:\[ R = \frac{ST}{S+T} \].

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In algebra, these symbols often represent numbers, and they help us describe relationships between them. These symbols are called variables, and they allow us to formulate equations that can be solved to find unknown quantities.

When solving equations, the main goal is to isolate the variable you are solving for, usually on one side of the equation. This involves using algebraic operations like addition, subtraction, multiplication, and division. These operations help in simplifying the equation and eventually solve it to obtain the value of the unknown variable.

In the given exercise, algebra is used to rearrange the equation and solve for the desired variable, \( R \). By applying algebraic principles correctly, you can find a solution that provides a clear relationship between the variables involved in the equation.

Variables

Variables are a fundamental concept in algebra. They act as placeholders for unknown values within mathematical equations. Variables are usually represented by letters such as \( x \), \( y \), or in this case \( R \), \( S \), and \( T \).

Understanding variables is crucial when solving equations. Essentially, an equation is a statement that two expressions are equal. When it includes variables, one or more values are unknown and need to be found. In the exercise we are looking at, the goal is to express \( R \) in terms of \( S \) and \( T \).

To manipulate equations involving variables, one must use the properties of equality. These properties allow you to add, subtract, multiply, or divide both sides of the equation by the same amount, ensuring you maintain a valid equation throughout the process.

Fraction operations

Fraction operations are essential in solving algebraic equations, particularly when equations involve fractional expressions. Fractions consist of a numerator and a denominator, and understanding how to manipulate these can help simplify and solve equations effectively.

In this exercise, you perform fraction operations to combine the fractions \( \frac{1}{S} \) and \( \frac{1}{T} \) into a single fraction with a common denominator \( ST \). This allows the equation to be combined and simplified. The steps include finding a common denominator, combining the fractions, and then inverting the result to solve for the variable.

Understanding the basics of how to combine, add, subtract, multiply, and divide fractions is essential for solving many types of algebraic equations. Once you've mastered these operations, equations like the one given can be tackled confidently.