Problem 53 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ \frac{1}{r}+\frac{1}{s}=\frac{1}{t} \text { for } r $$

Step-by-Step Solution

Answer

\( r = \frac{ts}{s-t} \)

Step 1: Understand the Equation

We have the equation \( \frac{1}{r} + \frac{1}{s} = \frac{1}{t} \). The goal is to solve for the variable \( r \).

Step 2: Isolate the Term with r

Subtract \( \frac{1}{s} \) from both sides of the equation to isolate the term involving \( r \) on one side:\[ \frac{1}{r} = \frac{1}{t} - \frac{1}{s} \]

Step 3: Find a Common Denominator

The common denominator of \( t \) and \( s \) is \( ts \). Rewrite \( \frac{1}{t} \) and \( \frac{1}{s} \) with this common denominator:\[ \frac{1}{t} = \frac{s}{ts} \quad \text{and} \quad \frac{1}{s} = \frac{t}{ts} \]

Step 4: Simplify the Right Side Expression

Now substitute back into the equation:\[ \frac{1}{r} = \frac{s}{ts} - \frac{t}{ts} \]Simplify the right-hand side expression:\[ \frac{1}{r} = \frac{s - t}{ts} \]

Step 5: Solve for r

Take the reciprocal of both sides to solve for \( r \):\[ r = \frac{ts}{s-t} \]

Key Concepts

Formula Rearrangement

In algebra, rearranging a formula means manipulating it so you express a particular variable in terms of others. To rearrange a formula, you need to apply basic algebraic operations systematically. Rearrangement can involve operations like addition, subtraction, multiplication, division, and, importantly, finding common denominators. Breaking down the original exercise, we aim to isolate the desired variable, \( r \), on one side of the equation.

• Start by identifying terms associated with the target variable.
• Use inverse operations to shift these terms appropriately.
• Ensure to maintain the balance of the equation by performing the same operations on both sides.

Rearranging the initial formula involves isolating the \( \frac{1}{r} \) term, demonstrating the principle of formula rearrangement.

Equations

An equation is a mathematical statement showing that two expressions are equal, involving an equals sign \( = \). Solving an equation requires finding the value of the variable that makes the equation true. In the given exercise, the equation \( \frac{1}{r} + \frac{1}{s} = \frac{1}{t} \) is dealt with step by step.

• First, examine the equation structure and the position of each term.
• Perform algebraic operations to modify the equation while preserving equality. This includes adding, subtracting, multiplying, or dividing both sides.
• The aim is to express the equation in a form that makes solving for the desired variable straightforward.

Equations are like puzzles where you must adjust pieces coherently to uncover the unknown number.

Variables

Variables represent unknown values or quantities in an equation. They are typically denoted by letters, such as \( r \), \( s \), and \( t \) in our problem. Solving for a variable involves determining the value that satisfies the given equation.
To solve for \( r \):

• Recognize \( r \) as the unknown to be isolated.
• Shift and adjust other terms to clearly express \( r \) in terms of \( s \) and \( t \).

A variable can be seen as a placeholder waiting to be decoded. In our equation, \( r \) becomes the focal point, with each step driving towards simplifying and clarifying its role and value.