In Exercises 35–54, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? S = C/(1 - r) for r

Verified step by step guidance

1. Identify the variable to solve for, which is \( r \) in the formula \( S = \frac{C}{1 - r} \).
2. Start by isolating the denominator on one side. Multiply both sides of the equation by \( 1 - r \) to get rid of the fraction: \( S(1 - r) = C \).
3. Distribute \( S \) on the left side: \( S - Sr = C \).
4. Next, isolate the term containing \( r \) by subtracting \( S \) from both sides: \( -Sr = C - S \).
5. Finally, solve for \( r \) by dividing both sides by \( -S \): \( r = \frac{S - C}{S} \).

Key Concepts

Solving Formulas for a Specific Variable

This involves manipulating an equation to isolate the desired variable on one side. Techniques include using inverse operations such as addition, subtraction, multiplication, division, and factoring. The goal is to rewrite the formula so the specified variable is expressed explicitly in terms of the other variables.

Understanding Rational Expressions

A rational expression is a ratio of two polynomials, like C/(1 - r). When solving for a variable in the denominator, it is important to consider restrictions (e.g., the denominator cannot be zero) and use algebraic steps carefully to avoid undefined expressions.

Recognizing Geometric Series Sum Formula

The formula S = C/(1 - r) represents the sum of an infinite geometric series with first term C and common ratio r, where |r| < 1. Understanding this helps interpret the formula’s meaning and the conditions under which it applies.