In Exercises 35–54, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? 1/R = 1/R1 + 1/R2 for R1

Verified step by step guidance

1. Start with the given formula: \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \).
2. To solve for \( R_1 \), first isolate \( \frac{1}{R_1} \) by subtracting \( \frac{1}{R_2} \) from both sides: \( \frac{1}{R_1} = \frac{1}{R} - \frac{1}{R_2} \).
3. Find a common denominator for the right side of the equation to combine the fractions: \( \frac{1}{R_1} = \frac{R_2 - R}{R \cdot R_2} \).
4. Take the reciprocal of both sides to solve for \( R_1 \): \( R_1 = \frac{R \cdot R_2}{R_2 - R} \).
5. This formula is recognized as the formula for the total resistance \( R \) in a parallel circuit, where \( R_1 \) and \( R_2 \) are the resistances of individual resistors.

Key Concepts

Reciprocal Relationships

In mathematics, a reciprocal relationship involves the inverse of a number or expression. For example, the reciprocal of a number 'x' is '1/x'. In the context of the given formula, the relationship between resistances in parallel circuits is expressed through reciprocals, indicating how the total resistance 'R' is affected by the individual resistances 'R1' and 'R2'.

Solving for a Variable

Solving for a variable means rearranging an equation to isolate the desired variable on one side. This process often involves using algebraic operations such as addition, subtraction, multiplication, and division. In the given formula, we need to manipulate the equation to express 'R1' in terms of 'R', 'R2', and their relationships, which is a fundamental skill in algebra.

Electrical Resistance in Parallel Circuits

The formula provided describes the total resistance in a parallel circuit, where multiple resistors are connected across the same voltage source. The total resistance 'R' is less than the smallest individual resistance, and the formula shows how the total resistance can be calculated using the individual resistances 'R1' and 'R2'. Understanding this concept is crucial for analyzing electrical circuits and their behavior.