Solve each equation for the specified variable. (Assume all denominators are nonzero.) 1/R=1/r_1 + 1/r_2, for R

Verified step by step guidance

1. Start with the given equation: \(1\/R = 1\/r_1 + 1\/r_2\).
2. To combine the terms on the right side, find a common denominator, which is \(r_1 r_2\), so rewrite the right side as \(\frac{r_2}{r_1 r_2} + \frac{r_1}{r_1 r_2}\).
3. Add the fractions on the right side: \(\frac{r_2 + r_1}{r_1 r_2}\), so the equation becomes \(1\/R = \frac{r_1 + r_2}{r_1 r_2}\).
4. To solve for \(R\), take the reciprocal of both sides, remembering that taking the reciprocal flips the fraction: \(R = \frac{r_1 r_2}{r_1 + r_2}\).
5. This expression gives \(R\) in terms of \(r_1\) and \(r_2\), completing the solution.

Key Concepts

Solving Equations for a Specific Variable

This involves isolating the desired variable on one side of the equation. Techniques include using inverse operations such as addition, subtraction, multiplication, division, and applying algebraic manipulations to rewrite the equation in terms of the specified variable.

Working with Rational Expressions

Rational expressions are fractions that contain variables in the numerator, denominator, or both. Understanding how to manipulate these expressions, including finding common denominators and simplifying complex fractions, is essential for solving equations involving sums of reciprocals.

Reciprocals and Their Properties

A reciprocal of a number is 1 divided by that number. Recognizing how to combine and manipulate sums of reciprocals, such as 1/R = 1/r_1 + 1/r_2, helps in rewriting the equation and solving for the variable of interest by inverting or rearranging terms.