College Algebra, Chapter 1, 1.1, Section 1.1, Problem 86

Solve the equation $\displaystyle \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$ for $R_1$

$\begin{equation} \begin{aligned} \frac{1}{R} &= \frac{1}{R_1} + \frac{1}{R_2} && \text{Subtract both sides by } \frac{1}{R_2}\\ \\ \frac{1}{R} - \frac{1}{R_2} &= \frac{1}{R_1} + \frac{1}{R_2} - \frac{1}{R_2} && \text{Simplify}\\ \\ \frac{1}{R} - \frac{1}{R_2} &= \frac{1}{R_1} && \text{Get the LCD of the left side}\\ \\ \frac{R_2 - R}{RR_2} &= \frac{1}{R_1} && \text{Multiply both sides by } R_1(RR_2)\\ \\ R_1 \cancel{(RR_2)} & \left[ \frac{R_2 - R}{\cancel{RR_2}} = \frac{1}{\cancel{R_1}} \right] \cancel{R_1} (RR_2) && \text{Cancel out like terms}\\ \\ R_1 (R_2 - R) &= RR_2 && \text{Divide both sides by } (R_2 -R)\\ \\ \frac{R_1 \cancel{(R_2 - R)}}{\cancel{R_2-R}} &= \frac{RR_2}{R_2 - R} && \text{Simplify}\\ \\ R_1 &= \frac{RR_2}{R_2 - R} \end{aligned} \end{equation}$