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}
$