College Algebra, Chapter 1, 1.1, Section 1.1, Problem 90

Solve the equation $\displaystyle \frac{a+1}{b} = \frac{a-1}{b} + \frac{b+1}{a}$ for $a$

$\begin{equation} \begin{aligned} \frac{a+1}{b} &= \frac{a-1}{b} + \frac{b+1}{a} && \text{Subtract both sides by } \left( \frac{a-1}{b} \right)\\ \\ \frac{a+1}{b} - \frac{a-b}{b} &= \frac{a-1}{b} + \frac{b+1}{a} - \frac{a-1}{b} && \text{Simplify}\\ \\ \frac{\cancel{a}+1-\cancel{a}+1}{b} &= \frac{b+1}{a} && \text{Get the LCD and combine like terms}\\ \\ \frac{2}{b} &= \frac{b+1}{a} && \text{Multiply both sides by } (ab)\\ \\ a\cancel{b} & \left[ \frac{2}{\cancel{b}} = \frac{b+1}{\cancel{a}} \right] \cancel{a}b && \text{Cancel out like terms} \\ \\ 2a &= b(b+1) && \text{Divide both sides by 2}\\ \\ \frac{\cancel{2}a}{\cancel{2}} &= \frac{b(b+1)}{2} && \text{Simplify}\\ \\ a &= \frac{b(b+1)}{2} \end{aligned} \end{equation}$