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}
$