Answer
$a = \frac{b^2+b}{2}$
Work Step by Step
$Solve$ $the$ $equation$ $for$ $the$ $indicated$ $variable:$
$\frac{a+1}{b} = \frac{a-1}{b}+ \frac{b+1}{a};$ $for$ $a$
Solve for a
Multiply both sides by $ab$ [Note: We combine the steps for multiply a and b for both sides to speed up the process]
$ ab(\frac{a+1}{b}) = ab(\frac{a-1}{b}+ \frac{b+1}{a})$
$ab(\frac{a+1}{b}) = ab(\frac{a-1}{b})+ ab(\frac{b+1}{a})$
Simplify
$a(a+1) = a(a-1)+ b(b+1)$
$a^2 + a = a^2-a + b^2 + b$
Subtract $a^2$ from both sides
$a^2 + a - a^2 = a^2 -a + b^2 + b - a^2$
Simplify
$a = -a + b^2 + b$
Add $a$ to both sides
$a+a = -a + b^2 + b + a$
Simplify
$2a = b^2 + b$
Divide both sides by 2
$\frac{2a}{2} = \frac{b^2+b}{2}$
$a = \frac{b^2+b}{2}$
