Solve for a in terms of b & c .

so we can multiply both sides by abc

bc + ac = ab

ac - ab = -bc

a(c-b) = -bc

-a(c-b) = bc

-a = bc/(c-b)

so 

a = bc/(c-b)

Solution:..

Answer:  

$\large a=-\dfrac{bc}{c-b}\quad |\quad \:b\ne \:c$

GA

$\dfrac{1}{a} +\dfrac{1}{b}=\dfrac{1}c\\ \dfrac{1}a = \dfrac{1}c-\dfrac{1}b\\ \dfrac{1}a = \dfrac{b-c}{bc}\\ a = \dfrac{bc}{b-c}$

Max, your way may be better. You need exceptions: $b \ne c$ & $(b * c) \ne 0$ .   

...Solution: 

$\Large \frac{1}{a} \large abc+\Large \frac{1}{b} \large abc=\Large \frac{1}{c}\large abc\\ \large bc+ac=ab\\ \large bc+ac-bc=ab-bc\\ \large ac=ab-bc\\ \large ac-ab=ab-bc-ab\\ \large ac-ab=-bc\\ \large a\left(c-b\right)=-bc\\ \large \dfrac{a\left(c-b\right)}{c-b}=\dfrac{-bc}{c-b}\quad | \quad \:b\ne \:c\\ \text{ }\\ \LARGE a=-\dfrac{bc}{c-b}\quad | \quad \:b\ne \:c\\ $

GA

a =bc / (b - c) is eqivalent to a =- bc /(c - b) !!!.........(1)
Let a=2, b =3, c =? 

1/2 + 1/3 = 1/c                                                                                 
c = 6/5 . Sub these into (1) above:
2 =(3 * 6/5) /(3 - 6/5)
2 =3.6 / 1.8
2 = 2. 
Now sub into the 2nd solution:
2=-(3 *6/5) / (6/5 - 3)
2 =-3.6 / - 1.8
2 = 2