(1/a) - (1/b) = (1/c)

Solve for a:

1/a - 1/b = 1/c

Bring 1/a - 1/b together using the common denominator a b:

(b - a)/(a b) = 1/c

Cross multiply:

c (b - a) = a b

Expand out terms of the left hand side:

b c - a c = a b

Subtract a b + b c from both sides:

a (-b - c) = -b c

Divide both sides by -b - c:

a = (bc) / (b + c)

(1/a) - (1/b) = (1/c) 

make a the subject

$\begin{array}{|rcll|} \hline \dfrac{1}{a}-\dfrac{1}{b} &=& \dfrac{1}{c} \quad & | \quad +\dfrac{1}{b} \\\\ \dfrac{1}{a} &=& \dfrac{1}{c} +\dfrac{1}{b} \\\\ \dfrac{1}{a} &=& \dfrac{1}{c}\cdot \dfrac{b}{b} +\dfrac{1}{b}\cdot \dfrac{c}{c} \\\\ \dfrac{1}{a} &=& \dfrac{b}{bc} +\dfrac{c}{bc} \\\\ \dfrac{1}{a} &=& \dfrac{b+c}{bc} \\\\ \dfrac{a}{1} &=& \dfrac{bc}{b+c} \\\\ \mathbf{a} & \mathbf{=} & \mathbf{\dfrac{bc}{b+c}} \\\\ \hline \end{array}$