# How do you solve for a in 1/a + 1/b = 1/f ?

Aug 7, 2017

See a solution process below:

#### Explanation:

First, subtract $\textcolor{red}{\frac{1}{b}}$ from each side of the equation to isolate the $a$ term while keeping the equation balanced:

$\frac{1}{a} + \frac{1}{b} - \textcolor{red}{\frac{1}{b}} = \frac{1}{f} - \textcolor{red}{\frac{1}{b}}$

$\frac{1}{a} + 0 = \frac{1}{f} - \frac{1}{b}$

$\frac{1}{a} = \frac{1}{f} - \frac{1}{b}$

Next, subtract the fractions on the right side of the equation after putting each fraction over a common denominator by multiplying each fraction by the appropriate form of $1$:

$\frac{1}{a} = \left(\frac{b}{b} \times \frac{1}{f}\right) - \left(\frac{1}{b} \frac{f}{f}\right)$

$\frac{1}{a} = \frac{b}{b f} - \frac{f}{b f}$

$\frac{1}{a} = \frac{b - f}{b f}$

We can now "flip" the fraction on each side of the equation to solve for $a$ while keeping the equation balanced:

$\frac{a}{1} = \frac{b f}{b - f}$

$a = \frac{b f}{b - f}$

If you require the more rigorous process to solve for $a$ see below:

Multiply each side of the equation by $a b f$ to eliminate the fractions while keeping the equation balanced:

$a b f \times \frac{1}{a} = a b f \times \frac{b - f}{b f}$

$\textcolor{red}{\cancel{\textcolor{b l a c k}{a}}} b f \times \frac{1}{\textcolor{red}{\cancel{\textcolor{b l a c k}{a}}}} = a \textcolor{red}{\cancel{\textcolor{b l a c k}{b f}}} \times \frac{b - f}{\textcolor{red}{\cancel{\textcolor{b l a c k}{b f}}}}$

$b f = a \left(b - f\right)$

Now, divide each side of the equation by $\textcolor{red}{b - f}$ to solve for $a$ while keeping the equation balanced:

$\frac{b f}{\textcolor{red}{b - f}} = \frac{a \left(b - f\right)}{\textcolor{red}{b - f}}$

$\frac{b f}{b - f} = \frac{a \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(b - f\right)}}}}{\cancel{\textcolor{red}{b - f}}}$

$\frac{b f}{b - f} = a$

$a = \frac{b f}{b - f}$

Aug 7, 2017

color(magenta)(a=(bf)/(b-f)

#### Explanation:

$\frac{1}{a} + \frac{1}{b} = \frac{1}{f}$

$\therefore \frac{b f + a f = a b}{a b f}$

multiply both sides by $a b f$

$\therefore b f + a f = a b$

$\therefore a b - a f = b f$

$\therefore a \left(b - f\right) = b f$

:.color(magenta)(a=(bf)/(b-f)