How to solve $1/x = 1/a + 1/b$ to get $x = ab/(a + b)$?
I'm working through the first problem set in a text book. I have the question and the solution, however the solution gives no in between steps, only the final result.
The question is solve for $x$ in :
$$\frac{1}{x} = \frac{1}{a} + \frac{1}{b}$$
I could only get one step in:
$$x = x\left(\frac{1}{a} + \frac{1}{b}\right)$$
The solution by the text book answers section is:
$$x = \frac{ab}{a + b}$$
How was this arrived at? Seeking the steps between to arrive at this?
$\endgroup$ 22 Answers
$\begingroup$$$\frac{1}{x}=\frac{1}{a}+\frac{1}{b}$$
$$\frac{1}{x}=\frac{\color{red}b}{a\color{red}b}+\frac{\color{red}a}{b\color{red}a}$$
$$\frac{1}{x}=\frac{a+b}{ba}$$
$$x=\frac{ab}{a+b}$$
$\endgroup$ 9 $\begingroup$$$\frac{1}{a}+\frac{1}{b}=\frac{b}{ab}+\frac{a}{ab}=\frac{a+b}{ab}$$ So $$\frac{1}{\frac1a+\frac1b}=\frac{1}{\frac{a+b}{ab}}=\frac{\frac11}{\frac{a+b}{ab}}=\frac{ab}{a+b}$$
$\endgroup$
