i feel like i should know this

[Vanth]

but i cant for the life of me get it

solve for "a" : ƒ = 1 / (1/a + 1/b)


thanks in advance for any help

 

[Deleted member 4993]

but i cant for the life of me get it

solve for "a" : ƒ = 1 / (1/a + 1/b)


thanks in advance for any help

First find:

1/a + 1/b = ?

 

[soroban]

Hello, Vanth!

\(\displaystyle \text{Solve for }a\!:\;\;f \:=\:\dfrac{1}{\frac{1}{a}+\frac{1}{b}} \)

\(\displaystyle \text{Multiply the fraction by }\frac{ab}{ab}:\)

. . \(\displaystyle f \;=\;\dfrac{ab}{ab}\cdot\dfrac{1}{\frac{1}{a} + \frac{1}{b}} \quad\Rightarrow\quad f \;=\;\dfrac{ab(1)}{ab(\frac{1}{a}) + ab(\frac{1}{b})} \quad\Rightarrow \quad f\;=\;\dfrac{ab}{b+a} \)


\(\displaystyle \text{Multiply by }(b+a):\;\; f(b+a) \:=\:ab \quad\Rightarrow\quad bf + af \:=\:ab \quad\Rightarrow\) . . \(\displaystyle ab - af \:=\:bf\)


\(\displaystyle \text{Factor: }\:a(b-f) \:=\:bf \quad\Rightarrow\quad\boxed{ a \:=\:\frac{bf}{b-f}} \)