Notes: Solve for a. c=(ab) / (a+b)

Saturday, May 30, 2015

$\displaystyle{c}=\frac{{{a}{b}}}{{{a}+{b}}}$ solve for
a

first multiply both sides by (a+b) to get rid of
fraction

$\displaystyle{c}{\left({a}+{b}\right)}={a}{b}$

Now
distribute

$\displaystyle{a}{c}+{b}{c}={a}{b}$

Now we
want to get all the terms with a as a factor on one side and everything else on the
other

$\displaystyle{a}{c}-{a}{b}=-{b}{c}$

now factor
out the a

$\displaystyle{a}{\left({c}-{b}\right)}=-{b}{c}$

Now
divide by
$\displaystyle{c}-{b}$

$\displaystyle{a}=-\frac{{{b}{c}}}{{{c}-{b}}}$

And we
can get rid of the negative sign by noting
$\displaystyle{\left({c}-{b}\right)}=-{\left({b}-{c}\right)}$

To get our final answer $\displaystyle{a}=$
(bc)/(b-c)

Notes