Question

if $\left \{ {{a+b=3ab} \atop {a+c=4ac}} \right.$ and b+c=5bc then find the
$\frac{1}{a} + \frac{2}{b} + \frac{3}{c}$

Answer 1

solving the system of equations will give us two solutions either
a=0,b=0,c=0 which is not the answer we are looking for since this will be undefined since a/0 is undefined$\frac{1}{a} + \frac{2}{b} + \frac{3}{c}$
or a=1 and b=0.5 and c=$\frac{1}{3}$

which will make
$\frac{1}{a} + \frac{2}{b} + \frac{3}{c}$
by substituting
$1+ \frac{2}{ \frac{1}{2} } + \frac{3}{ \frac{1}{3} } =1+4+9=14$
OR
you can reverse engineer it
$\frac{1}{a} + \frac{2}{b} + \frac{3}{c}$
$\frac{1}{a} + \frac{1}{b}+ \frac{1}{b} + \frac{3}{c}$
by substituting
$\frac{a+b}{ab} + \frac{c+3b}{bc}$
$\frac{3ab}{ab} + \frac{5bc+2b}{bc}$
$3+ \frac{5c+2}{c}$