Question

The value of x satisfying
\[\Large \frac{ 6x+2a+3b+c }{ 6x+2a-3b-c }=\frac{ 2x+6a+b+3c }{ 2x+6a-b-3c }\]

Answer 1

apply componendo and dividendo rule both sides to simplify and then further simplify

Answer 2

\( \huge \frac{ 6x+2a+3b+c }{ 6x+2a-3b-c }=\frac{ 2x+6a+b+3c }{ 2x+6a-b-3c } \)

there is two cases :-
i) \(6x+2a+3b+c =2x+6a+b+3c=0 \)

ii) or , there is k s.t
\(2x+6a+b+3c = k(2x+6a+b+3c )\)
\(6x+2a-3b-c =k(2x+6a-b-3c )\)

Answer 3

i made a typo , in second case/first line
6x+2a+3b+ c = k(2x+6a+b+3c )

Answer 4

(6x+2a)/(3b+c) = (2x+6a)/(b+3c)
2*(3x+a) /(3b+c) = 2(x+3a)/(b+3c)
(3x+a)/(x+3a) = (3b+c)/(b+3c)
(3x+a)/(x+3a) - 3 = (3b+c)/(b+3c) -3
(3x+a-3x-9a)/(x+3a) = (3b+c-3b -9c) /(b+3c)
-8a/(x+3a) = -8c/(b+3c)
(x+3a)/a=(b+3c)/c
x/a +3 =b/c +3
x/a = b/c
x=ab/c

Answer 5

\[ \frac{ 6x+2a+3b+c }{ 6x+2a-3b-c }=\frac{ 2x+6a+b+3c }{ 2x+6a-b-3c } \stackrel{\color{red}{C\&D}}{\iff} \frac{ 3x+a}{ 3b+c }=\frac{ 3a+x }{ 3c+b } \]

Answer 6

+1 for C&D :)

Answer 7

what is C&D ??

Answer 8

Componendo and Dividendo :
\[\large \dfrac{a}{b} = \dfrac{c}{d} \iff \dfrac{a+b}{a-b} = \dfrac{c+d}{c-d}\]

http://en.wikipedia.org/wiki/Componendo_and_dividendo

Answer 9

it just follows from adding 1 to both sides, but really more powerful than it appears when you look at it first time !

Answer 10

hehe cool xD
ik this one though if 0<c , a<b
then
a-c<b+c
hehe

Answer 11

true, but hows that related here ?

Answer 12

hehe nothing xD i just remmembered it :P

Answer 13

that wiki link has a short+nice proof :)

Answer 14

i read it , was cute to know ;)
nice addition to my day lol

Answer 15

Sorry about late reply
i got it nice

Answer 16

So simple, and yet I don't know this. Cool haha, this is just the kind of thing that shows up with triangles all the time and sometimes chemistry.