A Simple Linear equation in 1 Variable!

From $abc=1$ , the given equation can be changed in the form

$\frac{2abcx}{(ab)(bc)+a(bc)+bc} + \frac{2bx}{bc+b+1} +\frac{2bcx}{(ca)(b)+c(b)+b}=1$ ,

$\frac{2x}{b+1+bc} + \frac{2bx}{bc+b+1} + \frac{2bcx}{bc+b+1} = 1$ ,

$\frac{2(bc+b+1)x}{bc+b+1}=1$ ,

Yielding $x= \frac{1}{2} = 0.5$

$\begin{aligned} \frac {2ax}{ab+a+1} + \frac {2bx}{bc+b+1} + \frac {2cx}{ca+c+1} & = 1 \\ \frac {2ax\blue{\div a}}{(ab+a+\blue{abc})\blue{\div a}} + \frac {2bx}{bc+b+1} + \frac {2cx\red{\times b}}{(ca+c+1)\red{\times b}} & = 1 & \small \blue{\text{Note that }abc = 1} \\ \frac {2x}{b+1+bc} + \frac {2bx}{bc+b+1} + \frac {2bcx}{(1+bc+b)} & = 1 \\ \frac {2x(1+b+bc)}{bc+b+1} & = 1 \\ 2x & = 1 \\ \implies x & = \frac 12 = \boxed{0.5} \end{aligned}$