Find a, b, c if three equations are given?

Question

I was given three equations in term of $a, b$ and $c$.
Equations are as follows
$ab (a+b+c)=1001$
$bc(a+b+c)=2002$
$ac(a+b+c)=3003$
Find $a, b, c$.

MY ATTEMPT
I took tue ratio and I got relation as follows
$2a=c ,2a=3b$
I dont know how to do further to solve this

Can anyone help . Shortcut would be appreciated more.

Answer 1

So, we replace $c$ with $2a,$ and $b$ with $\dfrac{2a}3$

to get $a\cdot\dfrac{2a}3\left(a+2a+\dfrac{2a}3\right)=1001\iff\dfrac{22a^3}9=1001$

Answer 2

For nonzero $d$ the system of equations \begin{align*} ab(a+b+c)-d & = 0, \\ bc(a+b+c)-2d & = 0,\\ ca(a+b+c)-3d & =0 \end{align*} has exactly the solutions $c=2a$, $3b=2a$ and $22a^3=9d$. If you want rational or integer solutions, then you can choose $d$ accordingly. So, for example if $d=66$, a solution is $(a,b,c)=(3,2,6)$.