Let $F\_n$ be a free group, $u, v \in F\_n$, where $[u,v] \neq 1 $. Trying to show that the following equation does not have a solution in $F\_n$ : 

$$ [v,x] = [u,v] .$$

Any ideas are appreciated. I seem to have a proof but it's ugly (combinatorial, cancellation theory etc.) and not short (so, more likely to have a silly mistake :) )

Immediate thing to notice is that if $x=w$ is a solution, then so is the "line" $x=C(v)w$, where $C(v)$ is a centralizer of $v$. 

Also, not difficult to show that there are no solution in a free group of rank two generated by $u$ and $v$. 

thanks!