Let $l$ and $b$ be the length and breadth of the room respectively.
Then, the perimeter of the room is $2(l+b)$ metres.
From the question, 
$l=6+b$    ...(1)
$\frac{1}{2}\times 2(l+b)=46$
$\Rightarrow l+b=46$    ...(2)
Let's solve these two equations using substitution method.
On substituting the value of $l$ from (1) in (2), we get
$6+b+b=46$
$\Rightarrow$ $6+2b=46$
$\Rightarrow 2b=40$
$\Rightarrow b=20m$
$\Rightarrow l=20+6=26m$
Therefore, length and breadth are 26 m and 20 m long respectively.