the length of a room exceeds its breadth by 3m. If the length is increased by 3m and the breadth is decreased by 2m, the area remains same. Find the dimensions of the room

Suppose the length and breadth of room are l and b respectively.
So the area of rectangle = lb
Now length of the room exceeds its breadth by 3 m.
So; l = b + 3 ...(i)
Now when length is increased by 3 m and breadth is decreased by 2 m; then we have
New area of the room = $\left(l+3\right)\left(b-2\right)$
and according to the question we have;
$$\begin{gathered} lb = \left(l+3\right)\left(b-2\right) \\ \Rightarrow lb = lb-2l+3b-6 \\ \Rightarrow 3b-2l = 6 \\ \Rightarrow 3b-2\left(b+3\right) = 6 \{\mathrm{using} (\mathrm{i}\left)\right\} \\ \Rightarrow 3b-2b-6 = 6 \\ \Rightarrow b = 12 \end{gathered}$$
Then from (i) we have;
l = 12 + 3 = 15
Therefore length and breadth of the room are 15 m and 12 m respectively.