Let $T$ be an affine transformation defined by $T(\mathbf{v}) = M\mathbf{v} + \mathbf{b}$ where $M = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}$. If a triangle $\mathcal{T}$ has an area of $5$ square units, what is the area of the transformed triangle $T(\mathcal{T})$?