Matrices & Systems · Hard problem

A two-stage production line applies $R = \begin{pmatrix}2 & 1\\1 & 3\end{pmatrix}$ in stage $1$, then stage $2$ applies $F(k) = \begin{pmatrix}k & 1\\1 & 2\end{pmatrix}$, where $k$ is an adjustable calibration setting. The combined effect of both stages is $T = F(k)R$. Quality control requires the row 1, column 1 entry of $T$ to equal $15$. Once $k$ is set to meet that requirement, what is the row 1, column 2 entry of $T$?