Row Reduction & Rank · Hard problem

A square metal plate has its four interior grid points' steady-state temperatures $T_1,T_2$ (top row) and $T_3,T_4$ (bottom row) each equal to the average of their four neighbors. The top edge is held at $100°$, the bottom edge at $0°$, and both side edges at $50°$ (so the setup is symmetric left-to-right, meaning $T_1=T_2$ and $T_3=T_4$). The resulting equations are $4T_1=100+50+T_2+T_3$, $4T_2=100+50+T_1+T_4$, $4T_3=T_1+50+T_4$, $4T_4=T_2+T_3+50$. Find $T_1$.