$(1)$ A solution containing $30wt.\%$ benzene $($B$),45wt.\%$ toluene $($T$)$ and $25wt.\%$ xylene $($X$)$ is fed into a distillation process as shown in the attached diagram. The overhead product from the first column contains $($by wt$.)91\mathrm{.}40\%B,8\mathrm{.}30\%T,$ and $0\mathrm{.}30\%x.$ The overhead product from the second column contains $4\mathrm{.}25\%$ B$,91\mathrm{.}60\%T,$ and $4\mathrm{.}15\%x.$ One$-$third of the bottom product of the second column is recycled into the first column. It is also required that the benzene in the first column overhead product be $20$ times that of the second column overhead product, and $2$ times that of the second column bottom product. Assuming a basis of $100kg$ fresh feed, carry out the following operations:

Mass Balance around two distillation columns

$($a$)$ Carry out DOF analysis for all possible units and 'mixing points' that mass balance can be carried out.

$($b$)$ Derive and show all the independent simultaneous linear mass balance equations for the system using all the information provided.

$($c$)$ Solve the mass balance equations using the Excel Solver only $($as demonstrated in class$/$recitation$).$ The formula used in each cell must be shown in the submitted work as well as the computed cell result.