$3\mathrm{.}2-1.$ The following table summarizes the key facts about two products, A and B$,$ and the resources, Q$,$ R$,$ and S$,$ required to produce them.

$\backslash$table$[[$Resource$,\backslash$table$[[$Resource Usage$],[$per Unit Produced$]],],[$Product A$,$Product B$,\backslash$table$[[$Amount of Resource$],[$Available$]]],[2,1,2],[1,2,2],[$S$,3,3,4],[$Profit per unit,$3,2,]]$

All the assumptions of linear programming hold.

$($a$)$ Formulate a linear programming model for this problem.

D$\mathrm{.}1($b$)$ Solve this model graphically.

$($c$)$ Verify the exact value of your optimal solution from part $($b$)$ by solving algebraically for the simultaneous solution of the relevant two equations.