Consider the following question of practical importance. Suppose a company makes products $N_1$ and $N_2$ using resources $R_1$ and $R_2.$ Moreover, to make one unit of the

product $N_j,$ where jin$\{1,2\},$ the company must use $a_{ij}$ units of the resource $R_i,$ where iin$\{1,2\}.$ If $b_i$ units of resource $R_i$ are available, then how many units $x_j$ of

product $N_j$ should be made so as to exactly use up the available resources? That is$,$ what is an optimal production plan?

Interpret this question as a system of equations. Is an optimal production plan guaranteed to exist?