Minimize the objective function $Z = 3x + 2y$ subject to the constraints: $x + y \geq 8$,$x + y \leq 5$,$x \geq 0$,$y \geq 0$.

For the following shaded region,the linear constraints are:

The graphical solution set of the system of inequations $x+y \geq 1$,$7x+9y \leq 63$,$y \leq 5$,$x \leq 6$,$x \geq 0$,$y \geq 0$ is represented by:

$A$ dietician wishes to mix two types of foods in such a way that vitamin contents of the mixture contain at least $8$ $units$ of vitamin $A$ and $10$ $units$ of vitamin $C$. Food $'I'$ contains $2$ $units/kg$ of vitamin $A$ and $1$ $unit/kg$ of vitamin $C$. Food $'II'$ contains $1$ $unit/kg$ of vitamin $A$ and $2$ $units/kg$ of vitamin $C$. It costs Rs $50$ per $kg$ to purchase Food $'I'$ and Rs $70$ per $kg$ to purchase Food $'II'$. Formulate this problem as a linear programming problem to minimise the cost of such a mixture.

The graphical solution set of the system of in-equations $x+y \leq 70, x+2y \leq 100, 2x+y \leq 120, x \geq 0, y \geq 0$ is given by:

The feasible region of an $LPP$ is shown in the figure. If $z = 3x + 9y$,then the minimum value of $z$ occurs at