If $Z = 7x + y$ subject to $5x + y \geq 5$,$x + y \geq 3$,$x \geq 0$,$y \geq 0$,then the minimum value of $Z$ is

The constraints $-x_{1} + x_{2} \leq 1$,$-x_{1} + 3x_{2} \leq 9$,$x_{1}, x_{2} \geq 0$ define:

The maximum value of $z=50x+15y$ subject to the constraints $x+y \leq 60$,$5x+y \leq 100$,$x \geq 0$,$y \geq 0$ is at the point:

The shaded area in the figure given below is a solution set of a system of inequations. The minimum value of the objective function $Z = 3x + 5y$,subject to the linear constraints given by this system of inequations,is:

$A$ diet is to contain at least $80$ units of vitamin $A$ and $100$ units of minerals. Two foods $F_{1}$ and $F_{2}$ are available. Food $F_{1}$ costs $Rs. 4$ per unit and food $F_{2}$ costs $Rs. 6$ per unit. One unit of food $F_{1}$ contains $3$ units of vitamin $A$ and $4$ units of minerals. One unit of food $F_{2}$ contains $6$ units of vitamin $A$ and $3$ units of minerals. Formulate this as a linear programming problem. Find the minimum cost for a diet that consists of a mixture of these two foods and also meets the minimal nutritional requirements.

The shaded area in the given figure is a solution set for some system of inequations. The maximum value of the function $z=10x+25y$ subject to the linear constraints given by the system is