The feasible region represented by the given constraints $2x + 3y \geqslant 12$,$-x + y \leqslant 3$,$x \leqslant 4$,$y \geqslant 3$ is denoted by

The objective function $Z = 4 x_1 + 5 x_2$,subject to $2 x_1 + x_2 \geq 7$,$2 x_1 + 3 x_2 \leq 15$,$x_2 \leq 3$,$x_1, x_2 \geq 0$ has minimum value at the point

The maximum value of the objective function $z=4x+6y$ subject to the constraints $3x+2y \leq 12$,$x+y \geq 4$,$x, y \geq 0$ is

The maximum value of $z=4x+2y$,subject to the constraints $3x+4y \geqslant 12$,$x+y \leqslant 5$,$x, y \geqslant 0$ is

The difference between the maximum value and minimum value of the objective function $z = 3x + 5y$ subject to the constraints $x + 3y \leqslant 60$,$x + y \geqslant 10$,$x - y = 0$,and $x, y \geqslant 0$ is:

The point at which the maximum value of $Z = x + y$ subject to the constraints $x + 2y \leq 70$, $2x + y \leq 95$, $x \geq 0$, $y \geq 0$ occurs is: