For the linear programming problem, $x + 2y \leq 10,\ 3x + y \leq 12,\ x, y \geq 0$, the maximum value of $z = 5x + 10y$ occurs at every point on the line segment joining the points..

The region satisfying the inequalities $y - x \geq 2,\ x + y \leq 5,\ x \geq 0$ and $y \geq 0$ is

The minimum value of $z = 3x + 5y$, subject to constraints $x \leq 80$, $y \geq 60$, $x + y \leq 200$ & $x, y \geq 0$ occurs at the point...

The feasible region represented by the constraints $y - 2x \leq 4, x + y \geq 5, x \leq 4, y \geq 2, x, y \geq 0$ is ...........