For the $LPP$,maximize $z=x+4y$ subject to the constraints $x+2y \leq 2$,$x+2y \geq 8$,$x, y \geq 0$.

Solution of the following $LP$ problem
Minimize $z = -3x + 2y$
subject to $0 \leq x \leq 4, 1 \leq y \leq 6, x + y \leq 5$ is $.....$

$A$ manufacturer of electronic circuits has a stock of $200$ resistors,$120$ transistors and $150$ capacitors and is required to produce two types of circuits $A$ and $B$. Type $A$ requires $20$ resistors,$10$ transistors and $10$ capacitors. Type $B$ requires $10$ resistors,$20$ transistors and $30$ capacitors. If the profit on type $A$ circuit is $Rs. 50$ and that on type $B$ circuit is $Rs. 60$,formulate this problem as a $LPP$ so that the manufacturer can maximize his profit.

The maximum value of the objective function $Z = 3x + 2y$ for the linear constraints $x + y \leq 7$,$2x + 3y \leq 16$,$x \geq 0$,$y \geq 0$ is

The optimal solution of the $L$.$P$.$P$. Maximize $Z = 8x + 3y$ subject to the constraints $x + y \leq 3, 4x + y \leq 6, x \geq 0, y \geq 0$ is

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..