Solve the following linear programming $($LP$)$ problems:

Problem $1$:

Solve the following linear programming $($LP$)$ problem using Graphical method:

Maximize $Z=36x1+40x2$

Subject $to$:

$3x1+5x245$

$x110$

$x28$

and $x10,x20$

a$-$ Plot the feasible region

b$-$ Determine all corner point feasible solutions,

c$-$ Determine all basic feasible solutions,

d$-$ Determine all optimal solutions,

Problem $2$:

Find the optimal solution for following linear programming $($LP$)$ problem using Simplex method:

Maximize $Z=10x1+8x2$

Subject $to$:

$21+4x236$

$4x1+2x248$

and $x10,x20$

Problem $3$:

Find the optimal solution for following linear programming $($LP$)$ problem using Simplex method:

Maximize $Z=5x1+6x2$

Subject $to$:

$2x1+3x230$

$5x1+4x260$

and

$x10,x20$