Graphical Methods of Solving Linear Programming Problems

By graphical method, the solution of linear programming problem

Maximize $\mathrm{Z}=3x_1+5x_2$

Subject to $3x_1+2x_2\le 18$

$x_1\le 4$

$x_1\ge 0,x_2\ge 0,\text{ is }$

The objective function $z=4 x+3y$ can be maximised subjected to the constraints $3x+4y\le 24,8x+6y\le 48,x\le 5,y\le 6;x,y\ge 0$

Consider a LPP given by
Minimum $Z=6 x+10y$
Subjected to $x\ge 6; y\ge 2; 2x+y\ge 10; x,y\ge 0$
Redundant constraints in this $LPP$ are

The maximum value of $Z=4 x+3 y$ subjected to the constraints $3x+2y\ge 160,5x+2y\ge 200,x+2y\ge$ $80;x,y\ge 0$ is 

The maximum value of $Z=4 x+2y$ subjected to the constraints $2x+3y\le 18,x+y\ge 10;x,y\ge 0$ is

The point at which the maximum value ofx+y, subject to the constraintsx+ 2y≤ 70, 2x+y≤ 95,x,y≥ 0 is obtained, is

• (30, 25)
• (20, 35)
• (35, 20)
• 
  (40, 15)

The region represented by the inequation systemx,y≥ 0,y≤ 6,x+y ≤ 3 is

• unbounded in first quadrant
• unbounded in first and second quadrants
• bounded in first quadrant
• 
  none of these

A manufacturer has three machine $I, II, III$ installed in his factory. Machines I and II are capable of being operated for at most $12$ hours whereas machine $III$ must be operated for atleast $5$ hours a day. She produces only two items $M$ and $N$ each requiring the use of all the three machines.
The number of hours required for producing $1$ unit each of $M$ and $N$ on the three machines are given in the following table: