The L.P.P. to maximize $z=x+y,$ subject to $x+y\le 30,x\le 15,y\le 20,x+y\ge 15, x,y\ge 0$ has

A furniture trader deals in only two items - chairs and tables. He has $50,000$ rupees to invest and a space to store at most $35$ items. A chair costs him $1000$ rupees and a table costs him $2000$ rupees . The trader earns a profit of $150$ rupees and $250$ rupees on a chair and table, respectively. Formulate the above problem as an LPP to maximise the profit and solve it graphically.

The maximum value of $z=6x+8y$ subject to $x-y\ge 0, x+3y\le 12, x\ge 0, y\ge 0$ is

The feasible region of an $\mathrm{LPP}$ is shown in the figure. If $z=3x+9y$, then the minimum value of $z$ occurs at 

Solve graphically the following linear programming problem:

Maximize or minimize $Z=1000x+600y$ subject to constraints $x+y\le 200, x\ge 20, y\ge 4x, x\ge 0 \& y\ge 0$.

Solve graphically the following linear programming problem:

Maximize or minimize $Z=x+2y$ subject to constraints $x+2y\ge 100, 2x-y\le 0, 2x+y\le 200 \& x\ge 0, y\ge 0$.