Question

Minimize $Z=x+2y$
Subject to the constraints
$2x+y\ge 3$
$x+2y\ge 6$
and $x\ge 0,y\ge 0$

Answer 1

Converting the given inequations into equations
$2x+y=3$ ….(1)
$x+2y=6$ ….(2)
Region represented by $2x+y\ge 3$ : The line $2x+y-3$ meets the coordinate axis at points $A(,0)$ and $B\left(0.3\right)$.
$2x+y=3$

X	3/2	0
y	0	3

$A(,0);B(0,3)$

Join the points, A and B to get the line. Clearly $(0,0)$ does not satisfy the inequation $2x+y\ge 3$. So the region opposite to the origin represents the solution set of the inequation.

Region represented by $x+2y\ge 6$ : The line $x+2y=6$ meets the coordinate axis at point $C(6,0)$ and $D(0,3)$.
$x+2y=6$

X	6	0
y	0	3

$C(6,0);D(0,3)$

Join point C to D to obtain the line. Clearly $(0,0)$ does not satisfy the inequation $x+2y\ge 6$. So the region opposite to the origin represents the solution set of the inequation.

Region represented by $x\ge 0$ and $y\ge 0$: Since every point in the first quadrant satisfies the inequations. So the first quadrant is the region represented by the inequations $x\ge 0$ and $y\ge 0$.

In the shaded region on CB, each and every point on line CB is satisfying the given inequations.
So the values of the objective function on these points are given in the following table :

Point	x-coordinate	v-coordinate	Objective function $Z=x+2y$
5	0	3	$Z_B=0+2\left(3\right)=6$
C	6	0	$Z_c=6+2\left(0\right)=6$

It is clear from the table that required solution of given L.P.P. every point is on line BC and the minimum value of $Z=6$.