Maximise $Z=x+y$.Subject to $x+4y\leq 8,2x+3y\leq 12,3x+y\leq 9,x\geq 0,y\geq 0$.

Toolbox:
• First formulate the objective function and identify the constraints from the problem statement, To solve a Linear Programming problem graphically, first plot the constraints for the problem. This is done by plotting the boundary lines of the constraints and identifying the points that will satisfy all the constraints.
• Let $R$ be the feasible region for a linear programming problem and let $Z=ax+by$ be the objective function.When $z$ has an optimum value (maximum or minimum),where variables $x$ and $y$ are subject to constraints described by linear inequalities,this optimum value must occur at a corner point of the feasible region.
• If R is bounded then the objective function Z has both a maximum and minimum value on R and each of these occur at corner points of R
Step 1:
The objective function which is to maximized is given as
$Z=x+y$,subject to constraints $x+4y \leq 8,2x+3y\leq 12,3x+y\leq 9,x\geq 0,y\geq 0$
Now let us draw the line AB :$x+4y=8$,CD : $2x+3y=12$,EF :$3x+y=9$ on the graph.
Step 2:
Consider the line $x+4y=8$
Put $x=0,y=0$
$\Rightarrow 0\leq 8$
Which is true.
Hence the region $x+4y\leq 8$ lies below the line AB:$x+4y=8$
Consider the line$2x+3y=12$
Put $x=0,y=0$
$\Rightarrow 0\leq 12$
Which is true.
Hence the region $2x+3y\leq 12$ lies below the line CD :$2x+3y=12$
Consider the line$3x+y=9$
Put $x=0,y=0$
$\Rightarrow 0\leq 9$
Which is true.
Hence the region $3x+y\leq 9$ lies below the line $3x+y=9$
Step 3:
Clearly the origin $O$ is also included in the feasible region.
$\therefore$ The feasible region is $OEQA$,which is the shaded portion.
Step 4:
Let us obtain the maximum value of $Z=x+y$,subjected to the constraints as follows :
At the points $(x,y)$ the value of the objective function subjected to $Z=x+y$
At $O(0,0)$ the value of the objective function $Z=0+0=0$
At $A(0,2)$ the value of the objective function $Z=0+2=2$
At $Q(2.5,1.4)$ the value of the objective function $Z=2.5+1.4=3.9$
At $E(3,0)$ the value of the objective function $Z=3+0=3$
Clearly the maximum value is at $Q(2.5,1.4)$ and equal to 3.9