Step 1:

The objective function function given is $Z=3x-4y$ which is subjected to the constraints $x-2y\leq 0,-3x+y\leq 4,x-y\leq 6$ and $x,y\geq 0$

Now let us plot the points in the graph ,to find out the feasible region .

The line AB represents the equation $x-2y=0$,the line CD represents the equation $-3x+y=4$,the line EF represents the equation $x-y=6$

Step 2:

The feasible region bounded the above lines is OFE

The coordinates of O is the region (0,0)

The coordinates of F is the intersection of the line EF and the x-axis(6,0).

The coordinates of E is the intersection of the line EF and AB (i.e) (12,6)

Step 3:

Now let us find the optimal solutions of the objective function $Z=3x-4y$

At the points $(x,y)$ the values of the objective function subjected to $Z=3x-4y$

At $O(0,0)$ the value of the objective function $Z=3(0)-4(0)=0$

At $E(12,6)$ the value of the objective function $Z=3(12)-4(6)=12$

At $F(6,0)$ the value of the objective function $Z=3(6)-4(0)=18$

Hence the maximum value is 18.

The maximum value is 12.