In coordinate plane, construct the feasibility domain for the given constraints.


So, in the first quadrant, you plot the lines y = 4-x  and y = 5-2x.

They are shown below in the Figure as red line and green line, respectively.


    


        The lines  y = 4-x (red)  and   y = 5-2x (green).


The feasibility domain is the quadrilateral in QI, which is below these lines.

It is the area, where you look and search for the maximum of the objective function.


The quadrilateral has vertices 

    P0 = (0,0)    - the origin of the coordinate system;

    P1 = (0,4)    - y-intercept of the red line;

    P2 = (1,3)    - intersection of red line and green line;

    P3 = (2.5,0)  - x-intercept of the green line.


The given objective function is a linear function of x and y over the feasibility quadrilateral.

It it obvious that it gets its maximum value in some of the four vertices.


So you check the value of the objective fubction in vertices and find one of them where it has the maximum.

See the Table below, which I prepared for you:


    Point     coordinates   calculation of the      the value of the
                 (x,y)      objective function      objective function
                            Z(x,y) = 3x + 4y        at the vertex


     P0         (0,0)       3*0 + 4*0                    0

     P1         (0,4)       3*0 + 4*4                   16

     P2         (1,3)       3*1 + 4*3                   15

     P3         (2.5,0)     3*2.5 + 4*0                  7.5


From the table, you see that the objective function has the maximum value in vertex P1.

It means that the objective function has the maximum at the point P(0,4) at x= 0, y= 4.


ANSWER.  The solution to the given LP problem is the point P(0,4) at x= 0, y= 4.

         The maximum value is 16.