First inequality 9x + 12y >= 61 is THIS restriction

     y >= .


Second inequality 8x + 4y >= 32 is THIS restriction

    y >= .


Together with the inequalities x >= 0, y >= 0 they form THIS feasibility domain in the first quadrant QI, shown in the Figure below:





Plot y >=  (over the read line) and  y >=  (over the green line)

Feasibility domain is INFINITE AREA in Q1 OVER the both red and green lines.



Feasibility domain has 3 vertices:

    P1 = (0,8)        (y-intercept to green line);

    P2 = (7/3,10/3)   (intersection point of the red and green line);

    P3 = (61/9,0)     (x-intercept to red line).


You should calculate the value of the objective function Z = 5x + 30y  at these three points:

    at  P1:   Z = 5*0      + 30*8       = 240;

    at  P2:   Z = 5*(7/3)  + 30*(10/3)  = 111.667;

    at  P3:   Z = 5*(61/9) + 30*0       = 33.889.


The minimum is achieved at the point P3 = (61/9,0), where x= 61/9, y=0,  and is equal to 305/9 = 33.889 (approximately).