Answer:
1. Draw all graphs and determine the shaded region:
a) for you should take 1 and 4 quadrants (including y-axis);
b) for you should take 1 and 2 quadrants (including x-axis);
c) for you should draw the graph of the line y=-x+3 and take the bottom half-plane;
d) for you should draw the graph of the line y=1/3x+1 and take the bottom half-plane.
Then the intersection of all shaded parts is needed shaded region, quadrilateral OABC (see diagram for details).
2. Th graphs of the lines C=5x-4y will intersect this shaded region at different points. The maximum of the function could be only at vertices.
3. Find the coordinates of all vertices:
a) O(0,0) - the origin;
b) A(0,1) - the y-intercept of line y=-x+3;
c) C(3,0) - the x-intercept of line y=1/3x+1;
d) B: solve the system y=1/3x+1, y=-x+3.
-x+3=1/3x+1,
-x-1/3x=1-3,
-4/3x=-2,
x=3/2,
y=-3/2+3=3/2,
then B(3/2,3/2).
4) F(x)=5x-4y and
F(O)=0,
F(A)=5·0-4·1=-4,
F(C)=5·3-4·0=15,
F(B)=5·3/2-4·3/2=7.5-6=1.5
The maximal value of F is at point C.
Answer: Maximum value is 15 at point (3,0)
