Step 1: graph the constraint boundary lines to determine the feasibility region.
--> (RED line) --> (GREEN line)
The feasibility region is in the first quadrant, below the red line and above the green line. So the corners of the feasibility region are (300,0), (1000,0), and the point of intersection of the two constraint boundary lines.
Step 2: Determine the corner of the feasibility region at which the objective function is maximized.
With the standard linear programming method, you evaluate the objective function at each corner of the feasibility region. However, that is not necessary.
The objective function is for some constant C; that function has a slope of -4/7. The maximum value of the objective function will be where a line with that slope just touches the feasibility region.
Comparing the slope of the objective function with the slopes of the constraint boundary lines will determine the point where the objective function is maximized.
Of course we can make that determination with simple calculations. -4/7 is between -1/2 and -8/3, so the objective function will be maximized at the intersection point of the two constraint boundary lines.
Graphing a couple of lines with slope -4/7 (blue and purple lines) will make it easy to see that the intersection of the constraint boundary lines is the point in the feasibility region where a line with slope -4/7 just touches the feasibility region.
Step 3: Evaluate the objective function at the point you have determined.
Solve the pair of equations of the constraint boundary lines and evaluate the objective function using the coordinates of the intersection point.
