Question 818226
<pre>
Maximize z=2x+4y
 subject to x+4y &#8806; 18
 4x+2y &#8806; 16
 x &#8807; 0
 y &#8807; 0

We form the equations of graphs of the boundary lines
lines by replacing the inequality symbols by equal 
signs.

x+4y = 18    <---line thru points (0,4.5) and (6,3) 
4x+2y = 16   <---line thru points (4,0) and (0,8)
x = 0        <---the y-axis 
y = 0        <---the x-axis

{{{drawing(2000/11,400,-1,6,-1,10, graph(2000/11,400,-1,6,-1,10),
green(line(0,4.5,14,1),line(0,8,4,0)), locate(.7,1.5,FEASIBLE),
locate(.7,1,REGION),locate(2.5,4.5,"(2,4)"), locate(4.1,.7,"(4,0)"),
locate(-1,5,"(0,4.5)")

  )}}}

We evaluate the objective function to maximize
at each corner point:

Corner     z = 2x+4y
 (0,0)     z = 2(0)+4(0) = 0      <---- minimum value
 (0,4.5)   z = 2(0)+4(4.5) = 18
 (2,4)     z = 2(2)+4(4) = 20     <---- maximum value
 (4,0)     z = 2(4)+4(0) = 8

The maximum value of z is 20, and will occur at (2,4), when x=2 and y=4

Edwin</pre>