SOLUTION: Find the maximum value of y subject to; {{{x>= 0}}}, {{{y>=0}}}, {{{y<=x+5}}} and {{{y>=2x}}} Please help me!

We draw the lines , ,  and ,

by finding the intercepts and the point where   and 
intersect by solving the system:

  by substitution to see where they intersect.

x+5=2x 
  5=x
y=5+5
y=10, so they intersect at (5,10) 

And we find the feasible region: 

shade to the right of the line , the y-axis, because it's .
shade above the line , the x-axis, because it's .
shade below the line , because it's .
shade above the line , because it's .
[So the shading will automatically be above the x-axis].



The shaded region, including its boundaries, is called 
the "feasible region".

The maximum and minimum values of any linear expression
in x and y will occur at one of the three corner points:

y is a very simple linear expression so its minimum value 
occurs at the corner point (0,0) where y=0, and its maximum 
value occurs at the corner point (5,10) where y=10.

Answer: the maximum value of y is 10.

--------------------------------------------

Normally you have a more complicated expression to maximize 
than just the simple letter y.  

Suppose instead of having to maximize and minimize just 
simply y, you had to maximize and minimize 3y-4x instead.

Then you'd substitute each corner point in 3y-4x to see which
is the largest and smallest.

Substituting corner point (0,0) in 3y-4x = 3(0)-4(0) = 0-0 = 0
Substituting corner point (0,5) in 3y-4x = 3(5)-4(0) = 15-0 = 15
Substituting corner (5,10) in 3y-4x = 3(10)-4(5) = 30-20 = 10

So in that case the minimum value of 3y-4x would be 0 at (0,0)
and the maximum value of 3y-4x would be 15 at the corner point (0,5).

 Edwin