Linear Programming optimization with multiple optimal solutions I am trying to solve the following optimization problem using linear programming (deterministic operations research). According to the book, there are multiple optimal solutions, I don't understand why. I'll show you what I have done.
The problem is:
$max Z=500x_{1}+300x_{2}$
s.t.
$15x_{1}+5x_{2}\leq 300$
$10x_{1}+6x_{2}\leq 240$
$8x_{1}+12x_{2}\leq 450$
$x_{1},x_{2}\geqslant 0$
I have plotted the lines graphically to get:

The intersection points are:
$(15,15)$
$(\frac{135}{14},\frac{435}{14})$
$(\frac{5}{2},\frac{215}{6})$
The target function equals to 12000 for two of these points. If this value was the maximum, then I would say that the entire line, edge, between these intersection points is the optimal solution (multiple solutions). However, this is not the maximum. The second intersection I wrote gives a higher value, and therefore is the maximum. So I think there is a single solution.
What am I missing ?
And generally speaking, what is the mathematical justification for having multiple solutions when two points gives the maximum (or minimum)?

The second intersection is the one of the red and blue lines. This point does not satisfy the constraint given by the green line.
The optimal solutions are given by the line segment of the green line with its intersection points as end points. This is because the constraint puts the same relative weights on $x_1,x_2$ as the objective function. 
