So i have this LP problem:
The variables are x1 and x2 (like x and y).
max 3x1 + 2x2 :
2x1 + x2 ≤ 4
−2x1 + x2 ≤ 2
x1 − x2 ≤ 1
x1 ≥ 0, x2 ≥ 0
Where the second, third and fourth lines are the constraints.
The constraints can be re-written as
x2=4-2x1
x2= 2+2x1
x2=x1-1 .
This question arrives to the point were i need to find the feasibility constraint given the problem.
On the answer sheet it states that in order to find the optimal value, we set
3x1+2x2=V
and to solve for x1 and x2 we set the first constraint equal to the second constraint, then
4-2x1=2+2x1
that gives us
x1= 1/2 and x2=3
I plug the values back in the objective function to get an optimal value of
3(0.5)+2(3)=7.5.
The feasibility set will then be shown on the graph as the shaded area.
Now my question is: why can't I equate the second and the third constraint to find the values of x1 and x2?
E.g 2+2x1=x1-1
then x1=-3 and x2=-4 and the optimal value will be
3(-3)+2(-4)=-17.
Can you please explain this in the simplest way possible?
The answer sheet states that :
The point where the objective function takes maximal value is found at the intersection between first and second constraint.
Please explain in the simplest way possible
which constraints to equate to find x1 and x2.
Thanks! Greatly appreciate it.