# How to solve a system of linear equations with strict inequalities and equalities?

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Terri Kneeland on 21 Jan 2020
Answered: David Goodmanson on 22 Jan 2020
Hi all. I am looking to solve systems of linear inequalites that is similiar to the below:
F1x = 10x1 + 2x2 + 10x3
F2x = 15x1 + 20x2 + 8x3
F3x = 6x1 + 4x2 + 12x3
System:
F1x - F2x > 0
F1x - F3x > 0
x1, x2, x3 >=0
x1, x2, x3 <= 1
x1 + x2 + x3 = 1
I do not need to find all solutions, I simply need to know whether there is a solution or not.
Thanks!

David Goodmanson on 22 Jan 2020
Hi Terri,
Looks like it is not possible. You have
[F1;F2;F3] = [10 2 10
15 20 8
6 4 12] * [x1;x2;x3]
Now if F1-F2 > 0 ident2
and F1-F3 > 0 ident3
then 2*F1-F2-F3 > 0 ident4 necessary but not sufficient
but that last quantity is [2 -1 -1]*[F1;F2;F3]
= [2 -1 -1]*[10 2 10
15 20 8
6 4 12] * [x1;x2;x3];
= [-1 -20 0] * [x1;x2;x3]
and that quantity can't be greater than zero. If you had allowed equality with the F's
F1-F2 >= 0 and F1-F3 >= 0
then x1 = x2 = 0, x3 = 1 might have been possible but it gives
F1 = 10 F2 = 8 F3 = 12
which satisfies the equality-allowed version of ident4 but not the equality-allowed version of ident3. So it goes.