Russ - did you notice that you have only odd numbers to work with? If you are trying to combine five odd numbers using only addition and subtraction, you are going to end up with an odd result. Hence, you cannot get 14 as an answer.It may help to write your available numbers this way:
13 = 2 * 6 + 1
11 = 2 * 5 + 1
9 = 2 * 4 + 1
7 = 2 * 3 + 1
5 = 2 * 2 + 1
3 = 2 * 1 + 1
1 = 2 * 0 + 1
If you add any two of these numbers, you will have an even result:
(2 * x + 1) + (2 * y + 1) = (2 * x) + 1 + (2 * y) + 1
= (2 * x) + (2 * y) + 2
= 2 * (x + y) + 2
= 2 * (x + y + 1)
= even!
for example, 13 + 5 = 18 (even)
If you subtract any two of them, you end up even again:
(2 * x + 1) - (2 * y + 1) = (2 * x) + 1 - (2 * y) - 1
= (2 * x) - (2 * y) + 1 - 1
= 2 * (x - y) + 0
= 2 * (x - y)
= even again!
for example, 13 - 5 = 8 (even)
Extending this thought a little, you can see that adding or subtracting four odd numbers is going to give you an even result. So will six. In fact, using ANY even number of odd integers is going to give you an even result. In your problem, you state that you must use any FIVE of the given integers. When you use the first FOUR of them, no matter which ones and no matter whether you add or subtract, you'll have an even number at that point. And when you either add or subtract that fifth odd number, you are going to have an odd final result.
14 is an even number. Your puzzle has no solution.
Hope that helps.