Jamie has 14 bottles, each of a different height, in inches. If he adds the heights of two more bottles having the same height, then their average height increases by 1 inch. What is the height of each bottle he added so that the average height after including them is 15 inches?

Let S be the sum of the 14 original heights. The average is S/14. After adding two bottles of height x, the new sum is S + 2x and the new average is (S + 2x)/16 = 15, so S + 2x = 240. We are given that (S + 2x)/16 = S/14 + 1. Substituting S = 240 - 2x into this equation gives (240)/16 = (240 - 2x)/14 + 1, which simplifies to 15 = (120 - x)/7 + 1. Solving for x gives 14 = (120 - x)/7, so 98 = 120 - x, resulting in x = 22.