The given points (a, b) and (a + k, b - 3) lie on the line defined by y = 2x - 5. For the point (a, b) to satisfy this line equation, we have b = 2a - 5. For the point (a + k, b - 3) to also lie on this line, substituting into the equation gives b - 3 = 2(a + k) - 5. We already know that b = 2a - 5. Substituting in this value for b, we have (2a - 5) - 3 = 2(a + k) - 5. Simplifying, we get 2a - 8 = 2a + 2k - 5. Subtracting 2a from both sides, we get -8 = 2k - 5. Adding 5 to both sides, we get -3 = 2k. Dividing by 2, we get k = -3/2.