Final answer:
To find the numbers, we can set up a system of equations using the sum of the numbers and the sum of their reciprocals. Simplifying the equations and solving the quadratic equation, we find that the numbers are 6 and -3.
Step-by-step explanation:
To solve this problem, we can set up a system of equations. Let's assume the two numbers are x and y. We know that the sum of the two numbers is 9, so we can write the equation x + y = 9. We are also given that eighteen times the sum of their reciprocals is 9, so we can write the equation 18(1/x + 1/y) = 9.
To simplify the second equation, we can find a common denominator for 1/x and 1/y, which is xy. This gives us the equation 18(x+y)/xy = 9. Multiplying both sides of the equation by xy/9, we get 2(x+y) = xy.
Now, we can substitute x + y = 9 into the equation 2(x+y) = xy. This gives us the equation 2(9) = xy, which simplifies to 18 = xy. We now have a quadratic equation xy - 18 = 0.
To solve this quadratic equation, we can factor it. The only two numbers that multiply to -18 and add up to 9 are 6 and -3. So, the numbers x and y are 6 and -3, respectively.