The sum of two numbers is 9. and the sum of their reciprocals is 1/2.find the numbers

Set up a system of equations:

x + y = 9
1/x +1/y = 1/2

The easiest way to solve would be substitution.
x + y = 9 can be solved for y:

y = 9 - x

Then plug this into the second equation and solve for x:

1/x + 1/(9-x) = 1/2
(9-x) + x = x(9-x)/2
9 = (9x-x²)2
18 = 9x - x²
x² - 9x + 18 = 0
(x-6)(x-3) = 0
x = 6, 3

You can check your answer by plugging the two numbers back into both of the equations:

6 + 3 = 9
1/6 + 1/3 = 1/2

The two numbers are 6 and 3.