Fraction equations with word problems

Question

Find 2 numbers whose sum is 8 if the sum of their reciprocals is 2/3

Answer 1

x + y = 8

1/x + 1/y = 2/3

solving for y in first equation, to plug into second equation: y = 8 - x

1/x + 1/(8-x) = 2/3

Multiply both sides by LCM to clear denominators:

(x)(8-x) * (1/x + 1/(8-x)) = 2/3 (x)(8-x)

8-x + x = 2/3 (8x - x^2)

8 = 2/3 (8x - x^2)

3/2 ( 8) = 2/3 * 3/2 (8x - x^2)

12 = 8x - x^2

x^2 -8x + 12 = 0

(x-6)(x-2) = 0

x = 2 or 6

y = 6 or 2

numbers are 2 and 6