Question 1204922
let x equal the larger number
let y equal the smaller number


your equation is 3/x - 1/y = 1/12


that equation states that 3 times the reciprocal of the larger number minus the reciprocal of the smaller number is equal to 1/12.


another way of saying that is that the reciprocal of the smaller number is subtracted from 3 time the reciprocal of the larger number.


you start with 3/x - 1/y = 1/12


multiply both sides of the equation by xy to get:


3y - x = xy/12


multiply both sides of the equation by 12 to get:


12 * (3y - x) = xy


simplify to get:


36y - 12x = xy


you are given that y = x-5, so replace y with x-5 in the equation to get:


36 * (x-5) - 12x = x * (x-5)


simplify to get:


36x - 180 - 12x = x^2 - 5x


combine like terms to get:


24x - 180 = x^2 - 5x


subtract 24x from both sides of the equation and add 180 to both sides of the equation to get:


0 = x^2 - 29x + 180


factor this quadratic equation to get:


(x - 20) * (x - 9) = 0


solve for x to get:


x = 20 or x = 9


when x = 20, y = 15
when x = 9, y = 4


your initial equation was 3/x - 1/y = 1/12


when x = 20 and y = 15, the equation becomes 3/20 - 1/15 = 1/12
multiply both sides of this equation by 300 to get 45 - 20 = 25
this results in 25 = 25, confirming the equation is true.


when x = 9 and y = 4, the equation becomes 3/9 - 1/4 = 1/12
multiply both sides of this equation by 36 to get 12 - 9 = 36/12
this results in 3 = 3, confirming the equation is true.


your value pairs of x = 20 and y = 15 or x = 9 and y = 4 both satisfy the requirements of the problem.


you actually have two answers.


the first answer is your integers are 20 and 15.


the second answer is your integers are 9 and 4.