The sum of two numbers is 51. If the larger is doubled and the smaller is tripled, the difference is 12. Find the numbers.

Let two numbers be x and y, where x > y.

Given,

Sum of numbers = 51

⇒ x + y = 51

⇒ x = 51 - y     ……(1)

Given,

Difference of numbers when larger is doubled and the smaller is tripled = 12

⇒ 2x - 3y = 12     ……(2)

Substituting value of x from equation (1) in (2), we get :

⇒ 2(51 - y) - 3y = 12

⇒ 102 - 2y - 3y = 12

⇒ -5y = 12 - 102

⇒ -5y = -90

⇒ y = $\dfrac{90}{5} = 18$.

Substituting value of y in equation (1), we get :

⇒ x = 51 - y

⇒ x = 51 - 18

⇒ x = 33.

Hence, the numbers are 33 and 18.