For solving each pair of equations, use the method of elimination by equating coefficients :

$\dfrac{x - y}{6} = 2(4 - x)$

2x + y = 3(x - 4)

Simplifying first equation :

⇒ $\dfrac{x - y}{6} = 2(4 - x)$

⇒ x - y = 12(4 - x)

⇒ x - y = 48 - 12x

⇒ x + 12x - y = 48

⇒ 13x - y = 48

⇒ 13x - y - 48 = 0 …….(1)

Simplifying second equation :

⇒ 2x + y = 3(x - 4)

⇒ 2x + y = 3x - 12

⇒ y + 2x - 3x + 12 = 0

⇒ y - x + 12 = 0 …….(2)

Adding equations (1) and (2), we get :

⇒ (13x - y - 48) + (y - x + 12) = 0

⇒ 13x - x - y + y - 48 + 12 = 0

⇒ 12x - 36 = 0

⇒ 12x = 36

⇒ x = $\dfrac{36}{12}$ = 3.

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

⇒ y - 3 + 12 = 0

⇒ y + 9 = 0

⇒ y = -9.

Hence, x = 3 and y = -9.