Solve the following pair of linear equations by the elimination method and the substitution method :

$\dfrac{x}{2} + \dfrac{2y}{3} = -1 \text{ and } x - \dfrac{y}{3} = 3$.

Given,

$\dfrac{x}{2} + \dfrac{2y}{3} = -1$ ………..(1)

$x - \dfrac{y}{3} = 3$ ……..(2)

Multiplying equation (2) by 2, we get :

$2x - \dfrac{2y}{3} = 6$ …………(3)

Adding equation (1) and (3), we get :

$\Rightarrow \dfrac{x}{2} + \dfrac{2y}{3} + 2x - \dfrac{2y}{3} = -1 + 6 \\[1em] \Rightarrow \dfrac{x + 4x}{2} = 5 \\[1em] \Rightarrow \dfrac{5x}{2} = 5 \\[1em] \Rightarrow x = \dfrac{2 \times 5}{5} \\[1em] \Rightarrow x = 2.$

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

$\Rightarrow 2 - \dfrac{y}{3} = 3 \\[1em] \Rightarrow \dfrac{y}{3} = 2 - 3 \\[1em] \Rightarrow \dfrac{y}{3} = -1 \\[1em] \Rightarrow y = -3.$

Hence, x = 2 and y = -3.