Solve the following system of simultaneous linear equations by the substitution method:

$\dfrac{2x}{a} + \dfrac{y}{b} = 2$

$\dfrac{x}{a} - \dfrac{y}{b} = 4$

Given,

$\dfrac{2x}{a} + \dfrac{y}{b} = 2$ …….(i)

$\dfrac{x}{a} - \dfrac{y}{b} = 4$ ……..(ii)

Multiplying both side of eqn (i) by ab we get,

$\Rightarrow ab\Big(\dfrac{2x}{a} + \dfrac{y}{b}\Big) = 2ab \\[1em] \Rightarrow 2bx + ay = 2ab ……(iii)$

Multiplying both side of eqn (ii) by ab we get,

$\Rightarrow ab\Big(\dfrac{x}{a} - \dfrac{y}{b}\Big) = 4ab \\[1em] \Rightarrow bx - ay = 4ab \\[1em] \Rightarrow bx = 4ab + ay ……(iv)$

Substituting value of bx in eq. (iii) we get,

⟹ 2(4ab + ay) + ay = 2ab

⟹ 8ab + 2ay + ay = 2ab

⟹ 8ab + 3ay = 2ab

⟹ 3ay = 2ab - 8ab

⟹ 3ay = -6ab

⟹ y = $\dfrac{-6ab}{3a}$ = -2b.

Substituting value of y in eqn. (iv) we get,

⟹ bx = 4ab - 2ab

⟹ bx = 2ab

⟹ x = 2a.

Hence, x = 2a and y = -2b.