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

3x - 5y = 4

9x - 2y = 7

Given,

3x - 5y = 4 …….(i)

9x - 2y = 7 …….(ii)

Solving eqn. (i) we get,

⟹ 3x - 5y = 4

⟹ 3x = 4 + 5y

⟹ x = $\dfrac{4 + 5y}{3}$.

Substituting above value of x in eqn. (ii) we get,

$\Rightarrow 9x - 2y = 7 \\[1em] \Rightarrow 9\Big(\dfrac{4 + 5y}{3}\Big) - 2y = 7 \\[1em] \Rightarrow 3(4 + 5y) - 2y = 7 \\[1em] \Rightarrow 12 + 15y - 2y = 7 \\[1em] \Rightarrow 12 + 13y = 7 \\[1em] \Rightarrow 13y = 7 - 12 \\[1em] \Rightarrow 13y = -5 \\[1em] \Rightarrow y = -\dfrac{5}{13}.$

Solving for x by substituting value of y,

$\Rightarrow x = \dfrac{4 + 5y}{3} \\[1em] = \dfrac{4 + 5 \times \dfrac{-5}{13}}{3} \\[1em] = \dfrac{4 - \dfrac{25}{13}}{3} \\[1em] = \dfrac{\dfrac{52 - 25}{13}}{3} \\[1em] = \dfrac{27}{39} \\[1em] = \dfrac{9}{13}.$

Hence, x = $\dfrac{9}{13}\text{ and y } = -\dfrac{5}{13}.$