Solve $2x - \dfrac{3}{y} = 9, 3x + \dfrac{7}{y} = 2.$ Hence, find the value of k if x = ky + 5.

Given,

$2x - \dfrac{3}{y} = 9$ …….(i)

$3x + \dfrac{7}{y} = 2$ …….(ii)

Multiplying (i) by 3 and (ii) by 2 we get,

$6x - \dfrac{9}{y} = 27$ …….(iii)

$6x + \dfrac{14}{y} = 4$ …….(iv)

Subtracting (iii) from (iv) we get,

$\Rightarrow 6x + \dfrac{14}{y} - \Big(6x - \dfrac{9}{y}\Big) = 4 - 27 \\[1em] \Rightarrow \dfrac{14}{y} + \dfrac{9}{y} = -23 \\[1em] \Rightarrow \dfrac{23}{y} = -23 \\[1em] \Rightarrow y = -1.$

Substituting value of y in (iv),

$\Rightarrow 6x + \dfrac{14}{-1} = 4 \\[1em] \Rightarrow 6x - 14 = 4 \\[1em] \Rightarrow 6x = 18 \\[1em] \Rightarrow x = 3.$

Substituting values of x and y in x = ky + 5 we get,

⇒ 3 = k(-1) + 5

⇒ 3 = -k + 5

⇒ k = 5 - 3 = 2.

Hence, x = 3, y = -1 and k = 2.