The value of expression mx - ny is 3 when x = 5 and y = 6 and its value is 8 when x = 6 and y = 5. Find the values of m and n.

Given,

Value of expression mx - ny is 3 when x = 5 and y = 6.

∴ 5m - 6n = 3

⇒ 5m = 3 + 6n

⇒ m = $\dfrac{3 + 6n}{5}$ …………(1)

Value of expression mx - ny is 8 when x = 6 and y = 5.

∴ 6m - 5n = 8

Substituting value of m from equation (1) in above equation.

$\Rightarrow 6\Big(\dfrac{3 + 6n}{5}\Big) - 5n = 8 \\[1em] \Rightarrow \dfrac{18 + 36n}{5} - 5n = 8 \\[1em] \Rightarrow \dfrac{18 + 36n - 25n}{5} = 8 \\[1em] \Rightarrow 18 + 11n = 5 \times 8 \\[1em] \Rightarrow 18 + 11n = 40 \\[1em] \Rightarrow 11n = 40 - 18 \\[1em] \Rightarrow 11n = 22 \\[1em] \Rightarrow n = \dfrac{22}{11} \\[1em] \Rightarrow n = 2.$

Substituting value of n in equation (1), we get :

$\Rightarrow m = \dfrac{3 + 6 \times 2}{5} \\[1em] = \dfrac{3 + 12}{5} \\[1em] = \dfrac{15}{5} \\[1em] = 3.$

Hence, m = 3 and n = 2.