Solve the following simultaneous equations:

10x + 3y = 75, 6x - 5y = 11

Given,

Equations : 10x + 3y = 75, 6x - 5y = 11

⇒ 10x + 3y = 75

⇒ 10x = 75 - 3y

⇒ x = $\dfrac{75 - 3y}{10}$     ….(1)

Substituting value of x from equation (1) in 6x - 5y = 11, we get :

$\Rightarrow 6\Big(\dfrac{75 - 3y}{10}\Big) - 5y = 11 \\[1em] \Rightarrow 3\Big(\dfrac{75 - 3y}{5}\Big) - 5y = 11 \\[1em] \Rightarrow \Big(\dfrac{225 - 9y}{5}\Big) - 5y = 11 \\[1em] \Rightarrow \Big(\dfrac{225 - 9y - 25y}{5}\Big) = 11 \\[1em] \Rightarrow \Big(\dfrac{225 - 34y}{5}\Big) = 11 \\[1em] \Rightarrow 225 - 34y = 11 \times 5\\[1em] \Rightarrow -34y = 55 - 225 \\[1em] \Rightarrow -34y = -170 \\[1em] \Rightarrow y = \dfrac{170}{34} \\[1em] \Rightarrow y = 5.$

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

$\Rightarrow x = \dfrac{75 - 3y}{10} \\[1em] \Rightarrow x = \dfrac{75 - 3(5)}{10} \\[1em] \Rightarrow x = \dfrac{75 - 15}{10} \\[1em] \Rightarrow x = \dfrac{60}{10} \\[1em] \Rightarrow x = 6.$

Hence, x = 6, y = 5.