Solve :

$\dfrac{34}{3x + 4y} + \dfrac{15}{3x - 2y} = 5$

$\dfrac{25}{3x - 2y} - \dfrac{8.50}{3x + 4y}$ = 4.5

Given, equations :

$\dfrac{34}{3x + 4y} + \dfrac{15}{3x - 2y} = 5$ …….(1)

$\dfrac{25}{3x - 2y} - \dfrac{8.50}{3x + 4y}$ = 4.5 ………(2)

Multiplying equation (2) by 4, we get :

$\Rightarrow 4\Big(\dfrac{25}{3x - 2y} - \dfrac{8.50}{3x + 4y}\Big) = 4 \times 4.5 \\[1em] \Rightarrow \dfrac{100}{3x - 2y} - \dfrac{34}{3x + 4y} = 18 \text{ ………(3)}$

Adding equation (1) and (3), we get :

$\Rightarrow \dfrac{34}{3x + 4y} + \dfrac{15}{3x - 2y} + \Big(\dfrac{100}{3x - 2y} - \dfrac{34}{3x + 4y}\Big) = 5 + 18 \\[1em] \Rightarrow \dfrac{115}{3x - 2y} = 23 \\[1em] \Rightarrow 3x- 2y = \dfrac{115}{23} \\[1em] \Rightarrow 3x - 2y = 5 \text{ ………(4)} \\[1em] \Rightarrow 3x = 5 + 2y \\[1em] \Rightarrow x = \dfrac{5 + 2y}{3} \text{ ……(5)}$

Substituting value of 3x - 2y from equation (4) in (2), we get :

$\Rightarrow \dfrac{25}{5} - \dfrac{8.50}{3x + 4y} = 4.5 \\[1em] \Rightarrow \dfrac{25(3x + 4y) - 42.50}{5(3x + 4y)} = \dfrac{45}{10} \\[1em] \Rightarrow \dfrac{75x + 100y - 42.50}{15x + 20y} = \dfrac{9}{2} \\[1em] \Rightarrow 2(75x + 100y - 42.50) = 9(15x + 20y) \\[1em] \Rightarrow 150x + 200y - 85 = 135x + 180y \\[1em] \Rightarrow 150x - 135x + 200y - 180y = 85 \\[1em] \Rightarrow 15x + 20y = 85 \text{ ……..(6)}$

Substituting value of x from equation (5) in (6), we get :

$\Rightarrow 15 \times \dfrac{5 + 2y}{3} + 20y = 85 \\[1em] \Rightarrow 5(5 + 2y) + 20y = 85 \\[1em] \Rightarrow 25 + 10y + 20y = 85 \\[1em] \Rightarrow 30y = 85 - 25 \\[1em] \Rightarrow 30y = 60 \\[1em] \Rightarrow y = \dfrac{60}{30} = 2.$

Substituting value of y in equation (5), we get :

$\Rightarrow x = \dfrac{5 + 2y}{3} \\[1em] = \dfrac{5 + 2 \times 2}{3} \\[1em] = \dfrac{5 + 4}{3} \\[1em] = \dfrac{9}{3} \\[1em] = 3.$

Hence, x = 3 and y = 2.