Find the equation of the line passing through the origin and the point of intersection of the lines 5x + 7y = 3 and 2x – 3y = 7.

⇒ 5x + 7y = 3 ….(1)

⇒ 2x - 3y = 7 ….(2)

Multiplying equation (1) by 3, we get :

⇒ 3(5x + 7y) = 3.3

⇒ 15x + 21y = 9 ….(3)

Multiplying equation (2) by 7, we get :

⇒ 7(2x - 3y) = 7.7

⇒ 14x - 21y = 49 ….(4)

Adding equations (3) and (4) we get,

⇒ 15x + 21y + 14x - 21y = 9 + 49

⇒ 29x = 58

⇒ x = $\dfrac{58}{29}$

⇒ x = 2.

Substituting x = 2 in (1), we get :

⇒ 5(2) + 7y = 3

⇒ 10 + 7y = 3

⇒ 7y = 3 - 10

⇒ 7y = -7

⇒ y = $\dfrac{-7}{7}$

⇒ y = -1.

Hence, the point of intersection of lines is (2, -1).

The equation of the line joining (2, -1) and (0, 0) will be given by two-point form i.e.,

$y - y$1 = \dfrac{y2 - y1}{x2 - x1} (x - x1)

Substituting values in above equation we get,

⇒ y - (-1) = $\dfrac{0 - (-1)}{0 - 2}$ (x - 2)

⇒ y + 1 = $\dfrac{1}{-2}$ (x - 2)

⇒ -2(y + 1) = (x - 2)

⇒ -2y - 2 = x - 2

⇒ x + 2y - 2 + 2 = 0

⇒ x + 2y = 0.

Hence, equation of line is x + 2y = 0.