Find the equation of the line passing through the point of intersection of the lines 2x + y = 5 and x - 2y = 5 and having y-intercept equal to $-\dfrac{3}{7}$.

Equation of the lines are
⇒ 2x + y = 5 …(i)
⇒ x - 2y = 5 …(ii)

Multiplying (i) by 2, we get
⇒ 4x + 2y = 10 ….(iii)

Adding (iii) and (ii) we get,
⇒ 4x + 2y + x - 2y = 10 + 5
⇒ 5x = 15
⇒ x = 3.

Substituting the values of x in (i)
⇒ 2(3) + y = 5
⇒ 6 + y = 5
⇒ y = -1.

∴ Coordinates of point of intersection are (3, -1).

Hence, line passes through (3, -1). So it will satisfy y = mx + c.
⇒ -1 = 3m + c

Given, y-intercept is $-\dfrac{3}{7}$ so, c = $-\dfrac{3}{7}.$

⇒ -1 = 3m + $\Big(-\dfrac{3}{7}\Big)$

⇒ -7 = 21m - 3

⇒ -7 + 3 = 21m

⇒ -4 = 21m

⇒ m = $-\dfrac{4}{21}.$

Putting value of m and c in y = mx + c,

$\Rightarrow y = -\dfrac{4}{21}x + \Big(-\dfrac{3}{7}\Big) \\[1em] \Rightarrow y = \dfrac{-4x - 3 \times 3}{21} \\[1em] \Rightarrow 21y = -4x - 9 \\[1em] \Rightarrow 4x + 21y + 9 = 0.$

Hence, the equation of the line is 4x + 21y + 9 = 0.