Write down the equation of the line whose gradient is $\dfrac{3}{2}$ and which passes through P, where P divides the line segment joining A(-2, 6) and B(3, -4) in the ratio 2 : 3.

By section-formula,

P = $\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big)

Substituting values we get,

$P = \Big(\dfrac{2 \times 3 + 3 \times -2}{2 + 3}, \dfrac{2 \times -4 + 3 \times 6}{2 + 3}\Big) \\[1em] = \Big(\dfrac{6 + (-6)}{5}, \dfrac{-8 + 18}{5}\Big) \\[1em] = \Big(\dfrac{0}{5}, \dfrac{10}{5}\Big) \\[1em] = (0, 2).$

By point-slope form,

Equation of line with slope = $\dfrac{3}{2}$ and passing through (0, 2) is :

⇒ y - y₁ = m(x - x₁)

⇒ y - 2 = $\dfrac{3}{2}$(x - 0)

⇒ 2(y - 2) = 3x

⇒ 2y - 4 = 3x

⇒ 2y = 3x + 4.

Hence, equation of required line is 2y = 3x + 4.