Find the gradient and the equation of the line passing through the points:

(i) A(–2, 1) and B(3, –4)

(ii) A(4, –2) and B(2, –3)

(i) Slope of AB = $\dfrac{y$2 - y1}{x2 - x1}

= $\dfrac{-4 - 1}{3 - (-2)}$

= $\dfrac{-5}{5}$

= -1.

Equation : y - y₁ = m(x - x₁)

⇒ y - 1 = -1(x - (-2))

⇒ y - 1 = -1(x + 2)

⇒ y - 1 = -x - 2

⇒ x + y - 1 + 2 = 0

⇒ x + y + 1 = 0.

Hence, slope = -1, equation of AB is x + y + 1 = 0.

(ii) Slope of AB = $\dfrac{y$2 - y1}{x2 - x1}

= $\dfrac{-3 - (-2)}{2 - 4}$

= $\dfrac{-1}{-2}$

= $\dfrac{1}{2}$.

Equation : y - y₁ = m(x - x₁)

⇒ y - (-2) = $\dfrac{1}{2}$ (x - 4)

⇒ 2(y + 2) = (x - 4)

⇒ 2y + 4 = x - 4

⇒ x - 2y - 4 - 4 = 0

⇒ x - 2y - 8 = 0.

Hence, slope = $\dfrac{1}{2}$, equation of AB is x - 2y - 8 = 0.