What is the value of x for which the points A (-1, 3), B (2, x) and C (5, -1) are collinear?

Three points are collinear if they all lie on the same straight line. If three points lie on the same line, then the slope between any two pairs of points must be equal.

Given:

Point A = (-1, 3)

Point B = (2, x)

Point C = (5, -1)

The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

Slope $= \dfrac{y_2 - y_1}{x_2 - x_1}$

---

Slope of AC $= \dfrac{y_C - y_A}{x_C - x_A}$

$= \dfrac{-1 - 3}{5 - (-1)}$

$= \dfrac{-4}{6}$

$= -\dfrac{2}{3}$

---

Slope of AB $= \dfrac{y_B - y_A}{x_B - x_A}$

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

$= \dfrac{x - 3}{3}$

---

For the points to be collinear, Slope of AB = Slope of AC:

$\dfrac{x - 3}{3} = -\dfrac{2}{3}$

$x - 3 = -2$

$x = 1$

Therefore, the value of $x = 1$.