The problem asks us to find the gradient of the line passing through each of the given pairs of points. The gradient, also known as slope, is a measure of the steepness of a line.

1. Problem Description

The problem asks us to find the gradient of the line passing through each of the given pairs of points. The gradient, also known as slope, is a measure of the steepness of a line.

2. Solution Steps

The gradient $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

a) Given points are $(11, 7)$ and $(6, 3)$.

Let $(x_1, y_1) = (11, 7)$ and $(x_2, y_2) = (6, 3)$.

$m = \frac{3 - 7}{6 - 11} = \frac{-4}{-5} = \frac{4}{5}$

b) Given points are $(-1, -4)$ and $(4, 7)$.

Let $(x_1, y_1) = (-1, -4)$ and $(x_2, y_2) = (4, 7)$.

$m = \frac{7 - (-4)}{4 - (-1)} = \frac{7 + 4}{4 + 1} = \frac{11}{5}$

c) Given points are $(0, 0)$ and $(-3, 5)$.

Let $(x_1, y_1) = (0, 0)$ and $(x_2, y_2) = (-3, 5)$.

$m = \frac{5 - 0}{-3 - 0} = \frac{5}{-3} = -\frac{5}{3}$

d) Given points are $(4, 2)$ and $(-1, -6)$.

Let $(x_1, y_1) = (4, 2)$ and $(x_2, y_2) = (-1, -6)$.

$m = \frac{-6 - 2}{-1 - 4} = \frac{-8}{-5} = \frac{8}{5}$

3. Final Answer

a) $m = \frac{4}{5}$

b) $m = \frac{11}{5}$

c) $m = -\frac{5}{3}$

d) $m = \frac{8}{5}$