This question concerns the straight line that passes through the points (−1, 3) and (2, −6). Choose the three true statements from the following.

Options
A
The gradient of the line is 3.
B
The gradient of the line is −3.
C
The gradient of the line is
13.
D
The equation of the line is y = −3x.
E
The equation of the line is y =2x − 6.
F
The line cuts the y-axis at y =3.
G
The line passes through the origin.
H
The line passes through the point (1, −1).

To determine the true statements about the line passing through the given points, we need to find the gradient (slope) of the line and the equation of the line.

1. Find the gradient (slope) of the line:
The gradient (m) of a line passing through two points (x₁, y₁) and (x₂, y₂) can be calculated using the formula:
m = (y₂ - y₁) / (x₂ - x₁)

Given points: (-1, 3) and (2, -6)
Substituting the values:
m = (-6 - 3) / (2 - (-1))
m = (-9) / (3)
m = -3

Therefore, the gradient of the line is -3.

2. Find the equation of the line:
We can use the point-slope form of a line given the gradient (m) and a point (x₁, y₁):
y - y₁ = m(x - x₁)

Using point (-1, 3) and m = -3:
y - 3 = -3(x - (-1))
y - 3 = -3(x + 1)
y - 3 = -3x - 3
y = -3x - 3 + 3
y = -3x

Therefore, the equation of the line is y = -3x.

Now, let's evaluate the given options to determine the true statements:

A) The gradient of the line is 3.
This statement is false since we found the gradient to be -3, not 3.

B) The gradient of the line is -3.
This statement is true. We found the gradient to be -3.

C) The gradient of the line is 13.
This statement is false. The gradient is -3, not 13.

D) The equation of the line is y = -3x.
This statement is true. We found the equation of the line to be y = -3x.

E) The equation of the line is y = 2x - 6.
This statement is false. The equation of the line is y = -3x, not y = 2x - 6.

F) The line cuts the y-axis at y = 3.
This statement is false. To find where the line cuts the y-axis, we set x = 0 in the equation y = -3x, giving us y = 0. Therefore, the line cuts the y-axis at y = 0, not y = 3.

G) The line passes through the origin.
This statement is false. The origin is the point (0, 0), and substituting these values into the equation y = -3x gives us y = 0. Therefore, the line does not pass through the origin.

H) The line passes through the point (1, -1).
This statement is false. Substituting (1, -1) into the equation y = -3x, we get -1 = -3(1), which is not true. Therefore, the line does not pass through the point (1, -1).

In summary, the true statements are:
B) The gradient of the line is -3.
D) The equation of the line is y = -3x.