Sure, I can help you with that.
### Part (a)
To find the gradient (slope) of the line joining points [tex]\( A (-1,2) \)[/tex] and [tex]\( B (3,-2) \)[/tex], we use the formula for the gradient between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex]:
[tex]\[ \text{Gradient} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ x_1 = -1, \; y_1 = 2 \][/tex]
[tex]\[ x_2 = 3, \; y_2 = -2 \][/tex]
So the gradient is:
[tex]\[ \text{Gradient}_{AB} = \frac{-2 - 2}{3 - (-1)} = \frac{-2 - 2}{3 + 1} = \frac{-4}{4} = -1.0 \][/tex]
Thus, the gradient of the line joining points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( -1.0 \)[/tex].
### Part (b)
To find the gradient of the line joining points [tex]\( C (0, -1) \)[/tex] and [tex]\( D (4, 1) \)[/tex], we again use the formula for the gradient:
[tex]\[ \text{Gradient} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of points [tex]\( C \)[/tex] and [tex]\( D \)[/tex]:
[tex]\[ x_1 = 0, \; y_1 = -1 \][/tex]
[tex]\[ x_2 = 4, \; y_2 = 1 \][/tex]
So the gradient is:
[tex]\[ \text{Gradient}_{CD} = \frac{1 - (-1)}{4 - 0} = \frac{1 + 1}{4} = \frac{2}{4} = 0.5 \][/tex]
Thus, the gradient of the line joining points [tex]\( C \)[/tex] and [tex]\( D \)[/tex] is [tex]\( 0.5 \)[/tex].
So the gradients are:
a) The gradient of the line joining [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( -1.0 \)[/tex]
b) The gradient of the line joining [tex]\( C \)[/tex] and [tex]\( D \)[/tex] is [tex]\( 0.5 \)[/tex]
