Find the slope of the line through the points
`(i) (1,2)`, `(4,2)` `(ii) (0,-4)`, `(-6,2)`
`(iii) (4,-6)`, `(-2,-5)` `(iv) (4,6)`, `(2,12)`
`(v) (3,-2)`, `(3,4)`

Answer

To find the slope of the line through given points, we use the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

### Step-by-step Solution:

**(i) Points: (1, 2) and (4, 2)**

1. Identify the points: \( (x_1, y_1) = (1, 2) \) and \( (x_2, y_2) = (4, 2) \).
2. Calculate \( y_2 - y_1 \):
\[ y_2 - y_1 = 2 - 2 = 0 \]
3. Calculate \( x_2 - x_1 \):
\[ x_2 - x_1 = 4 - 1 = 3 \]
4. Substitute into the slope formula:
\[ m = \frac{0}{3} = 0 \]

**(ii) Points: (0, -4) and (-6, 2)**

1. Identify the points: \( (x_1, y_1) = (0, -4) \) and \( (x_2, y_2) = (-6, 2) \).
2. Calculate \( y_2 - y_1 \):
\[ y_2 - y_1 = 2 - (-4) = 2 + 4 = 6 \]
3. Calculate \( x_2 - x_1 \):
\[ x_2 - x_1 = -6 - 0 = -6 \]
4. Substitute into the slope formula:
\[ m = \frac{6}{-6} = -1 \]

**(iii) Points: (4, -6) and (-2, -5)**

1. Identify the points: \( (x_1, y_1) = (4, -6) \) and \( (x_2, y_2) = (-2, -5) \).
2. Calculate \( y_2 - y_1 \):
\[ y_2 - y_1 = -5 - (-6) = -5 + 6 = 1 \]
3. Calculate \( x_2 - x_1 \):
\[ x_2 - x_1 = -2 - 4 = -6 \]
4. Substitute into the slope formula:
\[ m = \frac{1}{-6} = -\frac{1}{6} \]

**(iv) Points: (4, 6) and (2, 12)**

1. Identify the points: \( (x_1, y_1) = (4, 6) \) and \( (x_2, y_2) = (2, 12) \).
2. Calculate \( y_2 - y_1 \):
\[ y_2 - y_1 = 12 - 6 = 6 \]
3. Calculate \( x_2 - x_1 \):
\[ x_2 - x_1 = 2 - 4 = -2 \]
4. Substitute into the slope formula:
\[ m = \frac{6}{-2} = -3 \]

**(v) Points: (3, -2) and (3, 4)**

1. Identify the points: \( (x_1, y_1) = (3, -2) \) and \( (x_2, y_2) = (3, 4) \).
2. Calculate \( y_2 - y_1 \):
\[ y_2 - y_1 = 4 - (-2) = 4 + 2 = 6 \]
3. Calculate \( x_2 - x_1 \):
\[ x_2 - x_1 = 3 - 3 = 0 \]
4. Substitute into the slope formula:
\[ m = \frac{6}{0} \]
Since division by zero is undefined, the slope is undefined.

### Final Answers:
1. (i) Slope = 0
2. (ii) Slope = -1
3. (iii) Slope = -\(\frac{1}{6}\)
4. (iv) Slope = -3
5. (v) Slope = Undefined