Find the slopes of the lines passing through the following points :
`(i) (1,5)` and `(3,2)`
`(ii) (-4,3)` and `(-6,3)`
`(iii) (1,3)` and `(1,4)`
`(iv) (2,-1)` and `(3,2)`

Answer

To find the slopes of the lines passing through the given points, we can use the formula for the slope \( m \) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\):

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

Now, let's solve each part step by step.

### (i) Points: (1, 5) and (3, 2)

1. Identify the points:
- \( (x_1, y_1) = (1, 5) \)
- \( (x_2, y_2) = (3, 2) \)

2. Substitute into the slope formula:
\[ m = \frac{2 - 5}{3 - 1} \]

3. Calculate the numerator and denominator:
\[ m = \frac{-3}{2} \]

4. Final slope:
\[ m = -\frac{3}{2} \]

### (ii) Points: (-4, 3) and (-6, 3)

1. Identify the points:
- \( (x_1, y_1) = (-4, 3) \)
- \( (x_2, y_2) = (-6, 3) \)

2. Substitute into the slope formula:
\[ m = \frac{3 - 3}{-6 - (-4)} \]

3. Calculate the numerator and denominator:
\[ m = \frac{0}{-2} = 0 \]

4. Final slope:
\[ m = 0 \]

### (iii) Points: (1, 3) and (1, 4)

1. Identify the points:
- \( (x_1, y_1) = (1, 3) \)
- \( (x_2, y_2) = (1, 4) \)

2. Substitute into the slope formula:
\[ m = \frac{4 - 3}{1 - 1} \]

3. Calculate the numerator and denominator:
\[ m = \frac{1}{0} \]

4. Since division by zero is undefined:
\[ m = \text{undefined} \]

### (iv) Points: (2, -1) and (3, 2)

1. Identify the points:
- \( (x_1, y_1) = (2, -1) \)
- \( (x_2, y_2) = (3, 2) \)

2. Substitute into the slope formula:
\[ m = \frac{2 - (-1)}{3 - 2} \]

3. Calculate the numerator and denominator:
\[ m = \frac{3}{1} = 3 \]

4. Final slope:
\[ m = 3 \]

### Summary of Slopes:
- (i) Slope = \(-\frac{3}{2}\)
- (ii) Slope = \(0\)
- (iii) Slope = \(\text{undefined}\)
- (iv) Slope = \(3\)