Find the slope of the line passing through the points
(i) (2,3) and (-1,-3)
(ii) (-2,3) and (3, -5)
(iii) (-2,-3) and (-3, 5)

Answer

To find the slope of the line passing through two 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.

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

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

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

3. Simplify the numerator and denominator:
\[ m = \frac{-6}{-3} \]

4. Calculate the slope:
\[ m = 2 \]

### (ii) Points: (-2, 3) and (3, -5)

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

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

3. Simplify the numerator and denominator:
\[ m = \frac{-8}{3 + 2} = \frac{-8}{5} \]

4. Calculate the slope:
\[ m = -\frac{8}{5} \]

### (iii) Points: (-2, -3) and (-3, 5)

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

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

3. Simplify the numerator and denominator:
\[ m = \frac{5 + 3}{-3 + 2} = \frac{8}{-1} \]

4. Calculate the slope:
\[ m = -8 \]

### Summary of Results:
- (i) Slope = 2
- (ii) Slope = -\(\frac{8}{5}\)
- (iii) Slope = -8