Find the equation of the st.line passing through the points :
`(i) (2,3)` and `(5,-2)` `(ii) (1,-1)` and `(3,5)`.

Answer

To find the equations of the straight lines passing through the given points, we will use the two-point form of the equation of a straight line. The formula for the equation of a straight line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]

### (i) For the points (2, 3) and (5, -2):

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

2. **Calculate the slope**:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 3}{5 - 2} = \frac{-5}{3} \]

3. **Substitute into the point-slope form**:
\[ y - 3 = \frac{-5}{3}(x - 2) \]

4. **Distribute the slope**:
\[ y - 3 = \frac{-5}{3}x + \frac{10}{3} \]

5. **Add 3 to both sides** (convert 3 to a fraction with a common denominator):
\[ y = \frac{-5}{3}x + \frac{10}{3} + \frac{9}{3} \]
\[ y = \frac{-5}{3}x + \frac{19}{3} \]

6. **Multiply through by 3 to eliminate the fraction**:
\[ 3y = -5x + 19 \]

7. **Rearranging gives the standard form**:
\[ 5x + 3y - 19 = 0 \]

### (ii) For the points (1, -1) and (3, 5):

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

2. **Calculate the slope**:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-1)}{3 - 1} = \frac{6}{2} = 3 \]

3. **Substitute into the point-slope form**:
\[ y - (-1) = 3(x - 1) \]
\[ y + 1 = 3(x - 1) \]

4. **Distribute the slope**:
\[ y + 1 = 3x - 3 \]

5. **Subtract 1 from both sides**:
\[ y = 3x - 4 \]

6. **Rearranging gives the standard form**:
\[ 3x - y - 4 = 0 \]

### Final Answers:
1. The equation of the line passing through the points (2, 3) and (5, -2) is:
\[ 5x + 3y - 19 = 0 \]

2. The equation of the line passing through the points (1, -1) and (3, 5) is:
\[ 3x - y - 4 = 0 \]