If the line through (3,y) and (2,7) is parallel to the line through `(-1,4)` and (0,6), then the value of y is (i) `-9` (ii) `-8` (iii) 8 (iv) 9

Options

A. `-9`
B. `-8`
C. 8
D. 9

Answer

To solve the problem, we need to find the value of \( y \) such that the line through the points \( (3, y) \) and \( (2, 7) \) is parallel to the line through the points \( (-1, 4) \) and \( (0, 6) \).

### Step-by-Step Solution:

1. **Find the slope of the line through the points (-1, 4) and (0, 6)**:
- The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
- Here, \( (x_1, y_1) = (-1, 4) \) and \( (x_2, y_2) = (0, 6) \).
- Substituting the values:
\[ m_2 = \frac{6 - 4}{0 - (-1)} = \frac{2}{1} = 2 \]

2. **Set up the slope of the line through the points (3, y) and (2, 7)**:
- Let \( (x_1, y_1) = (3, y) \) and \( (x_2, y_2) = (2, 7) \).
- Using the slope formula:
\[ m_1 = \frac{7 - y}{2 - 3} = \frac{7 - y}{-1} = -(7 - y) = y - 7 \]

3. **Since the lines are parallel, set the slopes equal**:
- We know \( m_1 = m_2 \):
\[ y - 7 = 2 \]

4. **Solve for \( y \)**:
- Rearranging the equation:
\[ y = 2 + 7 = 9 \]

Thus, the value of \( y \) is \( 9 \).

### Conclusion:
The correct answer is \( (iv) \, 9 \).