If the line joining the points `(5,y)` and `(4,9)` is parallel to the line joining the points `(0,5)` and `(1,7)`, find the value of `y`.

Answer

To solve the problem, we need to find the value of \( y \) such that the line joining the points \( (5, y) \) and \( (4, 9) \) is parallel to the line joining the points \( (0, 5) \) and \( (1, 7) \).

### Step-by-step Solution:

1. **Find the slope of the line joining the points \( (0, 5) \) and \( (1, 7) \)**:
- 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, \( (x_1, y_1) = (0, 5) \) and \( (x_2, y_2) = (1, 7) \).
- Substitute the values:
\[ m = \frac{7 - 5}{1 - 0} = \frac{2}{1} = 2 \]

2. **Find the slope of the line joining the points \( (5, y) \) and \( (4, 9) \)**:
- Using the same slope formula:
- Here, \( (x_1, y_1) = (5, y) \) and \( (x_2, y_2) = (4, 9) \).
- Substitute the values:
\[ m = \frac{9 - y}{4 - 5} = \frac{9 - y}{-1} = y - 9 \]

3. **Set the slopes equal since the lines are parallel**:
- Since the lines are parallel, their slopes must be equal:
\[ y - 9 = 2 \]

4. **Solve for \( y \)**:
- Rearranging the equation gives:
\[ y = 2 + 9 = 11 \]

### Final Answer:
The value of \( y \) is \( 11 \).