If the slope of a line joining the points `(7,3)and (k,2)` is -4, then the value of k is

Options

A. `29/4`
B. `(-29)/(4)`
C. `4/29`
D. `(-4)/(29)`

Answer

To solve the problem, we need to find the value of \( k \) given that the slope of the line joining the points \( (7, 3) \) and \( (k, 2) \) is \( -4 \).

### Step-by-Step Solution:

1. **Identify the points**: The points given are \( (x_1, y_1) = (7, 3) \) and \( (x_2, y_2) = (k, 2) \).

2. **Use the slope formula**: The formula for the slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

3. **Substitute the values**: Here, we know that \( m = -4 \), \( y_1 = 3 \), \( y_2 = 2 \), \( x_1 = 7 \), and \( x_2 = k \). Substituting these values into the slope formula gives:
\[ -4 = \frac{2 - 3}{k - 7} \]

4. **Simplify the equation**: The numerator simplifies to \( 2 - 3 = -1 \), so we have:
\[ -4 = \frac{-1}{k - 7} \]

5. **Cross-multiply**: To eliminate the fraction, we can cross-multiply:
\[ -4(k - 7) = -1 \]

6. **Distribute**: Distributing \( -4 \) gives:
\[ -4k + 28 = -1 \]

7. **Rearrange the equation**: Add \( 4k \) to both sides:
\[ 28 = 4k - 1 \]

8. **Add 1 to both sides**: This gives:
\[ 29 = 4k \]

9. **Divide by 4**: Finally, divide both sides by 4 to solve for \( k \):
\[ k = \frac{29}{4} \]

### Conclusion:
The value of \( k \) is \( \frac{29}{4} \).