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 find the value of \( k \) given that the slope of the line joining the points \( (7, 3) \) and \( (k, 2) \) is \( -4 \), we can follow these steps:

### Step 1: Use the Slope Formula
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In our case, the points are \( (7, 3) \) and \( (k, 2) \). Thus, we can assign:
- \( x_1 = 7 \), \( y_1 = 3 \)
- \( x_2 = k \), \( y_2 = 2 \)

### Step 2: Substitute the Values into the Slope Formula
We know the slope \( m \) is \( -4 \). Therefore, we can write:
\[ -4 = \frac{2 - 3}{k - 7} \]

### Step 3: Simplify the Equation
Now simplify the left side:
\[ -4 = \frac{-1}{k - 7} \]

### Step 4: Cross-Multiply
To eliminate the fraction, we can cross-multiply:
\[ -4(k - 7) = -1 \]

### Step 5: Distribute and Rearrange
Distributing the \( -4 \):
\[ -4k + 28 = -1 \]
Now, rearranging gives:
\[ -4k = -1 - 28 \]
\[ -4k = -29 \]

### Step 6: Solve for \( k \)
Dividing both sides by \( -4 \):
\[ k = \frac{29}{4} \]

### Final Answer
Thus, the value of \( k \) is:
\[ k = \frac{29}{4} \]

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