If slope of the line joining points P(k,0) and Q(-3,-2) is `2/7` , then find k.

Answer

To find the value of \( k \) given that the slope of the line joining points \( P(k, 0) \) and \( Q(-3, -2) \) is \( \frac{2}{7} \), we can follow these steps:

### Step 1: Write down the slope formula
The formula for the slope \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

### Step 2: Assign the coordinates
Here, we have:
- Point \( P(k, 0) \) gives us \( x_1 = k \) and \( y_1 = 0 \).
- Point \( Q(-3, -2) \) gives us \( x_2 = -3 \) and \( y_2 = -2 \).

### Step 3: Substitute the coordinates into the slope formula
Substituting the coordinates into the slope formula, we get:
\[ \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 0}{-3 - k} \]
This simplifies to:
\[ \frac{-2}{-3 - k} \]

### Step 4: Set the slope equal to \( \frac{2}{7} \)
We know from the problem statement that this slope equals \( \frac{2}{7} \):
\[ \frac{-2}{-3 - k} = \frac{2}{7} \]

### Step 5: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ -2 \cdot 7 = 2 \cdot (-3 - k) \]
This simplifies to:
\[ -14 = -6 - 2k \]

### Step 6: Solve for \( k \)
Now, we can isolate \( k \):
1. Add \( 6 \) to both sides:
\[ -14 + 6 = -2k \]
This simplifies to:
\[ -8 = -2k \]
2. Divide both sides by \( -2 \):
\[ k = \frac{-8}{-2} = 4 \]

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