A line with slope of `(3)/(14)` passes though points `(7, 3k) and (0, k)`. What is the value of k?

Answer

To find the value of \( k \), we will use the formula for the slope of a line that passes through two points. The points given are \( (7, 3k) \) and \( (0, k) \), and we know the slope of the line is \( \frac{3}{14} \).

### Step-by-Step Solution:

1. **Identify the coordinates:**
- Let \( (x_1, y_1) = (0, k) \)
- Let \( (x_2, y_2) = (7, 3k) \)

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

3. **Substitute the coordinates into the slope formula:**
\[ m = \frac{3k - k}{7 - 0} \]
Simplifying this gives:
\[ m = \frac{2k}{7} \]

4. **Set the slope equal to the given slope:**
We know the slope is \( \frac{3}{14} \), so we set up the equation:
\[ \frac{2k}{7} = \frac{3}{14} \]

5. **Cross-multiply to solve for \( k \):**
\[ 2k \cdot 14 = 3 \cdot 7 \]
This simplifies to:
\[ 28k = 21 \]

6. **Divide both sides by 28:**
\[ k = \frac{21}{28} \]

7. **Simplify the fraction:**
\[ k = \frac{3}{4} \]

### Final Answer:
The value of \( k \) is \( \frac{3}{4} \).