To solve the problem, we need to find the value of \( k \) such that the line \( 2x + y = k \) passes through the point that divides the line segment joining the points \( (1, 1) \) and \( (2, 4) \) in the ratio \( 3:2 \).
### Step-by-Step Solution:
1. **Identify the Points**:
The points given are \( A(1, 1) \) and \( B(2, 4) \).
2. **Use the Section Formula**:
The section formula states that if a point \( C(x, y) \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( C \) can be calculated as:
\[
C\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Here, \( m = 3 \), \( n = 2 \), \( A(1, 1) \) and \( B(2, 4) \).
3. **Calculate the Coordinates of Point \( C \)**:
- For the x-coordinate:
\[
x_C = \frac{3 \cdot 2 + 2 \cdot 1}{3 + 2} = \frac{6 + 2}{5} = \frac{8}{5}
\]
- For the y-coordinate:
\[
y_C = \frac{3 \cdot 4 + 2 \cdot 1}{3 + 2} = \frac{12 + 2}{5} = \frac{14}{5}
\]
Thus, the coordinates of point \( C \) are \( \left(\frac{8}{5}, \frac{14}{5}\right) \).
4. **Substitute into the Line Equation**:
Since point \( C \) lies on the line \( 2x + y = k \), we substitute \( x_C \) and \( y_C \) into the equation:
\[
2\left(\frac{8}{5}\right) + \frac{14}{5} = k
\]
5. **Calculate \( k \)**:
- Calculate \( 2 \cdot \frac{8}{5} \):
\[
2 \cdot \frac{8}{5} = \frac{16}{5}
\]
- Now add \( \frac{16}{5} \) and \( \frac{14}{5} \):
\[
k = \frac{16}{5} + \frac{14}{5} = \frac{16 + 14}{5} = \frac{30}{5} = 6
\]
Thus, the value of \( k \) is \( 6 \).
