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Step-by-Step Solution
Step 1: Identify the given points and the ratio
The two given points are $A(1, 1)$ and $B(2, 4)$. The ratio in which the line segment $\overline{AB}$ is divided is $3 : 2$.
Step 2: Apply the section formula to find the coordinates of the point
If a point $P$ divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m : n$, then
$$
P = \left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right).
$$
Here, $m = 3$, $n = 2$, $x_1 = 1$, $y_1 = 1$, $x_2 = 2$, and $y_2 = 4$.
Therefore,
$$
x_P = \frac{3 \times 2 + 2 \times 1}{3 + 2} = \frac{6 + 2}{5} = \frac{8}{5}, \quad
y_P = \frac{3 \times 4 + 2 \times 1}{3 + 2} = \frac{12 + 2}{5} = \frac{14}{5}.
$$
So the coordinates of the point $P$ are $P\left(\frac{8}{5}, \frac{14}{5}\right).$
Step 3: Substitute the coordinates of the dividing point into the line equation
The line is given by $2x + y = k$. Since $P$ lies on this line, we have
$$
2 \times \frac{8}{5} + \frac{14}{5} = k.
$$
Simplifying,
$$
\frac{16}{5} + \frac{14}{5} = \frac{30}{5} = 6.
$$
Hence, $k = 6.$
Step 4: Final Answer
Therefore, the value of $k$ for which the line $2x+y=k$ passes through the given point is $\boxed{6}.$
