Let's solve this problem step by step.

Step 1: Equation of the line parallel to $3x + 2y = 4$

The slope of a line in the form $Ax + By = C$ is given by $-\frac{A}{B}$.

For the line $3x + 2y = 4$, the slope is: $m_1 = -\frac{3}{2}$ So, the line through $(2, -1)$ and $(k, 4)$ must have the same slope because it is parallel.

Step 2: Find the slope between points $(2, -1)$ and $(k, 4)$

The slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the points $(2, -1)$ and $(k, 4)$, we get: $m = \frac{4 - (-1)}{k - 2} = \frac{5}{k - 2}$ Since this slope must be equal to $-\frac{3}{2}$, we set up the equation: $\frac{5}{k - 2} = -\frac{3}{2}$

Step 3: Solve for $k$

To solve for $k$, cross-multiply: $5 \cdot 2 = -3 \cdot (k - 2)$ $10 = -3k + 6$ Now solve for $k$: $10 - 6 = -3k$ $4 = -3k$ $k = -\frac{4}{3}$

Thus, the value of $k$ is $-\frac{4}{3}$.

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Would you like me to verify this solution further, or clarify any steps?

Here are some related questions:

1. What is the equation of a line parallel to $3x + 2y = 4$ through a different point?
2. How do you derive the general form of a line given a slope and a point?
3. What if the line through $(2, -1)$ and $(k, 4)$ was perpendicular to $3x + 2y = 4$?
4. How would the slope change if you had the equation $2x + 3y = 6$?
5. Can you solve for $k$ when a line is parallel to the y-axis?

Tip: To determine if two lines are parallel, always compare their slopes!