To determine the value of \( k \) such that the line \(-kx + 21y = 4\) is parallel to the line passing through \((5, -8)\) and \((2, 4)\), we need to find the slope of the given line and then equate it to the slope of the line passing through the given points.
The slope of the given line \(-kx + 21y = 4\) can be determined by rearranging it into the slope-intercept form \(y = mx + b\), where \(m\) represents the slope. We can rewrite the equation as:
\[21y = kx + 4 \Rightarrow y = \frac{k}{21}x + \frac{4}{21}\]
From this form, we can see that the slope of the given line is \(m = \frac{k}{21}\).
Now, let's find the slope of the line passing through \((5, -8)\) and \((2, 4)\). The slope of a line can be calculated using the formula:
\[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\]
Substituting the coordinates, we have:
\[m = \frac{{4 - (-8)}}{{2 - 5}} = \frac{{12}}{{-3}} = -4\]
Since the given line is parallel to the line passing through the given points, the slopes of both lines should be equal. Therefore, we can set \(\frac{k}{21} = -4\) and solve for \(k\).
\[\frac{k}{21} = -4 \Rightarrow k = -4 \times 21 = -84\]
Therefore, the value of \(k\) that satisfies the condition is \(k = -84\).