Final answer:
The value of k that makes the line kx+3y=4 parallel to the line through (2, -k) and (4, -1) is -3/5, as this ensures both lines have the same slope.
Step-by-step explanation:
To find the value of k that makes the graph of kx + 3y = 4 parallel to the line through (2, -k) and (4, -1), we first need to determine the slope of the line passing through these two points. The slope (m) is given by the rise over run, which is the difference in the y-coordinates divided by the difference in the x-coordinates.
For the points (2, -k) and (4, -1), the slope calculation is:
m = (y2 - y1) / (x2 - x1) = (-1 - (-k)) / (4 - 2) = (k + 1) / 2
Since parallel lines have equal slopes, the slope of the line kx + 3y = 4 must be the same. We can find the slope of this line by rewriting it in slope-intercept form (y = mx + b). This gives us:
3y = -kx + 4
y = (-k/3)x + 4/3
The slope of this line is therefore -k/3. Setting this equal to the slope calculated from the two points, we have:
-k/3 = (k + 1) / 2
To find k, solve the equation by cross-multiplication:
-2k = 3(k + 1)
-2k = 3k + 3
5k = -3
k = -3/5
Hence, the value of k that makes the lines parallel is -3/5.