Final answer:
The value of k for which the line through (8,-3) and (4,k) is perpendicular to the line y = -2/5x - 1 is -13. The slope of the perpendicular line is 5/2, and using the point-slope form with the point (8, -3), we solve for k to find that k=-13.
Step-by-step explanation:
To find the value of k for which the line through (8,-3) and (4,k) is perpendicular to y = -2/5x - 1, we first need to determine the slope of the perpendicular line. The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. Thus, the slope of the perpendicular line should be 5/2 (since the negative reciprocal of -2/5 is 5/2). Next, using the point-slope form of a line equation y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line, we use (8, -3) as our point and 5/2 as our slope to find k. The equation becomes:
y + 3 = (5/2)(x - 8)
Plugging in the x-coordinate of the other point, 4, we get:
k + 3 = (5/2)(4 - 8)
k + 3 = (5/2)(-4)
k + 3 = -10
k = -13
Thus, the value of k for which the line is perpendicular to y = -2/5x - 1 is -13.