Final answer:
To find the value of k such that the line through (-5,k) and (4,6) is parallel to the line through (5,4) and (1,-3), we equate their slopes and solve for k, obtaining k = -39/4.
Step-by-step explanation:
The subject of the question is Mathematics, specifically dealing with the concept of parallel lines and slopes in coordinate geometry. To find the value of k for which the new line is parallel to the given line, we need to equate the slopes of both lines because parallel lines have equal slopes.
First, we calculate the slope of the line passing through points (5,4) and (1,-3). The slope formula is (y2 - y1) / (x2 - x1), which gives us (4 - (-3)) / (5 - 1) = 7/4 as the slope of the line.
The slope of the line passing through (-5,k) and (4,6) would be (6 - k) / (4 - (-5)) = (6 - k) / 9. To make this line parallel to the first line, its slope must also be 7/4.
We set up the equation 7/4 = (6 - k) / 9 and then solsolvedr k. Multiplying both sides by 9 gives us 63/4 = 6 - k. Finally, solving for k yields k = 6 - 63/4 = 24/4 - 63/4 = -39/4.