Problem 76
Question
Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. $$(a-b, c) \text { and }(a, a+c)$$
Step-by-Step Solution
Verified Answer
The slope of the line passing through the points (a-b, c) and (a, a+c) is \(a / b\) and the line rises.
1Step 1: Identify the coordinates
Two points are given: (a-b, c) and (a, a+c). Therefore the first point's coordinates are (x1, y1) = (a-b, c) and the second point's coordinates are (x2, y2) = (a, a+c).
2Step 2: Calculate the Differences
Subtract the y-coordinates and the x-coordinates of the two points. This gives: \(\Delta y = (a+c) - c = a\) for the difference in y-values and \(\Delta x = a - (a-b) = b\) for the difference in x-values.
3Step 3: Calculate the Slope
Plug these differences into the slope formula: \(m = \Delta y / \Delta x\) which gives \(m = a / b\). So the slope of the line passing through (a-b, c) and (a, a+c) is \(a / b\)
4Step 4: Determine the line's direction
Since the slope is a positive number (all variables represent positive real numbers), the line rises.
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