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Question: Refer to the linear function z = f(x, y) whose values are in Table.
Each column of Table is linear with the same slope, m = Δz/Δx = 4/5. Each row is linear with the same slope, n = Δz/Δy = 3/2. We now investigate the slope obtained by moving through the table along lines that are neither rows nor columns.
(a) Move down the diagonal of the table from the upper left corner (z = 3) to the lower right corner (z = 31). What do you notice about the changes in z? Now move diagonally from z = 6 to z = 27. What do you notice about the changes in z now?
(b) Move in the table along a line right one step, up two steps from z = 19 to z = 9. Then move in the same direction from z = 22 to z = 12. What do you notice about the changes in z?
(c) Show that Δz = mΔx + nΔy. Use this to explain what you observed in parts (a) and (b).
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