Final answer:
The student's question about a direction field can be answered by observing the slopes indicated by the field lines. For the equations y' = y - x and y' = y + x, one would examine the direction field to see how the slope (y') behaves as the values of x and y change, to identify which equation the field represents.
Step-by-step explanation:
The student has asked to identify which of the two first-order equations a given direction field represents. These equations are (A) y' = y - x, and (B) y' = y + x. A direction field graphically represents all the possible slopes of the solution curves for a differential equation at any given point in the plane. To determine which equation corresponds to the direction field, one would look at the behavior of the field lines. For example, if at the point where x and y are equal, the slope (y') is zero, this would suggest that the direction field represents equation (A), where y' = y - x.
If, however, the slope continues to be positive as x and y increase together, this might be indicative of equation (B), where y' = y + x.
Without the given direction field, it is impossible to definitively answer which equation it represents. However, for the first-order equations provided, the direction field would have slopes calculated based on the x and y coordinates substituted into each respective differential equation to determine the slope at that point.