which slope field fits the differential equation: dy/dx=x - y.

Explanation:

Step1: Identify the condition for zero slope.

The slope is given by the differential equation $\frac{dy}{dx} = x - y$. The slope is zero when $x - y = 0$, which means $y = x$. We need to find the slope field where the line segments are horizontal along the line $y=x$.

Step2: Evaluate slope at specific points.

Let's check the slope at a few points not on the line $y=x$. At $(1, 0)$, the slope is $\frac{dy}{dx} = 1 - 0 = 1$. The line segment should have a positive slope. At $(0, 1)$, the slope is $\frac{dy}{dx} = 0 - 1 = -1$. The line segment should have a negative slope. At $(-1, 0)$, the slope is $\frac{dy}{dx} = -1 - 0 = -1$. The line segment should have a negative slope. At $(0, -1)$, the slope is $\frac{dy}{dx} = 0 - (-1) = 1$. The line segment should have a positive slope.

Step3: Compare with the given options.

We examine each slope field option based on the conditions found in Step 1 and Step 2. Option 1: Slopes are horizontal along the line $y=x$. At $(1,0)$, the slope is positive (approx. 1). At $(0,1)$, the slope is negative (approx. -1). This matches our calculations. Option 2: Slopes are not horizontal along $y=x$. Option 3: Slopes are horizontal along $y=x$. However, at $(0,1)$, the slope appears positive, which contradicts our calculation ($dy/dx = -1$). Option 4: Slopes are not horizontal along $y=x$.

Step4: Conclude the correct option.

Based on the comparison in Step 3, only the first slope field matches the properties derived from the differential equation $\frac{dy}{dx} = x - y$.

Answer:

The first option (topmost graph) is the correct slope field.