Sketching Slope Fields AB & BC

A slope field draws, at each lattice point, a short segment whose slope is what $\frac{dy}{dx} = f(x, y)$ gives there: for $\frac{dy}{dx} = x + y$ the segment at $(1, 2)$ has slope $3$, at $(1, -1)$ it is horizontal, and at $(-2, 0)$ it falls with slope $-2$. The segments are directions rather than pieces of any one solution, and a solution curve is any curve that stays tangent to the ones it passes. Sketching efficiently means finding where $f(x, y) = 0$, where it is undefined, and which side of that set is positive.

The fastest structural test asks which variable the slope uses: depending on $x$ alone makes every column of segments parallel, depending on $y$ alone makes every row parallel, and depending on both makes neither true. When structure does not decide it, substitute points chosen to separate the candidates, since $(1, 0)$ tells $\frac{dy}{dx} = x - y$ from $\frac{dy}{dx} = y - x$ while $(0, 0)$ tells you nothing. Matching a field by overall impression is the error this topic exists to prevent: different equations produce fields that look alike until a coordinate pair goes in.