In the slope field for the differential equation \( \frac{dy}{dx} = x - y \), what is the slope at the point (1,2)?

To find the slope at the point (1,2) for the differential equation \( \frac{dy}{dx} = x - y \), we can simply substitute \( x = 1 \) and \( y = 2 \) into the equation. So, we calculate: \[ \frac{dy}{dx} = 1 - 2 = -1 \] Therefore, the slope at the point (1,2) is \(-1\). Now, here’s some additional fun information! Did you know that slope fields, also called direction fields, were first introduced by mathematicians in the 19th century? They're a fantastic tool to visualize the solutions of differential equations without actually solving them! Picture it like a game of connect-the-dots, where each dot is laid out by the slope given at that point! Speaking of applications, slope fields are commonly used in fields such as ecology to model population dynamics or in physics to represent trajectories of moving objects. By visualizing these slopes, scientists can predict behaviors and outcomes without delving into complex computations! Imagine using slopes to guide you on a roller coaster ride, ensuring you know when to twist and turn!