\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Consider the following system of coupled differential equations:$$\begin{cases}\dfrac{dx}{dt} = x + 2y \\ \dfrac{dy}{dt} = 2x + y\end{cases}$$Show that the functions defined by \(x(t) = e^{3t}\) and \(y(t) = e^{3t}\) form a valid solution to this system.
Capture an image of your work. AI teacher feedback takes approximately 10 seconds.