For "normal" equations in one variable we have several techniques for solving equations, such as $\sin(5x) = 5\pi\cos(5x)$ or $\ln(x + 2) = 4$. However, imagine we have the following equations:
$\sin(5x + 2y) = 2x + 3y$
$\ln(x + y) = 4x - y$
Do techniques exist to solve them? I've taken a basic number theory course, and most of the time we dealt with linear or special nonlinear Diophantine equations. Is there any solution technique for, say, trigonometric equations in general? Or at least can we prove that solutions do/don't exist?
Any thoughts on this would be really appreciated.