Confusion about definition of linear equation re: Lorenz system & algebraic equations

Question

I am confused about the definition of a linear equation.

First, why is the Lorenz system considered to contain three nonlinear equations? (https://en.wikipedia.org/wiki/Lorenz_system#Overview) They all look linear to me. I don't see any powers on any of the terms in any of the 3 equations (much less all of them).

Second, I just realized the obvious fact that any linear equation can become nonlinear by simply multiplying both sides by the variable (e.g., $y=x$ can become $xy=x^2$). So how can we make a distinction between linear and nonlinear equations?

(I am a high school Calculus student, by the way - we haven't gone anywhere remotely close to the Lorenz equations in class but I'm interested in them.)

Answer 1

The terms $xz$ and $xy$ in the second and third equation make the system nonlinear.

You are right that you can transform any linear equation to a nonlinear one (the opposite is not always true though). But be careful: The way you did it, the two equations are not equivalent. For the second equation $x=0$ is a solution regardless of the value of $y$.

Answer 2

The distinction is simple:

In a linear equation, the operations performed are no more than multiplication of variables by numbers alone, and addition of such products.

Anything other than this is nonlinear.