On page 2 of this PDF from Standord, which describes the Method of Characteristics for first-order PDEs, it is written at the end of the page:
"In doing so, we see that $z(x,t)$ is constant along the lines $x-at=x_0$. That is, $z(x,t)=f(x-at)$."
I don't see one thing here: when they write that $z(x,t)$ is constant along $x-at$, do they mean that, since $x-at=x_0$, where $x_0$ is a constant, and $z(s) = k$, where $k$ is also some constant, we can express $z(x,t) = k$ in terms of $x-at=k$? Moreover, what does $f(x-at)$ do to $z(x,t)$? That is, I don't understand how this function is exactly justified.
Would appreciate a clarification.