How do you write text at the end of an equation?

Question

Like how do they get the bit where it says (By linearity of expectation) whilst still aligning the equations

Answer 1

Use the \tag*{} from amsmath as mentioned here:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
    \mathbf{A} \mathbf{x} &= \mathbf{b} \tag*{(Linear)} \\
    \mathbf{y} &= \mathbf{c} \tag*{(Trivial)}
\end{align*}
\end{document}

Result: