! Package amsmath Error: \tag not allowed here

Question

I want to use \tag in my LaTex file at https://www.sharelatex.com/.

For example, I want to do:

If $x \equiv x' \pmod{N}$ and $y \equiv y' \pmod{N}$, then: $xy \equiv x'y' \pmod{N}\tag{Substitution rule}$.

I look at a previous question on this website, \numcases with \tag, and found the answer insufficient for my problem.

I still get the ! Package amsmath Error: \tag not allowed here. error even though I am only importing the following:

\usepackage{textcomp,geometry,graphicx,hyperref,empheq}

I have two questions:

(1) Why does the amsmath package cause an error when \tag is used?
(2) How do I add customized tags to my equations without causing errors?

Thank you for any help you can provide on this!

EDIT:

This is a made-up example as requested:

Now solve for $E[X]$:\newline

    \hspace{30pt} $E[X] = 1 + E[X] - p \cdot E[X]$\newline

    \hspace{30pt} $0 = 1 - p \cdot E[X]\tag{"xyz"}$\newline

    \hspace{30pt} $p \cdot E[X] = 1\tag{"xyy"}$\newline

    \hspace{30pt} $E[X] = \frac{1}{p}\tag{"xzy"}$\newline\newline

Note the \tag parts were added in afterwards because they caused errors.

Then I want to say, based on "xyz" and "xyy", I can prove "abc".

Answer 1

You're still using inline math. (La)TeX distinguishes between math that is supposed be written on a line of text, delimited by $ ... $ or \( ... \), and displayed math, which is placed on its own paragraph.

For a single unnumbered, displayed equation you can use \[ ... \], for a numberered equation there is \begin{equation} ... \end{equation}. For sets of equations, or multiline equations, amsmath provides several environment, including align and gather, as well as the starred forms align* and gather* that are unnumbered.

Displayed equations are by default centered, to make them left-aligned add fleqn as an option to amsmath or the document class, e.g. \usepackage[fleqn]{amsmath}.

For more information about amsmath, read the manual. For math typesetting in general, you could take a look at Herbert Voss' Mathmode.

A demonstration with your example:

\documentclass{article}
\usepackage{amsmath}

\begin{document}
Now solve for $E[X]$:
\begin{gather}
E[X] = 1 + E[X] - p \cdot E[X] \\
0 = 1 - p \cdot E[X]\tag{"xyz"} \\
p \cdot E[X] = 1\tag{"xyy"} \\
E[X] = \frac{1}{p}\tag{"xzy"}
\end{gather}

Now solve for $E[X]$:
\begin{align}
E[X] &= 1 + E[X] - p \cdot E[X] \\
0 &= 1 - p \cdot E[X]\tag{"xyz"} \\
p \cdot E[X] &= 1\tag{"xyy"} \\
E[X] &= \frac{1}{p}\tag{"xzy"}
\end{align}
\end{document}

Answer 2

These are still inline equations. You need to use

\[ formula \]

of

\begin{equation}
formula
\end{equation}

to make your formulas display style. Read about it here.