Is “implies” the best symbol when rewriting equations?

In my mathematical homework, I usually indicate algebraic rewrites of equations using implication, and the symbol "$\implies$" (LaTeX \implies). For instance, I might write $$ 3 x - y = 0 \implies 3 x = y \implies x = \frac{y}{3} $$ to mean that, since $3 x - y = 0$, the equivalent equation $3 x = y$ is also true, which then indicates that $x = y/3$ is true. Is logical implication the correct facility to express rewriting an equation into an equivalent form? If not, what other concept and symbol would be correct here?

Another symbol which you can use is the "if and only if" symbol, in $\rm\LaTeX$, it is $\iff$ and is denoted by $\iff$(Also pointed out by @StefanOctavian)

So your equation re-write thus becomes : $$\begin{align} 3x-y=0 &\iff 3x = y\\ &\iff x = \frac{y}{3} \end{align}$$

The logical implication (and/or iff) symbol isn't inappropriate, but I sometimes find it a bit "heavy", both visually and cognitively. (After all, the reader is probably sophisticated enough to understand the implications of simple algebraic manipulations.)

I tend to use "$\to$" (\to) to provide a sense of flow from one version of an equation to another (with generous spacing (\quad or \qquad) to help indicate that I'm not constructing some formal logic expression); eg, $$3x-y=0\qquad\to\qquad 3x=y \qquad\to\qquad x = \frac{y}{3}$$

If it is a composition of multiple equations it makes sense to say: $$\therefore\tag{therefore}$$ you also have other arrows like: $$\Rightarrow\tag{Rightarrow}$$ which I prefer to use mostly because it is a shorter symbol. As others have mentioned, you also have: $$\iff\tag{iff}$$ $$\to\tag{to}$$ $$\rightarrow\tag{rightarrow}$$ as well as what you have mentioned: $$\implies\tag{implies}$$

It can also sometimes be useful to use symbols like: $$\forall\tag{forall}$$ $$\because\tag{because}$$

EDIT:

One thing I thought I would add because it is especially useful is spacing when writing these equations in one line. Try to use a combination of commands like "," for a short space and "\quad" "\qquad" for longer spaces

Further to the first two comments and Rob's answer:

1. When solving equations, to emphasise that your work is displaying a chain of equivalent equations, i.e., each step is “reversible”, the correct symbol is $\iff,$ not $\implies$: $$P(x)=Q(x)\implies x\in\{2,4,5\}$$ merely means that $2,4,5$ are candidate solutions, whereas $$P(x)=Q(x)\iff x\in\{2,4,5\}$$ means that the solution set is actually—not merely a subset of—$\{2,4,5\}.\\$

2. Outside of the context of writing proofs and specifying theorems, where statements like $$\text{“for each real $x,\quad$ <conditions> $\implies$ <implication>”}$$ are commonplace, usually the connective that is actually meant is “therefore” rather than “implies”, in which case the correct symbol is $\,\therefore\;,$ not $\implies.$ (Published work spell “therefore” out because peppering mathematical prose with symbols decreases readability.)

   For example, \begin{align}&x=3\\\therefore\; &x\in\mathbb R\end{align} and $$x=3\\x\in\mathbb R$$ are asserting the value of $x$ and (explicitly and implicitly, respectively) deducing that $x$ is real, whereas \begin{align}&x=3\\\implies &x\in\mathbb R\end{align} is merely specifying that $x$ being real is the consequence of a particular case: $x$'s value is not being established and $x$ possibly actually equals, say, $2i.$

The traditional symbol for "therefore" is $\therefore$ (\therefore in $\LaTeX$). I recommend it in your example, because "$A$ implies $B$" and "$A$ therefore $B$" don't mean the same thing: the former means that if $A$ holds, then so does $B$, the latter means that $A$ holds, and as a consquence of that so does $B$. In your example, with $A \equiv 3x - y = 0$, $B \equiv 3x = y$ and so on, it is the latter reading that you want. For a simpler example: "$0 = 1 \implies 1 = 1$" is true, but "$0 = 1 \therefore 1 = 1$" is false.

(Technically $\therefore$ is just logical conjunction, but with an implied hint that the right conjunct is easily derivable from the left conjunct.)