Implies sign in math I get myself confused sometimes when to use =&gt; or =

= x + y = x + 5 = x = -5


or

=&gt; x + y =&gt; x + 5 =&gt; x = -5
Neither of those are right.  It should be:
$$x + y = x + 5 \implies x = -5.$$
(Though I have to say, I'm not exactly sure how $x = -5$ directly follows from $x + y = x + 5$ to begin with.)



Let's first distinguish between expressions and equations.  Expressions are things like $x + 2$ or $37$ or $x^2 - 2x$.  On the other hand, an equation is an equality between two expressions, such as $x + 2 = 3$ or $x = 1$ or $8 + 9 = 17$ and so on.  So, an equality says that two expressions are the same, and we use the equals sign $=$.

Note that an equality is making a logical claim (true or false), whereas an expression is not.  For example, $6 = 6$ is a true equality, whereas $1 = 2$ is a false equality, whereas $17$ is just an expression (not true or false).

Now, the implies sign $\implies$ is used to show that one equation can be logically deduced from another.  So, the implication
$$x + 2 = 3 \implies x = 1.$$
means that from the equation $x + 2 = 3$, we can conclude that $x = 1$.

Note that it does not make sense to use the implication arrow between expressions.  For example $x + y \implies x = -5$ does not make any sense, because the $x + y$ is an expression.  After all, what is the truth value of $x+ y$ anyway?  There isn't one.
I think that equal sign is used for showing equality between first and second step but implies that is used for getting into the next step.