I get myself confused sometimes when to use =>
or =
= x + y = x + 5 = x = -5
or
=> x + y => x + 5 => x = -5
I get myself confused sometimes when to use =>
or =
= x + y = x + 5 = x = -5
or
=> x + y => x + 5 => 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.
The equal sign "=" should be used when you have two quantities which are equal. For example:
$x^2 + 3x - 4 = (x+4)(x-1).$
The "\Rightarrow" should be used when relating two statements. For example:
$ \text{Today is Tuesday} \Rightarrow \text{Tomorrow is Wednesday}.$
It is okay to mix them, say for instance you are solving an equation like the following:
$ 3x + 4 = 10 \Rightarrow 3x = 6 \Rightarrow x = 2.$
The quantities $3x + 4$ and $10$ are equal. However, the statement $3x + 4 = 10$ implies that $3x = 6$.
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.