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This question already has an answer here:

I have this equation

$$ 4y = y $$

If I divide by y in both sides I would get this: $$ 4 = 1$$

And this does not have sense.

I know that the solution is 0 but why I get this answer when dividing by y. What's the logic behind?

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marked as duplicate by Leucippus, TMM, Willie Wong, Community Sep 29 '16 at 2:30

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    $\begingroup$ The reason you cannot divide by $y$ is because $y$ may be 0 $\endgroup$ – Matt G Sep 28 '16 at 22:08
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    $\begingroup$ You can't divide by 0. y is 0. So you can't divide by y. $\endgroup$ – fleablood Sep 28 '16 at 22:17
  • $\begingroup$ Doesn't 0 - 0 equal 1? $\endgroup$ – OldBunny2800 Sep 29 '16 at 1:37
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You can't divide by $0$. Rigorously, there are two cases to consider:

Case 1: $y=0$

This is seen to be a solution.

Case 2: $y\neq 0$

In this case we can divide by $y$, getting

$$4=1$$

Since this is false regardless of $y$, we get a contradiction, and thus there are no solutions $y\neq 0$ to the equation. So, the only solution is $y=0$.

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Transposing $ 3y=0,\quad y=0.$ So division by 0 is forbidden.

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Division is better thought of as the multiplication by a multiplicative inverse. All real numbers except zero have multiplicative inverse. So for example in solving $$2y=2$$ you can multiply both sides by the multiplicative inverse of $2$, which is denoted by $1/2$, to get $$\left(\frac{1}{2}\times2\right)y=\frac{1}{2}\times 2$$ $$1\times y=1$$ $$y=1$$ For $0$, you cannot do the same thing because there is no multiplicative inverse. So if you divide both sides by an unknown $y$, you need to assume that it is not $0$. Hence you need to consider the case $y=0$ later as well.

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Whenever you do something like:

$y^2 = 4y $ so $y^2/y = 4y/y $

Or $y (y+1)=2 (y+1) $ so $y (y+1)/(y+1)=2 (y+1)/(y+1) $

Ever SINGLE time, you absolutely MUST, with no exceptions EVER, you must acknowledge, the possibility that what you are dividing by might be zero and thus you can not do it, and you must make a case for this.

You must do this EVERY time.

So for example:

If you are solving $y^2 = 7y $ and you say:

$y^2 = 7y$

$y^2/y= 7/y$

$y=7$

Then you have done it WRONG!!!

That is the wrong answer.

You must do it this way instead.

"$y^2 = 7y $"

"There are two cases we must consider.

Case 1: $y \ne 0$

And case 2: $y=0$.

If $y\ne 0$ then we can divide both sides by $y $ and

$y^2/y = 7y/y $

$y = 7$

BUT we must also consider case 2.

If $y= 0$ we can NOT divide by $y $. In this case $y=0$ so we are done.

Our answer is two possibilities: either $y=7$ or $y=0$.

So....

To do $4y = y $ we MUST do two cases:

Case 1: $y \ne 0$. IF so (and this is an "if"; it might not be true) we can divide by $y $ and get $4=1$. As this is impossible, this case is not true. It is NOT the case that $y\ne 0$.

Case 2: $y = 0$.

This must be true.

So our answer is $y=0$ and we can not EVER divide by 0.

You must do these two cases every single £¥₩@ing time. If you don't you are doing it wrong. Period.

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Take the number $a/b$. This, as you know, is $a$ divided by $b$. But what does this mean? Colloquially, this means the number of $b$'s that go into $a$'s.

For example, $6/3=2$. That is, it takes two $3$'s to fit into $6$. Generally, $a/b=c$ means $bc=a$. So what we mean by $a/b$ is what number $c$ we need to multiply $b$ by to get $a$, i.e. $bc=a$.

Now can we divide by $0$? That is, $a/0=?$ Well again, $a/0$ means what number do I multiply $0$ by to get $a$? Well, notice that any number would work. But then $a/0=$ any number? That doesn't make any sense! This is precisely why we can't divide by zero - what number would it mean?!

Now in your equation $4y=y$, $y$ could be any number so if we were to divide by it, we would need to know it is not zero. But we don't know that, so we can't divide by $y$.

We could, however, break this problem to cases.

Case $1$: $y \neq 0$: Well, since $y \neq 0$, we can divide by $y$ so that $4=1$, this makes no sense so certainly we don't get any solutions.

Case $2$: $y=0$. In this case, we just get $0=0$, which seems to work so $y=0$ is a solution for this.

Perhaps the best way of solving this is to just move $y$ to the other side. So we have $4y=y$, subtracting $y$ gives $3y=0$. Divide by $3$ to get $y=0$, as expected.

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I think you are asking for the logic not a solution for this equation.

If you divide by $y$ then you get.

$$4\frac yy=\frac yy$$

and if $y=0$ Then $\frac yy$ is undefined

So $$4\cdot undefined= undifined$$

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