the logic of why going from “y*x=z” to “z/x=y” is possible.

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I ask this in relation to ” /(x/y) ” = ” *(y/x) ”

My mathematical ignorance does not allow me to perceive exactly what it is that confuses me about these manoeuvres and so perhaps my question is vague.

I have no difficulty with it as a technique; as something through which I can put an expression, and out at the other end the right result will appear. What I am trying to understand is *why it works*, contrasted with remembering it as a kind of magical spell.

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**EDIT:**
It was very rewarding for me to read all of your comments. Thank you most kindly for enlightening me.

For those interested in the cause of my previous confusion:
The gaps in my understanding of going from y*x=z to y=z/x were definitions of the equal sign and division.

I can see now that I previously considered the = sign to mean «result» or «answer» in some sort of final sense, like a conclusion; I now see that it only states that this is equal to that.

Following this fundamental piece of knowledge, I can belatedly understand what an equation is. From there, via the definition of division as the opposite of multiplication, I can see that if I divide something while also multiplying it with the same number, these actions cancel each other out.

And so the magical spell between y*x=z and y=z/x is the logic above expressed mathematically as x/(y*x)=z/x.

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36 Answers

Anonymous 0 Comments

Consider 4 rows of 3 dots. It’s also 3 columns of 4 dots totalling 12:

. . .

. . .

. . .

. . .

So this illustrates that “4 X 3=12”, and “3 X 4=12”, and the general principle that “x * y” is the same as “y * x”.

If you then divide the dots among three people by giving each person one column, they would each get 4 dots. So 12/3=4. Or if you gave a row to each of 4 people, you get 12/4=3.

So if “x * y=z”, then “y * x=z”, “z/y=x”, and “z/x=y”.

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