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.
In: 25
If I understood the question correctly, it’s because of the “=” part of the equation. Suppose you have
x=y
Then you can freely manipulate these two **so long as you do it on both sides**. Twice X should be twice Y and half X should be half Y. Then if you add another element like
x=y*z
you just play around with that.
x=y*z (/y)
x/y) = y*z/y
x/y = z
or
x/y = a*b (*y)
(x/y)*y = a*b*y
x = a*b*y
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