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
y * x = z is equal to (y * x) / x = z / x if x is not zero . As long as do the same operation on both sides they are still equal
x / x = 1 so you (y * x) / x = z / x => y * x / x = z / x => y= z / x
The x is not zero is a required part that is often forgotten.
That 1 / (x / y) ” = ” 1* ( y / x) ” can be show the same way y / y = 1 and you can mupliply a equation by 1 and nothing changes so
(1 / (x / y)) * (y / y) is y /(x * y /y) that is easy so see if you write in on paper with the number in diffler rows on a paper.
The text next step is to look at (x * y / y) as rescue y / y = 1 so we get (x * y / y) =(x * 1) = x
No apply that to the full equation and y /(x * y /y) = y / x
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