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
When you multiply x times 1/x, you get 1. Hopefully you’re okay with that.
Let’s pick a new variable name for 1/x. Call it R. (This doesn’t change the logic, it just makes the symbols easier to follow.)
When you multiply x times R, you get 1. (This is the same as above, we’re just using our new name for 1/x.)
Then we multiply both sides by R and simplify.
y*x = z
-> (y*x)*R = z*R
-> y*(x*R) = z*R
-> y*1 = z*R
-> y = z*R
-> y = z*(1/x)
-> y = z/x
Think carefully about each step.
(I should note that some interesting things happen if x=0. But that may be a topic for another post.)
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