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

Question

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

Answer 1

In mathematics division is defined using multiplicative inverse. It is possible to inspect equations of the form a*x = b without defining multiplicative inverse. In this case you have definition for your own special multiplication *. You can solve equations involving your special multiplication by trial and error. When you introduce the concept of an inverse, solving these kind of equations becomes straightforward. There is no obvious connection why inverses makes solving equations straightforward meaning that there might be other ways to solve equations using some algorithm. We are fortunate that we can visualize numbers using number line and simultaneously solve equations of the form a*x = b using inverse elements which we can see in the number line. That is a fortunate coincidence.