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

True ELI5 (not a thorough mathematical explanation but an explanation by example, as you’d teach a kid)

A good example for y*x=z might be:
I have four kids (y=4)
Each kid has five apples (x=5)
This totals 5+5+5+5 apples or 20 apples (z=20)
So y*x means 5 apples 4 times, or 4 kids with 5 apples.

Now when somebody asks “what is z/x?” they’re saying:. “You have 20 apples and you want to divide them so that each kid gets 4 apples. How many kids can you feed?”

It’s backwards-engineering the prior problem. Before, I knew the number of kids (y) and how much each kid had (x) and had to figure out the total (z). Now, I have the total (z) and know how much each kid gets (y) and need to figure out the number of kids (x).

Similarly, if they ask “what is z/x?” they’re asking: “I have 20 apples and would like to share them evenly between 4 kids. How many apples should I give each kid?”.