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

We can think of multiplication as addition on steroids. Similarly, we can think of division as subtraction on steroids.

Any multiplication can be expressed as a series of additions. For example, 3*5 is a concise way of expressing 3+3+3+3+3. “Three times five” means “the sum of five instances of the value 3”.

To reverse the effect of addition, we can subtract. Start with 3; add 5 to get 8. Start with 8. Subtract 5 to get 3. (3+5=8, 8-5=3). We don’t often do this, but we can even think of the subtraction expression in terms of addition: what value, when added to five, produces the sum of eight.

In a smilar fashion we can reverse the effect of multiplication by dividing. Start with 3; multiply by 5 to get 15. Start with 15; divide by 5 to get 3. We can express the division in terms of multiplication: what value, multiplied by 3, produces the product of 15.