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
Note: this only works for x that isn’t 0 because you can’t divide by 0.
Multiplication and division are opposites.
If I have 5 groups of 6 things, that’s 30 things because 5*6=30
If I have 30 things and divide them into 5 groups, that’s 6 things per group because 30/5=6
If I have the equation y*x=z, if I do the same thing to both sides of the equation, it would make sense that the equality of both sides wouldn’t change. y*x+1=z+1 is still a true equation. The same holds true if we multiply or divide. y*x*3=z*3 is still true. y*x/5=z/5 is still true. So logically, y*x/x=z/x should be true as long as x isn’t 0. x/x = 1, therefore is the same thing as y*1=z/x, and y*1=y, so y=z/x.
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