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
At a very basic level an equation is telling you that the chunk to the left of the equal sign is the same value as the chunk on the right of the equal sign. `x=y` tells you that the value of x is the same as the value of y and `3x + 7y – z = 6x + z^2` tells you that `3x + 7y – z` has the same value as `6x + z^2`. Since they’re the same value, if you do exactly the same operation to both sides then they’re still equivalent. `3x + 7y – z + a` is necessarily the same value as `6x + z^2 + a`.
When you convert `y * x = z` into ‘z/x = y` what you’re actually doing is dividing each side by x. It’s the same operation in both sides so the values remain equivalent but it cancels out the x on the left side (because it’s multiplied by x and then divided by x). The result makes it much easier to solve for y (assuming that’s what you’re trying to accomplish).
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