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
When you’re given an equation you’re told that what’s on the left hand side is equal to what’s on the right hand side. So as long as you perform the same operation on both sides, they’ll remain equal. If you subtract 14 on both sides, they’ll still be equal. If you divide by three on both sides, they’ll still be equal. If you take the square root on both sides they’ll remain equal, and so on.
So what you’re doing is really finding operations that makes one side simpler, and then you perform that operation on both sides.
Like if you have `2x + 14 = y` then we’re told that `2x+14` is equal to `y`.
But that means that if we subtract 14 on both sides they’ll still be equal, right? In other words, `2x + 14 – 14 = y – 14`, or simplified a bit, `2x = y – 14`
Next, we can also divide by two on both sides:
`2x / 2 = (y – 14) / 2`, which we can simplify to `x = (y – 14) / 2`. So the “magic spell” as you call it is that we turn the `*` on one side into a `/` on the other side, but what *actually* happened is just that we performed the same division on both sides. On one side it canceled out the `*`, and on the other side, we now have a `/` operation left where there was nothing before.
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