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
Imagine the equation as a set of scales. Everything to the left of the = sign is on one side, everything on the right of the = sign is on the other. Now you don’t know the actual weight on each side because you’re using variables instead of numbers, but you do know that both sides weigh the same.
What happens if we take away half the weight on the left side? The right side now weighs twice as much since they were the same weight in the beginning. To equal out these scales, you’ll have to take away half of the right side’s weight. This applies to any change in weight. If you add, subtract, multiply, or divide one side’s weight, you need to do the same thing to the other side to keep it balanced.
Applying this to your example, y*x somehow became just y. This happened by dividing y*x by x, cancelling the two x’s out. Well, to make that change valid, you have to also divide the right side by x. This leads you to y = z/x.
This “rebalancing the scales” is the central premise behind algebra.
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