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
5 = 5.
The equal sign means that what’s on the left is exactly like what’s on the right.
If I were to say 5+3, what would that equal? How about:
5+3 = 5+3
You can see that, right? So we can do the same thing to both sides. This can be written
5+3 = 8
or even
8 = 8
And you can see how it makes sense. And can you see how it works even when you can’t just work out either side in your head?
7/3 = 7/3
Right? 7 = 7, so if you add “/3” to each side, they’re still equal.
Now we get into algebra. What if we don’t exactly know what the number is? We still know that
x = x
Right? Whatever x is, it’s the same both times, so it’s the same as 5 = 5, or 5+3 = 5+3, or 7/3 = 7/3.
And that means we can do the same thing to both sides, and it’s still equivalent. So,
x/3 = x/3
We know that, even if we don’t know what x is.
Up until now, these have all been something called a “tautology”. That means it’s just inherently true, it’s just always the case, universally. In any context, x will always be some given value, so no matter what, x will always equal x.
So, let’s branch out for a second. Let’s say I’ve told you that, for this equation, x is equal to z.
x = z
That’s not a tautology. In the next problem maybe x won’t equal z. But for now, this is what I’m giving you to solve this problem. In this equation, you know that whatever x is, it will always be the same as z.
Since they’re the same, that means we can keep things equal by doing the exact same thing to each side.
x+2 = z+2
x/7 = z/7
Following so far? As long as we know that x = z, we know that doing a thing to x is equal to doing the same thing to z.
The same works even for other variables; it doesn’t have to just be normal numbers. So,
x*y = z*y
Doesn’t matter that we don’t know exactly what x, y, or z means. Doesn’t matter that there’s no number whose value we exactly know. We have the “given” that x and z are equal, and we know y is the same every time it shows up. So that means that multiplying x by y, and multiplying z by y, will give us the same number.
So, what if we start from a different given? What if x doesn’t equal z. What if, instead, I told you that:
x*y = z
The same principles all apply. We can do something to the left, and as long as we do the same to the right, it’s all still equal. So:
2*x*y = 2*z
Right? We doubled whatever was on the left, and also doubled whatever was on the right. Since we know they were equivalent before, we know they still have to be equivalent. Anything doubled is always equal to itself doubled.
Same as above, this doesn’t have to be a known value to be the same. Just like we could multiply x and z both by y above and know they were still equal, now we can do:
x*y/y = z/y
Do you see why that follows? Whatever x*y was, it was the same as z. And dividing something by y will always equal that same thing divided by y, just like we can say: 6 = 6, 6/3 = 6/3.
At this point it’s only about simplifying. y/y is 1; that’s another tautology. Anything divided by itself is just 1. That works no matter what the value is. So since we know we can replace “y/y” with 1, that means we can write the equation like this:
x*1 = z/y
That’s just another way of writing the same equation, since y/y is the exact same thing as the number 1. Now, to simplify a bit more, we know that x*1 = x. Because literally any number, multiplied by 1, gives you that same number. So we can replace x*1 with x and we get:
x = z/y
So. Those are the steps we follow to understand why we can write these two equations, and they are the exact same thing.
x*y = z
x = z/y
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