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.
​
**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
An equation functions on the idea that the two sides are exactly equal. If that’s true, then they stay equal no matter what you do, as long as you do the same thing to both sides.
In your example, you’re dividing both sides by X, it just happens that on one side that cancels out with being multiplies by X so it disappears.
So 5*2 = 10.
(5*2)/2 = 10/2
5 = 10/2
5 = 5
Which is just as true as reducing the orignal function which works out to 10 = 10. As long as you do the same thing to each side you can do whatever mathematical operation you want, it’s still going to be equal.
You could also just start adding stuff:
5*2 = 10
(5*2) – 8 = (10-8)
2=2
The difference with this is just that it’s not usually helpful to do random stuff. You’re generally trying to get one variable to be a specifical function of the other variables. So let’s go back to X.
x*2 = 10. Okay, so I want to know what X is, which means I want to move things around until it’s alone on one side of the equation. Dividing both sides by 2 gives me:
(X*2)/2 = 10/2 or x = 5 which is helpful.
I could do other stuff, like the – 8:
(X*2) – 8 = (10-8) or (x*2) – 8 = 2, but that doesn’t get me any closer to isolating X so I wouldn’t choose to do that.
Latest Answers