the logic of why going from “y*x=z” to “z/x=y” is possible.

Question

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

Answer 1

Multiplication and division are opposites (for the most part).

Addition and subtraction are opposites.

If you are ok with this:

2+3=5 ➡️ 2=5-3

Then you should be okay with this:

2•3=6 ➡️ 2=6/3

Notice we are skipping a step if this was a workout and not a statement of fact, we need to be subtracting/dividing 3 on both sides.

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> I ask this in relation to ” /(x/y) ” = ” *(y/x) ”

What is 10 ➗ (1/2)?
It is the same as 10 • (2/1)

Keep Change Flip

Division is repeat subtraction. 10/2=5 because you need to -2 five times. 10/(1/2)=20 because you need to -0.5 twenty times.

Fractions can be viewed as proportions. 1/4 is 4 times smaller than 1, as such dividing by it leads to a number 4 times larger than dividing by 1.