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
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.
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.
You can go from xy=z to y=z/x in a single step, just by dividing both sides by x. If any two values are equal, then dividing both by the same number will maintain the equality (except in the case of dividing by zero).
At a very basic level an equation is telling you that the chunk to the left of the equal sign is the same value as the chunk on the right of the equal sign. `x=y` tells you that the value of x is the same as the value of y and `3x + 7y – z = 6x + z^2` tells you that `3x + 7y – z` has the same value as `6x + z^2`. Since they’re the same value, if you do exactly the same operation to both sides then they’re still equivalent. `3x + 7y – z + a` is necessarily the same value as `6x + z^2 + a`.
When you convert `y * x = z` into ‘z/x = y` what you’re actually doing is dividing each side by x. It’s the same operation in both sides so the values remain equivalent but it cancels out the x on the left side (because it’s multiplied by x and then divided by x). The result makes it much easier to solve for y (assuming that’s what you’re trying to accomplish).
y * x = z is equal to (y * x) / x = z / x if x is not zero . As long as do the same operation on both sides they are still equal
x / x = 1 so you (y * x) / x = z / x => y * x / x = z / x => y= z / x
The x is not zero is a required part that is often forgotten.
That 1 / (x / y) ” = ” 1* ( y / x) ” can be show the same way y / y = 1 and you can mupliply a equation by 1 and nothing changes so
(1 / (x / y)) * (y / y) is y /(x * y /y) that is easy so see if you write in on paper with the number in diffler rows on a paper.
The text next step is to look at (x * y / y) as rescue y / y = 1 so we get (x * y / y) =(x * 1) = x
No apply that to the full equation and y /(x * y /y) = y / x
1) Multiplication and division are inverses, so they “cancel out” when we do both at once:
(x*y)/y = x
2) When we have an equation, we can change it as long as we do the same thing to both sides, so if we have
6 = 6
We can divide each side by 3 and the statement remains true
6/3 = 6/3
2 = 2
So if we combine these two facts, we can do this:
y*x = z
(y*x)/x = z/x
y = z/x
When you multiply x times 1/x, you get 1. Hopefully you’re okay with that.
Let’s pick a new variable name for 1/x. Call it R. (This doesn’t change the logic, it just makes the symbols easier to follow.)
When you multiply x times R, you get 1. (This is the same as above, we’re just using our new name for 1/x.)
Then we multiply both sides by R and simplify.
y*x = z
-> (y*x)*R = z*R
-> y*(x*R) = z*R
-> y*1 = z*R
-> y = z*R
-> y = z*(1/x)
-> y = z/x
Think carefully about each step.
(I should note that some interesting things happen if x=0. But that may be a topic for another post.)
Assign them meaningful values and it gets much easier to explain:
x is the number of people
y is the number of apples per person
z is the total number of apples
y * x = z:
The number of apples per person, times the number of people, equals the total number of apples
z / x = y:
The total number of apples, divided by the number of people, equals the number of apples per person
Consider 4 rows of 3 dots. It’s also 3 columns of 4 dots totalling 12:
. . .
. . .
. . .
. . .
So this illustrates that “4 X 3=12”, and “3 X 4=12”, and the general principle that “x * y” is the same as “y * x”.
If you then divide the dots among three people by giving each person one column, they would each get 4 dots. So 12/3=4. Or if you gave a row to each of 4 people, you get 12/4=3.
So if “x * y=z”, then “y * x=z”, “z/y=x”, and “z/x=y”.
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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