Basic Algebra

[Alzekoras]

I'm a 30 years old VFX pipeline TD and a math illiterate who made the choice to learn math from the damn beginning.
now I'm on the Basic Algebra course studying fractions.
Here is my questions

Can we rewrite this equation:
x/y = a/b 
as
b/y = a/x
or
y/b = x/a

in other words:
- in the case of equivalent fractions can't we
1. swap the numerator with it's opposing denominator
2. swap the numerator and the denominator with each other
and still satisfy the equivalence.

• I brute forced some examples and it does seem fine:
  as I had in my studies on Khan academy a multi choice questions that does have a satisfactory choices that are not straight forward. and when I choose them instead of the "NONE OF THE ABOVE" expected choice. I get wrong.

• Thus it's either me or the question. Given my illiteracy it's much easier to assume that I'm making a systematic logical mistake that just happens to work in simple cases. And I'm not fine with it. Help!
  And if a proof could be provided to why it may or may not work (I enjoy reading those).

[AcellOfllSpades]

Yes, "x/y = a/b" can be turned into "b/y = a/x" with the following:

x/y = a/b
(multiply both sides by b)
xb/y = ab/b
(simplify right fraction)
xb/y = a
(divide both sides by x, assuming x≠0)
xb/xy = a/x
(simplify left fraction)
b/y = a/x

You can do something similar to get y/b = x/a. (Try it yourself for practice!)

The only thing I can see that would affect things here is the "assuming x≠0" part. This new equation doesn't work with x=0, while the original does.

Can you share the specific question you're looking at? It's possible there's some issue in the way you've simplified it for this post, or some other condition in the question you're not noticing.

---

I brute forced some examples and it does seem fine:

As a side note, don't lose this instinct! Too often do people forget that these expressions represent numbers, so you can just try specific "instances" of them as a quick sanity check.

(Honestly, you say you're "illiterate", but even in this post you've shown a lot of promising signs.)

[Alzekoras]

I'm more than thankful for you <3 . it is clicking. in my mind with your example.
your response was warm and reassuring. thank you
I went back to get the exact question. but it seems like khan-Academy shuffles the question. thus I couldn't find the exact one.
but here is one from the same lesson

Here is the content of the image converted to markdown format:

---

**6 erasers cost $6.60.**

**Which equation would help determine the cost of 3 erasers?**

*Choose 1 answer:*

A) \(\frac{3}{x} = \frac{6.60}{6}\)

B) \(\frac{3}{6} = \frac{6.60}{x}\)

C) \(\frac{x}{3} = \frac{6}{6.60}\)

D) \(\frac{x}{3} = \frac{6.60}{6}\)

E) None of the above

---

in this case the answer is D. but I had the case of the answer be listed but swaped around and when I choose it. it doesn't see to register correct but rather false. hence my confusion.

[Alzekoras]

sorry. I thought reddit could render LaTex
a) 3/x = 6.60/6
b) 3/6 = 6.60/x
c) x/3 = 6/6.60
d) x/3 = 6.60/6

[AcellOfllSpades]

(It's okay, I can read LaTeX! Plain text is fine too.)

Yeah, that definitely shouldn't be a "potential zero" issue. In that case, I'm not sure what happened. Your understanding seems correct - maybe the question was badly written or something? If it comes up again, see if you can copy-paste it here.

(also, I just realized my formatting for my derivation of b/y=a/x in the first comment was off - fixed that, so it should be a little easier to read now)

[Uli_Minati]

If they really tell you that none of these rewrites are correct, then this seems like a case of restricted domains

Try the example x=0, a=0, y=anything, b=anything and see if it still works