[jiaweihli]

This is how I learned it - in hindsight, I feel this is easier to reason about, as it forces you to apply operations synchronously. (As opposed to, say, subtract 5 from both sides, then add 8 => which we can reduce to add 3. But this reduction might just be noise to your brain.)

[iopq]

This way you get "proofs" like: http://www.math.hmc.edu/funfacts/ffiles/10001.1-8.shtml

If you have an intuitive understanding like "both sides represent a number, as long as I manipulate it the same way the equation stays true" then you wouldn't have problems with logic when dividing by zero

[jiaweihli]

Yes, I understand the concept that you quoted. Theoretically, it's more sound.

However, I want to point out that the purpose of the 'magic mirror' method is to ease mental math. It frees your brain from having to spend time / space on the extra step of applying the same operation to both sides.

Putting this in the context of 'showing your work':

Normal: x / 3 = 4 => 3 * x / 3 = 4 * 3 => x = 12

Mirror: x / 3 = 4 => x = 4 * 3 = 12

It's a bit complex to describe - I feel how you do basic math such as this is hardwired into your brain at a very young age.