Algebraic fractions, step by step

You may cancel a factor and you may not cancel a term, and (x² − 81) ÷ (x − 9) is where the difference bites.

The question

Check this line: (x² − 36) ÷ (x − 6) − x = 6. Which is correct?

(x² − 36) ÷ (x − 6) − x

(x² − 36) ÷ (x − 6) − x=(x + 6) − x

(x² − 36) ÷ (x − 6) − x=6

they wrote:6

Do not check by reading their line. Somebody else's working is the most persuasive thing on the page, and rereading it is how you end up agreeing with it — do the step yourself first, then compare.
On your own, and writing the step nobody writes down: (x² − 36) ÷ (x − 6) − x becomes (x + 6) − x.
Tidy that up and it is 6.
Now compare — and this time the two agree. The line on the page was already right, which is a finding, not a missing mistake. Some of them are.

How it works.

01
Do not check by reading their line. Somebody else's working is the most persuasive thing on the page, and rereading it is how you end up agreeing with it — do the step yourself first, then compare.
02
On your own, and writing the step nobody writes down: (x² − 36) ÷ (x − 6) − x becomes (x + 6) − x.
03
Tidy that up and it is 6.
04
Now compare — and this time the two agree. The line on the page was already right, which is a finding, not a missing mistake. Some of them are.