The short answer is that algebra doesn't work that way. You wouldn't divide *everything* by 2, but every term that contains a factor of 2.
In the expression
2 (6x - 1) + 2 (2x + 5)
both terms have a factor of 2 (the 2 out in front of them). They're the ones that get canceled when dividing by 2:
(2 (6x - 1) + 2 (2x + 5)) / 2 = 2/2 (6x - 1) + 2/2 (2x - 5)
… = 1 (6x - 1) + 1 (2x - 5)
… = (6x - 1) + (2x - 5)
and so on.
Looking ahead, it turns out that the equation is solved by x = 7. This makes 6x - 1 = 41 and 2x + 5 = 19. So the equation is saying that, if you make these replacements,
2×41 + 2×19 = 120
If you divide *everything* on the left by 2, you end up with fractions:
(2/2)×(41/2) + (2/2)×(19/2) = 41/2 + 19/2
but 41 + 19 = 60, so the end result would be 30, but that's not the same as 120/2 = 60.
