How are equivalent fractions used in writing the sum of two like fractions

Well because in order to get a common denominator you might have to multiply a fraction to get is, doesn't change the value of the fraction it's just written with greater numbers and is not in simplest form

meaning 1/2 is the same as 8/16 one is just not simplified.

ex.

3/4+5/6
(you can't add these yet because they don't have a common denominator)

but you can make a common denominator by finding the LCM of the two denominators

fx. 4 and 6 both go into twelve, so you'd multiply the entire 3/4 fraction with 3 (because 4*3 is 12) and the entire 5/6 fraction with 2 (because 6*2 is 12)

3/4 *3 is going to give us 9/12
5/6 *2 is going to give us 10/12

now that we have the same denominator we can add the fractions together

9+10= 19. so...

19/12