you're given background info, integer x has a remainder of 5 when divided by 9, integer y has a remainder of 7 when divided by 9

question: what is the remainder when 5x - y is divided by 9?

i solved this by picking numbers, but i think that it's probably inefficient.

i did 2 examples and made sure i got the same number

taking info from above, x has remainder of 5 so:

x = 14, remainder is 5

5x = 70, remainder is 7

x = 23, remainder is 5

5x = 115, remainder is 7

i recall in chapter 10, it only mentions that if you multiply the remainders of x and y, it is equivalent to finding the remainder of xy, but i dont recall it saying that multiplying x by 5 is the same as multiplying Rx by 5 (remainder of x multiplied by 5)

could someone provide the explanation/proof/algebra of why this works?