Allright one more i did the problem just wanted to make sure i got all the steps right. The problem was 57x + 7 (congruent to) 78mod53.
For my solution ill just use equal as congruence.
57x + 7 = 78mod53, then 57x = 71mod53, then 57x = 18mod53, then applying to euclidean algorithm you get 1 = 14(53) - 13(57) and then you get -13(57) = -13(18)mod 53 and x = -234mod53 giving x = 31
i think its right im just studying for an exam and want to make sure i got the steps all down and right