Find all integers x and y satisfying the equality 25x + 46y = 1.

Let us express the variable x through y, that is, we will leave 25x on the left side, and transfer the expression 46y to the right side, changing the sign to the opposite and get: 25x = 1 – 46y; x = (1- 46y) / 25. Then y = -19; x = 35 Therefore, we substitute our values into this equation and check whether it equals one. We get: 25 * 35 – 46 * (-19) = 1. Subtract one equation from another 25 * 35 – 25 + 46 * (-19) – 46y = 1; 25 (35 – x) + 46 (-19 – y) = 0; 25 (35 – x) = -46 (-19 – y). So 25 and -46 are coprime numbers, 35 – x = -46n, -19 – y = 25n.
Answer: x = 46n + 35, n belongs to the set of integers, y = -25n – 19, n belongs to the set of integers.