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Try

Solve[Det[({{1, x, y}, {1, a, b}, {1, c, d}})] == 0, y][[1]] //Collect[#, x, Simplify] &
(*{y -> (-b c + a d)/(a - c) + ((b - d) x)/(a - c)}*)

create equation

(% /. Rule -> Equal) [[1]]
(*y == (-b c + a d)/(a - c) + ((b - d) x)/(a - c)*)