BAD : We decided to make alternate arrangements in case the hotel was fully booked.
GOOD : We decided to make alternative arrangements in case the hotel was fully booked.
In British English alternate and alternative have different meanings.
alternate = (1) happening in turn, first one then the other: 'alternate periods of sun and rain' (2) every second (day, week, etc): 'Our local football team plays at home on alternate Saturdays.'
alternative = that may be used instead of the usual one or the one you had planned to use: 'In view of the roadworks, motorists are advised to use an alternative route.'
In American English alternate is used with the same meaning as alternative .