I would not use Google Translate as a benchmark for checking the validity of a translation. It deals with vocab fairly well but has trouble with the nuances and irregularities in grammar which makes it useless.
The English sentence and the Chinese are both perfectly fine, and exactly what we would be likely to say if we were to in this situation. Although the English version is not a literal translation, it is a perfect translation of meaning, which is more important than a direct translation.
The English translation leaves out the word 'airline' because I don't think I have ever included that specification when I'm buying plane tickets. The translation provided by Google Translate is a poor translation because it implies the tickets have already been bought and the sentence order is round the wrong way. If we wanted to say it this way, we would say something more like,
到/去北京 is a qualifier of the ticket and therefore must be placed before the noun, not afterwards.
I urge you not to use Google Translate for verifying the translation because it will only confuse you. It may be able to translate some very basic sentences accurately but all you have to do is take a basic sentence in English, translate it into Chinese using Google Translate, and then take the resulting Chinese and translate it back into English. You are likely to get a ridiculous result. Were it a good translation tool, then it would be able to take a sentence, translate it, then reverse translate it back into its original form without any problems. Unfortunately, this is not the case.
I hope this has cleared up some of your queries.
Keep up the good work and 加油！
- Lin Ping