Learn Turkish: Çekici olmak için sadece sebze yiyerek yaşıyor. - To be attractive, she lives by eating only vegetables.

It uses the suffix -(y)erek, which means "by doing" or "while doing" an action. The verb root is ye- (to eat). Because the suffix starts with a vowel, we add the buffer letter -y-. The tricky part here is that the root ye- changes to yi- before the -y-, making it yiyerek instead of yeyerek.

Why does the sentence use the singular sebze instead of the plural sebzeler for "vegetables"?

In Turkish, when you are talking about a general category of things rather than specific items, you usually leave the noun in the singular form. Since she is eating vegetables in general as a habit, sebze is used without the plural suffix -ler.

How does olmak için translate to "to be attractive"?

When you put the dictionary form of a verb (like olmak, meaning "to be") right before için (which normally means "for"), it translates to "in order to" or "for the purpose of". So çekici olmak için means "in order to be attractive".

I noticed yaşıyor comes from the verb yaşamak (to live). Where did the a go?