I have been wondering, why some japanese verb have a variety of ways to spell. Does it matter which spelling I used or does the variety of spelling serve a purpose. And if it doesn't matter, then why do they have it because I don't think I have see something like this before in other language
I think much of that is due to the shortage of standardization. That said, it serves some purpose.
送り仮名の付け方 tried to standardize it and succeeded to some extent for long. Many large newspapers and publishers have adhered to it. Before then, I believe there was no official rule and things were more chaotic.
The standard way, like 引き受けます, basically boils down to having kanji for the stem (the constant part in conjugation) and hiragana for the suffix (the variable part).
This distinction helps readers, because when the variable part is "hidden" in a kanji, it can create ambiguity and the reader may not know what is the reading the writer intends. If you write おりる as 降る, the reader might confuse it with ふる. (Although おり in おりる is the constant part now, the verb had forms like おるる historically.)
The non-standard way to write ひきうけます as 引受ます is economical in terms of the space it occupies. So it may be seen on an advertisement where you are charged per character or per square meter.
It doesn't matter.
Indeed, multiple patterns can be found in the Japanese language in the kun-yomi transcription of kanji, as in this problem.
This problem is caused by the combination of kanji (imported from China) and hiragana (invented in Japan).
Although many people are familiar with the kanji themselves, the use of the Okurigana is not precisely defined, and the Okurigana is used in various patterns.
No matter which pattern is used, the meaning is understood without confusion thanks to the kanji, so the word survives and has become a part of the Japanese language.
