Allah the Almighty commands:
“Neither their flesh, nor their blood reaches Allah; but it is your piety that reaches Him. He has put these animals into your service that you may glorify Allah for His guidance to you. Give glad tidings to the benefactors.” (22:37)
“The month of Ramadan in which was revealed the Qur’an, a guidance for mankind and clear signs of the right path. It lets discern right from wrong. So whoever of you sees that month, should spend it in fasting. Whoever is sick or on a journey, should fast the same number some other days. Allah desires ease for you. He does not desire for you difficulty for that you may complete the number of days, glorify Allah for His guidance to you and perhaps, you will be grateful.” (2:185)
The first verse is about the feast of sacrifice, and the second one is about the feast of fast-breaking. There is a same phrase which passes in both of the verses: “Glorifying Allah for His guidance”. The word translated as “glorify” is “takbeer” in Arabic. It means glorifying by saying “Allah-u Akbar”, which means Allah is Great.
Takbeer is also a phrase that we say in the beginning of any prayer. The Messenger of Allah understood these verses by both meanings of takbeer, which are praying and saying Allah-u Akbar. For this reason, he used to gather people in the first day of these holidays and offer a short prayer together, and then say “Allah-u Akbar” several times.
Both fasting and sacrificing an animal are the duties of Muslims regardless of their gender. So these holidays are the holidays of all Muslims, men and women. That’s why the messenger used to call every Muslim to these prayers. He even called the women who are unable to pray due to their special condition, and told them to stand in the back and participate in takbeers. He started the celebrations of these holidays by making everyone say “Allah-u Akbar” in the first hours of the feast morning according to these verses above.
Consequently, every Muslim should participate in these prayers if it is offered in the vicinity. If the person cannot find a mosque or a Muslim group offering this prayer, he/she should say Allah-u Akbar several times in the feast morning to celebrate the holiday and to obey the verses of Allah. Being together with some other Muslims is better.