Matthew follows the ancestors of Joseph, while Luke follows the ancestors of Mary. So why are there two different genealogies? Traditionally biblical genealogies are always traced through the male descendants, in this case Joseph. However there was an exception is Jesus case because while Joseph was the his legal father, he wasn't his biological father.
Now Jesus Himself began [His ministry at] about thirty years of age, being (as was supposed) [the] son of Joseph, [the son] of Heli, (Luk 3:23)Notice that Joseph is the "supposed" father of Jesus. "Son" can also refer to son-in-law or grandfather. In this case Heli was the father of Mary, Father-in-law of Joesph, and grandfather of Jesus. There is another reason why Mary's genealogy was included and it had to do with Jechonia:
[As] I live," says the LORD, "though Coniah the son of Jehoiakim, king of Judah, were the signet on My right hand, yet I would pluck you off; (Jer 22:24)Jechonia was a descendant of King David and the ancestor of Joseph. The Throne of David is promised to be established forever (1Ki 2:45). But because of Jechonia's sin, God said his decedents would never be allowed to rule on the Throne of David again. If Jesus bloodline was through Joesph he could not rule and reign during the Millennial Kingdom. However while Mary is a descendent King David, she is not a descendant of Jechonia. It was import for Luke to show that Jesus was not excluded as the Messiah because of Jechonia.
Thus says the LORD: 'Write this man down as childless, A man [who] shall not prosper in his days; For none of his descendants shall prosper, Sitting on the throne of David, And ruling anymore in Judah. (Jer 22:30)
