Question: Matthew 1:1 records Jesus' genealogy, but it differs from the genealogy in Luke 3:23. The names are different and it shows Joseph's father being Heli instead of Jacob as said in Matthew.
Response: Here is a footnote to Luke 3:23-28 in the Concordia Self-Study Bible (NIV): "From Abraham to David, the genealogies of Matthew and Luke are almost the same, but from David on they are different. Some scholars suggest that this is because Matthew traces the legal descent of the house of David using only heirs to the throne, while Luke traces the complete line from Joseph to David. A more likely explanation, however, is that Matthew follows the line of Joseph (Jesus' legal father), while Luke emphasizes that of Mary (Jesus' blood relative). Although tracing a genealogy through the mother's side was unusual, so was the virgin birth. Luke's explanation here that Jesus was the son of Joseph, "so it was thought" (v. 23), brings to mind his explicit virgin birth statement (1:34-35) and suggests the importance of the role of Mary in Jesus' genealogy."