With Matt 5:21-48, Jesus quotes several statements from the Law as He is describing what He expects from His followers. On six occasions Jesus refers to an OT command but then follows these with, "But I say to you" to indicate that He was about to intensify or alter the OT commands. Thus Jesus quotes the OT passages contextually but then offers more. We will call this usage of the OT—"Literal Restatement of OT Passage with Intensification or Alteration"
Matt 5:21 / Exod 20:13
"You have heard that the ancients were told, ‘YOU SHALL NOT COMMIT MURDER’ and ‘Whoever commits murder shall be liable to the court.’"
This quote is taken from Exod 20:13 and represents a contextual rendering of one of the Ten Commandments. But Jesus then intensifies this command by internalizing it and indicating that hating one’s brother makes one guilty (see Matt 5:22).
Matt 5:27 / Exod 20:14
"You have heard that it was said, ‘YOU SHALL NOT COMMIT ADULTERY.'"
Jesus quotes Exod 20:14 in a contextual way but and then intensifies the command by internalizing it and declaring that lust was the equivalent of adultery (see Matt 5:28).
Matt 5:31 / Deut 24:1, 3
"It was said, ‘WHOEVER SENDS HIS WIFE AWAY, LET HIM GIVE HER A CERTIFICATE OF DIVORCE.’"
Jesus quotes Deut 24:1, 3 to show that only unchastity was an allowable reason for divorce (see Matt 5:32). This may be an alteration to the original OT command.
Matt 5:33 / Lev 19:12; Num 30:2; Deut 23:21
"Again, you have heard that the ancients were told, ‘YOU SHALL NOT MAKE FALSE VOWS, BUT SHALL FULFILL YOUR VOWS TO THE LORD.'"
Jesus appears to combine two or three verses from the Law. Again this is a case of a contextual use of the OT but Jesus then alters this command by stating that oaths should not be made at all and that a simple "Yes" and "No" should suffice because of the integrity of our speech (see Matt 5:34, 37).
Matt 5:38 / Lev 24:20
"You have heard that it was said, ‘AN EYE FOR AN EYE, AND A TOOTH FOR A TOOTH.'"
This quote is from Exod 21:24 and Lev 24:20. Jesus alters the ‘eye for an eye’ principle by declaring that His people should not even resist those who resist them (see Matt 5:39-42).
Matt 5:43 / Lev 19:18
"You have heard that it was said, ‘YOU SHALL LOVE YOUR NEIGHBOR and hate your enemy.'"
The quote about loving one’s neighbor comes from Lev 19:18 although the statement to "hate your enemy" has no specific reference in the OT and may reflect a Jewish tradition that was added. With Matt 5:44-47 Jesus indicates that His people are to love their enemies and do good to them who wish them harm.
I have purposely avoided some of the more difficult theological issues concerning Jesus’ relationship to the Mosaic Law. I have not addressed what it means for Jesus to be the fulfillment of the Law (see Matt 5:17) or whether these intensifications or alterations in Matt 5:21-48 are related to an ongoing Mosaic Law or a new Law of Christ that is different from the Mosaic Law. What should be noted for our purposes, though, is that Jesus quoted the OT contextually and then used these contextual understandings to intensify or alter the previous Mosaic Law commands.