The Gospel of Luke does not contradict the Gospel of John, nor is there any contradiction between any of the Gospels. Each gospel record paints a different facet of the Messiah's life and ministry. Only by superimposing each gospel record over the other and synchronizing them with the instructions in the Torah do you get the full five-dimensional picture of the Messiah as He walked among men. All apparent contradictions lie either in our understanding or our interpretation. In the case of the apparent contradiction between Luke and John concerning the Last Supper, it is a combination of both.
It took me 70 hours of teaching to fully answer this question. You can get the abbreviated version in the 10-hour DVD series Volume 16 - The Jonah Code and the 70 Weeks Prophecy of Daniel. This series details the events surrounding the Last Supper, which was NOT the Passover meal, no matter how badly some people want it to be. Unless we get this detail correct, we will never be able to solve the Jonah Code, which is the "one sign" that authenticates Yahshua as the Messiah: three days and three nights in the grave, and yet raised on the third day. This detail is ignored by most because of their infatuation with Nimrod's reincarnated wife, the sex goddess Easter.