Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. Truly, I say to you, until heaven and earth pass away not an iota, not a dot will pass from the law until all is accomplished. Therefore, whoever relaxes one of the least of these commandments and teaches others to do the same will be called least in the kingdom of heaven, but whoever does them and teaches them will be called great in the kingdom of heaven
Why should we believe the Old Testament? Because Jesus did. Jesus seems to think that God's word should be the constant by which we evaluate the culture of the particular time and place we live in, not the other way around. This idea is unfashionable and counter-cultural, even in churches! But that's the attitude Jesus had.
We struggle to apply the Old Testament to our lives today and so tend to ignore it. If Jesus came to abolish the Old Testament and start again then it would be of no use to us and it shouldn't be in our bibles, but he makes it clear he has not come to do that. If the Old Testament law still applied to us in the same way as it did to Israel, we would be on a path to despair as we tried to keep it, or would be forced into picking and choosing the sections we want to obey (which sounds very familiar!). But Jesus says he came to fulfill the law and prophets, so it is not a burden and it is not pointless. The law and prophets are part of the gospel story that reaches it's climax in Jesus' life, death and resurrection.
"In Genesis, He is the seed of the woman. In Exodus, He is the passover lamb. In Leviticus, He is the high priest. In Numbers, He is the pillar of cloud by day and fire by night. In Deuteronomy, He is the prophet like unto Moses. In Joshua, He is the captain of our salvation. In Judges, He is the judge and lawgiver. In Ruth, He is the kinsman redeemer. In I & II Samuel, He is the trusted prophet. In Kings and Chronicles, He is the reigning king. In Ezra, He is the faithful scribe. In Nehemiah, He is the rebuilder of the broken wall. In Esther, He is the Mordecai. In Job, He is the ever-living redeemer. In Psalms, He is the Lord our shepherd. In Proverbs & Ecclesiastes, He is true wisdom. In Song of Solomon, He is the true lover and bridegroom. In Isaiah, He is the prince of peace. In Jeremiah and Lamentations, He is the weeping prophet. In Ezekiel, He is the wonderful four-faced man. In Daniel, He is the fourth man in the fiery furnace. In Hosea, He is the eternal husband, forever married to the backslider. In Joel, He is the baptizer with the Holy Spirit. In Amos, He is the burden-bearer. In Obadiah, He is the Savior. In Jonah, He is the great foreign missionary. In Micah, He is the messenger with beautiful feet. In Nahum, He is the avenger. In Habakkuk, He is God's evangelist pleading for revival. In Zephaniah, He is the Lord mighty to save. In Haggai, He is the restorer of the lost heritage. In Zechariah, He is the fountain opened in the house of David for sin and for cleansing. In Malachi, He is the sun of righteousness arising with healing in His wings. He is the theme of the Old Testament; every bit of it is His story." John Macarthur