Those with appetite for “solid food” are brought to understanding via in-depth study beginning with the last half of Hebrews 6 that “hope [is] like a sure and firm anchor of the soul” [v. 19 (HCSB)] if it rests in the “high priest forever in the order of Melchizedek” (v. 20) — Jesus — and that hope is illusory if it rests in other than Him, the Priest-King. As Dr. John MacArthur advocated via Hard To Believe: The High Cost and Infinite Value of Following Jesus, the wide gate and broad road that leads to destruction (Matthew 7:13) is prevalently marked “Jesus”; alas, it’s not Jesus the Priest-King which interests the unregenerate. Astonishingly, even among the regenerate elect, Truth regarding the Priest-King is quenched; evisceration of the church via concomitant woeful beliefs and practices has been and continues to be the result.
At Hebrews 5:6, remember, the Writer quoted Psalm 110:4: “You are a priest forever in the order of Melchizedek”. At 5:12, the Writer lamented: “You need milk, not solid food”. At 6:4 – 8, the Writer solemnly warns those satisfied with “milk” of their inevitable curse and burning; faith without appetite is dead (assuming mental capacity).
[Then], the author  related his readers’ condition to the purpose of God, as evidenced especially in his dealings with Abraham. In [6:20], the author completes his careful preparaton for the ‘teaching difficult to explain’ (5:11). He does this by a skilful combination of motifs: (1) traditional teaching about the resurrection or exaltation of Christ is re-expressed in terms of the entry of a high priest into the inner sanctuary; and (2) the contrast between Jesus’ ministry and that of the OT priesthood is expressed by use of the Melchizedek motif. This comparison and contrast, both based on exegesis of OT texts, will prove to be the heart of the epistle.
Paul Ellingworth, The New International Greek Testament Commentary ~ The Epistle to the Hebrews, pp. 347 – 348 (link previously provided). The “heart of the epistle” is, indeed, the expostion of its central theme: The New Covenant. The Writer
argues powerfully that a new priesthood signals a new covenant. You cannot graft Christ’s high-priesthood onto that of the Mosaic order. Nor can the Mosaic priesthood survive under the ‘better covenant’ established in Christ’s atoning blood. There is a new covenant and a new priesthood, and former things have passed away.
Edgar Andrews, A Glorious High Throne, p. 189 (emphasis sic) (link previously provided). John MacArthur astutely observed:
The accounts of Melchizedek in sacred history are one of the most remarkable proofs of the divine inspiration and unity of Scripture. The whole concept of Melchizedek is an amazing insight into the fact that God wrote the Bible. In Genesis we have only three verses about Melchizedek. Some thousand years later David makes a briefer mention of him in Psalm 110:4, declaring for the first time that the Messiah’s priesthood would be like Melchizedek’s. After another thousand years, the writer of Hebrews tells us even more of Melchizedek’s significance. He reveals things about Melchizedek that even Melchizedek, or his contemporary, Abraham, did not know — and of which David had only a glimpse. So we reason that the God who wrote the book of Hebrews wrote the book of Genesis and Psalm 110 — and all the rest of Scripture.
The MacArthur New Testament Commentary ~ Hebrews, p. 173 (link previously provided). Prior to asserting such, Dr. MacArthur explained:
In biblical study, a type refers to an Old Testament person, practice, or ceremony that has a counterpart, an antitype, in the New Testament. In that sense types are predictive. The type pictures, or prefigures, the antitype. The type, though it is historical, real, and of God, is nonetheless imperfect, and temporary. The antitype, on the other hand, is perfect and eternal. The study of types and antitypes is called, as one might expect, typology.
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Melchizecek is  a type of Christ. As mentioned earlier, the Bible gives very little historical information about Melchizedek. All that we know is located in Genesis 14, Psalm 110, and Hebrews 5 -7. The most detailed information is in Hebrews 7:1 – 3.
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Chapter 7 is the focal point of Hebrews. It concerns the central, the most important, part of Judaism — the priesthood. No sacrifices could be made except by the priest and no forgiveness of sins could be had apart from the sacrifices. Obedience to the law was exceedingly important, but the offering of sacrifices was even more important. And the priesthood was essential for offering them. Consequently, the priesthood was exalted in Judaism.
The law God gave Israel was holy and good, but because the Israelites, as all men, were sinful by nature, they could not keep the law perfectly. When they broke the law, fellowship with God was also broken. The only way of restoring fellowship was to remove the sin that was committed, and the only way to do that was through a blood sacrifice. When a person repented and made a proper offering through the priest, his sacrifice was meant to show the genuineness of his penitence by obedience to God’s requirement. God accepted that faithful act and granted forgiveness.
Id. at 172. Guthrie and Moo add:
The type of commentary found in 7:1 – 10 is known as midrash [running exposition on the Old Testament text]. J. A. Fitzmyer has noted that Hebrews 7 has features in common with a midrash: The Old Testament text is the point of departure, the exposition is homiletical, the author stresses details of the scriptural passage, the text is shown to be relevant to the contemporary audience, and the focus is on the narrative of the Old Testament situation, not just the individual characters.
The author of Hebrews may have been familiar with speculations about Melchizedek in various religious communities of his day. Yet, the author’s treatment of this priest can be explained wholly on his treatment of the two Old Testament texts in which Melchizedek is named. His treatment of Melchizedek in 7:1 – 10 can be explained as an expostion of Genesis 14:17 – 20 with Psalm 110:4 in mind.
George H. Guthrie, Douglas J. Moo (Clinton E. Arnold, Gen. Ed.), Hebrews, James; Zondervan Illustrated Bible Backgrounds Commentary, p. 43 – 44 (italics sic) (link previously provided). Ellingworth further observes:
The central problem in this passage is the status of Melchizedek in relation to Christ. Elsewhere in Hebrews, whenever OT figures are drawn into the argument, their place in the hierarchy is made crystal clear, usually by contrast with that of Jesus. The angels are ministering spirits (1:13), but Jesus is Son; ‘Moses was faithful in all God’s house as a servant …, but Christ was faithful over God’s house as a son’ (3:5f.); the levitical priests hold their office ‘according to a legal requirement concerning bodily descent’; Christ holds his ‘by the power of an indestructible life’ (7:17). * * *
Melchizedek is unique among OT figures in Hebrews in that his status is neither contrasted with that of Christ nor directly related to that of believers. * * * The passage as a whole excludes a temporal succession between Melchizedek and Christ, since both are priests “for ever”, and the argument thus cannot strictly be called typological.
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The awkwardness of the introduction of Melchizedek into the argument remains: he constitutes an unnecessary complication in the comparison and contrast between priesthood in the old and in the new dispensations. In an author whose argument is generally so well articulated, the awkwardness demands explanation, and one is driven to look for external factors which more or less obliged him to speak of Melchizedek.
Ellingworth at 350 – 351. While Dr. MacArthur’s observation regarding the “divine inspiration and unity of Scripture” (supra) is surely the correct view, we’ll consider the “external factors” next time (Deo volente). For now, please consider Dr. MacArthur’s observations regarding the Levitical Priesthood, as Hebrews 7:1 – 3 instructs us as to the superiority of Melchizedek’s priesthood (and, of course, that of Jesus) to the Levitical Priesthood:
First, as mentioned above, the entire tribe of Levi was dedicated by God for religious service. Although all priests were Levites, not all Levites were priests. All priests, in fact, not only had to be descended from Levi but also from Aaron, Moses’ brother. The nonpriestly Levites served as helpers to the priests, and probably as singers, instrumentalists, and the like. The priesthood was strictly national, strictly Jewish. Second, the Levites were subject to the king just as much as were the other tribes. Their priestly functions were not under the control of the king, but in all other matters they were ordinary subjects. They were in no way a ruling class. A Levite, in fact, could not be king. They were set aside as a first fruit to God for special priestly service (Num. 8:14 – 16). Third, the priestly sacrifices, including the one by the high priest on the Day of Atonement, were not permanent. They had to be repeated and repeated and repeated — continually. They had no permanence. They provided no permanent forgiveness, no permanent righteousness, no permanent peace. Fourth, the Levitical priesthood was hereditary. A man who served as a priest did so because he was born into the right family, not because he lived a right life. Fifth, just as the effects of the sacrifices were temporary, so was the time of priestly service. A priest served from the age of 25 until the age of 50, after which his ministry was over (Num. 8:24 – 25).
MacArthur/Hebrews at 174.