Brad Wilcox (First Counselor in the LDS Young Men General Presidency), writes in his book Born to Change the World: Your Part in Gathering Israel, on pages 36-37:
... Abraham's grandson was Jacob (Israel), and the members of his posterity were called Israelites. Thus, all Israelites were Hebrews and Semites but there were many Hebrews, Semites, and people who descended from Noah's other sons who were not Israelites. They were referred to as Gentiles. When the tribes were scattered, Israelites intermarried with Gentiles until the blood of Israel came to be found throughout the entire Earth. [1] Because of this mixing, many Gentiles today have the blood of Israel and just don't know it.
.... We [Latter-day Saints] know we are Israelites through revelation, and we have a covenant relationship with Jesus ...
The vast majority of Latter-day Saints--those born in the church and converts past and present--are literal descendants of Israel.[2] ...
Of course at this point it would be rare to find a pure-blood descendant of any one tribe because of all the mixing that has taken place. Almost everyone today has a combination of many bloodlines. This is why in a 2005 worldwide leadership training meeting for patriarchs, President Dallin H. Oaks taught, "A declaration of lineage is not a scientific pronouncement or an identification of genetic inheritance."[4] Patriarchs identify lineage "by the promptings of the Holy Ghost ... regardless of the race or nationality of the person receiving the blessing."[5] This clarification explains why two people in the same family--even twins--can be declared to be from different tribes.
Footnote [1] reads:
Along with referring to bloodlines, "Jew" and "Gentile" can also refer to people from different lands. Lehi and Nephi were from Manasseh by blood but referred to themselves as Jews because they had lived in the kingdom of Judah (see 1 Nephi 1:2). Joseph Smith was from Ephraim by blood (see Brigham Young, Jurnal of Discourses [London: Latter-day Saints' Book Depot], Vol. 2:269), but he is referred to on the title page of the Book of Mormon as a Gentile because he lived in a nation of Gentiles.
On page 49, Wilcox writes:
For most Christians, the tribes of Israel are a Bible story and nothing more. For Latter-day Saints, they are deeply personal, for we can do what no other Christians on earth can do: declare with confidence the tribes through which we will be blessed and bless others. Our patriarchal blessings assure us that the house of Israel is truly our home. [12]
When being interviewed by Richard N. Holzapfel, Joseph Fielding McConkey said, "The greatest evidence we have that the Book of Mormon is true is the way it picks up the promises of the Old Testament, continues the story, and extends the promise, ... the Abrahamic covenant. If you want the great evidence that Joseph Smith is a prophet, it is not that he restored New Testament Christianity. There were a host of small groups on the frontier seeking to replicate New Testament Christianity. What Joseph Smith restored was the Abrahamic covenant. That is the genius of the whole thing--that is who we are." [13].