Final answer:
An egalitarian (option b) family is where the mother and father share authority equally, differing from patriarchal or matriarchal systems where one gender predominates. Patrilocal and matrilocal systems refer to residence patterns, while patrilineal and matrilineal descent define lineage.
Step-by-step explanation:
The family structure where both the mother and the father share authority is described as an egalitarian family. This family system is characterized by an equal distribution of power and responsibilities between genders, as opposed to patriarchal or matriarchal systems where one gender holds a majority of the power. In egalitarian societies, both parents have equal say in decisions regarding their children and household affairs.
In contrast, a patrilineal descent system, which follows the father's line only, and a matrilineal descent system, which follows the mother's line only, define how inheritance and family ties are delineated. In patriarchal societies, which are often associated with patrilocal living arrangements, the male head typically assumes authority over the family. Similarly, in matrilineal societies, while descent may be traced through the female line, the senior men of the family often hold the decision-making power. To clarify the terms, matrilocality refers to a system in which it is customary for the husband to live with or near his wife's family, and matriarchy denotes a social system where women hold the majority of power, particularly in regard to leadership and property rights. These terms are specific to the lineage and residing customs rather than the distribution of authority within a household.