The purpose of the expedition buck takes with thornton and his partners is to
a. search for a gold mine.
b. find a new camp.
c. deliver mail.
d. hunt for food.

A. search for a gold mine

This is the sentence which contains an inappropriate shift in tense:
The dog is playing in the yard as Mrs. Smith worked in the garden.
It is not correct to use present continuous in one part of the sentence, and past simple in the other, in this case.
The possible correct forms of this sentence are:
The dog was playing in the yard as Mrs. Smith was working in the garden.
The dog played in the yard as Mrs. Smith worked in the garden.

They made a checks and balance system to equal power within the branches. the "check" meant that one branch would watch one branch and "balance" them if they were too powerful. if you want more info on this, i would just google checks and balance system.