Question

Given that (3, 2, –6) and (–2, 5, 1) are solutions of two equations in a system of three linear equations, which of the following is true about the system?

Answer 1

The system can be either inconsistent and independent or consistent and dependent.

Reasoning:

We have been given the solutions for only two equations in this system of three linear equations. To be consistent, all three equations must share either one or infinitely many solutions. We do not know whether all three equations share these solutions, which gives us the possibility of it being inconsistent. Inconsistent systems must be independent as they cannot be parallel and have infinitely many solutions at the same time.
There is also the possibility of all three equations sharing these solutions. Since there are two solutions, we must assume there are infinitely many solutions since our only choices are none, one, or infinitely many. We have already included the possibility that there are no solutions and it is not possible for there to be one as two have been listed. That means we are left to assume that if not inconsistent and independent, it must have infinitely many solutions, therefore being consistent and dependent.

Answer 2

We need more details about the equations themselves in order to determine the correct statement about the system of equations given the answers (3, 2, -6) and (-2, 5, 1). Which claims about the system is true cannot be determined without knowing the precise equations. The offered answers do not reveal details or features of the system as a whole; rather, they simply reveal information about certain places that meet the equations.