Given that (1, 0, 1) is a solution to a system of three linear equations, which of the following is true about the system? The system can be either inconsistent or consistent.
The system can be either independent or dependent.
The system can only be independent and consistent.
The system can only be dependent and inconsistent.

 Choice C) The system can only be independent and consistent

Explanation:

Let's go through the answer choices

A) This isn't possible. Either a system is consistent or inconsistent. It cannot be both at the same time. The term "inconsistent" literally means "not consistent". It's like saying a cup is empty and full at the same time. We can rule out choice A.B) This is similar to choice A and we cannot have a system be both independent and dependent. Either a system is independent or dependent, but not both. Independence means that the two equations are not tied together, while dependent equations are some multiple of each other. We can rule out choice B.C) We'll get back to this laterD) The independence/dependence status is unknown without the actual equations present. However, we know 100% that this system is not inconsistent. This is because the system has at least one solution. Inconsistent systems do not have any solutions at all (eg: parallel lines that never cross). We can rule out choice D because of this.

Going back to choice C, again we don't have enough info to determine if the system is independent or dependent, but we at least know it's consistent. Consistent systems have one or more solutions. So part of choice C can be confirmed. It being the only thing left means that it has to be the final answer.

If it were me as the teacher, I'd cross out the "independent" part of choice C and simply say the system is consistent.

The 3rd quadrant as 3x+g5ti=56i: 92d-543