Question

If a system of equations has linearly dependent equations and an equal number of equations and variables, does it have to be consistent?

Answer 1

No.

Explanation:

A linearly dependent system with an equal number of equations and unknowns is not necessarily consistent or inconsistent. It can be one or the other, depending upon the specific system. This is best proved by example.

Consider the system:
`x + y = 1`
`2x + 2y = 2`

The system is dependent because the second equation can be reduced to `2(x + y) = 2 -> x + y = 1`, which is identical to the first equation. However, the system has a solution whenever `x + y = 1`, so it is not inconsistent.

Consider the system:
`x + y + z= 1`
`2x + 2y + 2z = 2`
`x + y + z = 3`

This is a system with three equations and three unknowns. It is dependent because equation 2 is a constant multiple of equation 1. Additionally, it is inconsistent, because clearly `x + y + z` cannot equal both `1` and `3`.

We have seen two examples of a linearly dependent system with `n` unknowns and `n` equations. In one case, the system was consistent and in the other it was not. Thus, the answer to your question is no.