Determine whether the following statements are true or false and explain your reasoning.
If every variable is basic, then the linear system has exactly one solution.
If two augmented matrices are row equivalent to one another, then they describe two linear systems having the same solution spaces.
The presence of a free variable indicates that there are no solutions to the linear system.
If a linear equation has exactly one solution, then it must have the same number of equations as unknowns.
If a linear system has the same number of equations as unknowns, then it has exactly one solution.