While doing problem sets, I came across this question:

Given the linear system:

a) Determine a particular value of t so that the system has infinitely many solutions.

b) Determine a particular value of t so that the system has a unique solution.

According to the book, the value for t so that the system has infinitely many solutions is t = 0, and for a unique solution any real number that isn't 0.

That would imply that:

has infinitely many solutions, but if you multiply the first equation by 3 and then proceed to eliminate, nothing would remain. So, wouldn't this be a linear system with no solution (inconsistent system) rather than infinitely many? And for any nonzero number, wouldn't it likewise have no solution?