TITLE: For which sets of parameters will this system of equations have no solution?
QUESTION [10 upvotes]: If $S$ is the set of distinct values of '$b$' for which the following system of linear equations
\begin{align} 
 x + y + z = 1\\
 x + ay + z = 1\\ 
 ax + by + z = 0
 \end{align}
  has no solution, then $S$ is:

a singleton 
an empty set
an infinite set 
a finite set containing two or more elements


There's no information given about value of $a$.
You will be getting two cases:

For $a=1, b$ has one value $b=1$
For $a$ not equal to $1 , b$ has no value.

REPLY [2 votes]: When $a=1$ and $b=1$ The first two equations are coincident planes ($x+y+z=1$)and the last one ($x+y+z=0$) is a plane parallel to them . 
Therefore for $b=1$, these equations have no solutions at all.(Since parallel planes will intersect no where.)
Hence, your set $S=\{1 \}$