Answer
(a) there are two equation in two variables.
(b) the system is consistent if and only if $k\in R\backslash \{-3/4\}$.
If the matrix is the coefficient matrix of homogeneous system, we have
(a) there are two equation in three variables.
(b) the system is consistent for any value of $k$.
Work Step by Step
The augmented matrix is given by
$$\left[ \begin {array}{ccc} 1&k&2\\ -3&4&1
\end {array} \right]
$$
Multiply the first row by $3$ and adding it to the second row, we get
$$\left[ \begin {array}{ccc} 1&k&2\\ 0&3k+4&7\end {array} \right].
$$
Now, we have
(a) there are two equation in two variables.
(b) the system is consistent if and only if $3k+4\neq 0$. So, we have
$$3k+4= 0\Longrightarrow k=-\frac{3}{4}.$$
Hence, the system is consistent if and only if $k\in R\backslash \{-3/4\}$.
If the matrix is the coefficient matrix of homogeneous system, we have
(a) there are two equation in three variables.
(b) the system is consistent for any value of $k$. It has at least the trivial solution.
