## Elementary Linear Algebra 7th Edition

(a) there three two equations in two variables. (b) the system is consistent if and only if $k+6= 0$, that is $k=-6$. If the matrix is the coefficient matrix of homogeneous system, we have (a) there are three equations in three variables. (b) the system is consistent for any value of $k$. It has at least the trivial solution.
The augmented matrix is given by $$\left[ \begin {array}{ccc} 2&-1&3\\ -4&2&k \\ 4&-2&6\end {array} \right].$$ Multiply the first row by $2$ and adding it to the second row, adding the second row to the third row, we get $$\left[ \begin {array}{ccc} 2&-1&3\\ 0&0&k+6 \\ 0&0&k+6\end {array} \right].$$ Adding $-1$ times the second row to the third row we get $$\left[ \begin {array}{ccc} 2&-1&3\\ 0&0&k+6 \\ 0&0&0\end {array} \right].$$ Now, we have (a) there three two equations in two variables. (b) the system is consistent if and only if $k+6= 0$, that is $k=-6$. If the matrix is the coefficient matrix of homogeneous system, we have (a) there are three equations in three variables. (b) the system is consistent for any value of $k$. It has at least the trivial solution.