Answer:
k = 6
Step-by-step explanation:
The system of three equations has no solutions, when the determinant of the matrix determined by this system is equal to 0.
Write the determinant:
[tex]\left|\begin{array}{ccc}1&1&3\\-4&2&5\\k&0&1\end{array}\right|[/tex]
Find it:
[tex]\left|\begin{array}{ccc}1&1&3\\-4&2&5\\k&0&1\end{array}\right|\\ \\=1\cdot 2\cdot 1+1\cdot 5\cdot k+3\cdot (-4)\cdot 0-3\cdot 2\cdot k-5\cdot 0\cdot 1-(-4)\cdot 1\cdot 1\\ \\=2+5k+0-6k-0+4\\ \\=-k+6[/tex]
Equate it to 0:
[tex]-k+6=0\\ \\k=6[/tex]
