For a certain value of k, the system \(\begin{align*} x + y + 3z &= 10, \\ -4x + 2y + 5z &= 7, \\ kx + z &= 3 \end{align*}\)
has no solutions. What is this value of k ?
+128631 x + y + 3z = 10 multiply through by -2 ⇒ -2x - 2y - 6z = -20 (1)
-4x + 2y + 5z = 7 (2)
kx + z = 3 (3)
Add equations (1) and (2) and we get that
-6x -z = -13 multiply through by -1 ⇒ 6x + z = 13 (4)
So we have equations (3) and (4)
6x + z = 13
kx + z = 3
And when k = 6 we have no solutions
