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 $k$?
+128474 x + y + 3z = 10
-4x + 2y + 5z = 7
kx + z = 3 ⇒ z = 3 - kx (3)
Sub (3) into the first two equations
x + y + 3 [ 3 - kx] = 10
-4x + 2y + 5[3 - kx ] = 7 simplify these
(1 - 3k) x + y = 1
(-4 -5k)x + 2y = - 8
This will have no soution for k when this is true
( 1 - 3k) * 2 - ( - 4 - 5k)* 1 = 0
2 - 6k + 4 + 5k = 0
6 = k
