For a certain value of k, the system
x + y + 3z = 10
-4x + 8y + 5z = 7
kx + z = 3
has no solutions. What is this value of k?
+2668 Take the first equation, and multiply it by 8. This gives us \(8x + 8y + 24x = 80\).
Subtracting the second equation from this gives us \(12x + 19z = 73\).
Now, take the third equation and multiply it by 19. This gives us \(19kx + 19z = 57\)
These two equations won't have any solutions when the coefficient of x in both equations is equal. This will mean that two of the same things sum to two different values, which is impossible.
This means that \(12 = 19k\), meaning \(k = \color{brown}\boxed{12 \over 19}\)
