Answer: z = 55.
Explanation:
we want to find values of k that make this inconsistent.
x - 2y + 4z = 3
4x + 5y + kz = 9
y + 3z = 2
First, can you can see that k never can make some of the equations linearly dependent because of how constructed is the set. Now, let's see if there are values of k that give problems to our system.
To see it, let's solve the system.
from the third equation we can write y = 2 - 3z, and we can replace it into the first two equations:
x - 2(2 - 3z) + 4z = 3
4x + 5(2 - 3z) + kz = 9
simplify both equations and get
x + 10z = 7
4x + ( k - 15)*z = - 1
from the first equation, we have that:
x = 7 - 10z
we can replace it into the other equation:
4*(7 - 10z) + (k - 15)*z = -1
28 - 40z + (k -15)*z = -1
(k - 55)*z = -29
z = -29/(k - 55)
here you can see that the only value of z that has problems is z = 55, because we never can have a 0 in the denominator.