A) For what value (s) of k is the following SoLE inconsistent?

x - 2y + 4z = 3

4x + 5y + kz = 9

y + 3z = 2

Answer: z = 55.

Step-by-step 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.