Question

Determine the value of k for which the following system has no solutions. Write answer as an integer or a fraction in lowest terms.

x+y+4z=0

x+2y?4z=1

?2x?y+kz=2

Answer 1

Subtracting the first equation from second, we get:

(2y - y) - 4z - 4z = 1

y - 8z = 1

Adding twice the first equation to the third one, we get:

2x + 2y + 8z - 2x - y + kz = 2

y + (8 + k)z = 2

So now we have 2 equations here:
y - 8z = 1
y + (8 + k)z = 2

We wont have a solution, if we get something like 1 = 2 as it is not possible.

Therefore, for no solution, we get here:

y - 8z = y + (8 + k)z

8 + k = -8

k = -16

Therefore -16 is the required value of k here for the sytem to not have any solution.