SOLUTION: 8. Determine the values of k such that the system of linear equations is consistent -x+y+z=0 4y-2z=2 -4y+2z=k

Consider the second and the third equations of the system.


Their left sides are almost identical: they differ only by signs of their terms.


THEREFORE, the necessary condition for the system to be consistent is that their RIGHT sides should have the opposite signs.


So, the value of "k" must be -2:  k = -2.


Then the second and the third equation are, actually, IDENTICAL and represent two equivalent equations.


In other words, you can think that the second and the third equation represent the same equation 
and not two different equations.


Since these two equations include only two variables, one of them can take arbitrary values 
and the other variable can be expressed / calculated as the function of the other.


Next, having y and z with any assigned values, we always can fit the third variable x,
so the system is always consistent and have (infinitely many) solutions.


Thus the condition k = -2 is not only necessary for consistency of the system - it is a SUFFICIENT condition, too.


ANSWER.  k = -2 is the necessary and sufficient condition for the given system to be consistent.