X+ y+(k −5)z=k where k is an arbitrary constant. for which value(s) of k does this system have a unique solution? for which value(s) of k does the system have infinitely many solutions? for which value(s) of k is the system inconsistent?

Lets rewrite our equation.

x + y + kz - 5z =k

x + y -5z = k(1-z)

k =

so the only problem with the equation is that z can't be equal to one.

but x and y are free, so you have two free variables and one equation,

that means that for any k the system has infinitely many solutions.

180

step-by-step explanation: