Question

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?

Answer 1

Answer: Lets rewrite our equation.

x + y + kz - 5z =k

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

k = $\frac{x + y - 5z}{1 - z}$

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.