So I have that $\displaystyle xy + 4xz - A = 0, \lambda y + 4\lambda z - yz = 0, \lambda x - xz = 0, 4\lambda x - xy = 0$ and I want to solve for $\displaystyle x,y,z,\lambda$ in terms of $\displaystyle A$, but I am absolutely stuck and do not know how to solve this system of equations, any help would be appreciated.

Everytime I attempt to solve I get $\displaystyle z = \lambda$, which screws up my calculations (at least the way that I attempted)