Prove that the solutions of a linear system are the same when using RREF

I am asked to prove that for the linear system $x+2y+3z=6, 4x+5y+6z=15 text{ and } 7x+8y+(21+λ)z=24$ the solution set equals the solution set that you can come up using RREF.Using RREF I found that the linear system is Λ(Σ)= {$(x,y,z) in mathbb{R} | (x,y,z) = (0,3,0) text{ if } λneq-12 text{ or } (x,y,z) = (z,3-2z,z) text{ if } λ=-12 $}. How can I prove these two solution sets are the same? I thought of maybe proving that Gauss’ method works from the start, and that way we can prove these two are the same to begin with, but I wasn’t able to find a proof. Can anyone suggest anything?