# How to prove that the intersection of 3 planes is a line?

Consider the planes: $$P1:x - y = 0$$ $$P2:y-z = 0$$ $$P3:-x+z = 0$$ Prove that the intersection of the planes is a line.

My solution: Solving the system I've obtained that $$x=y=z$$ and I made the notation $$x=t$$. From here we get the parametric equations of a line $$d$$ and we can write the canonical form: $$d : {{x}\over1} = {{y}\over 1} = {{z}\over1}$$ Thus proving that the intersection of the 3 planes is a line. Is this correct? If so, are there any other ways to prove this? Thanks in advance!

• Obviously $(0,0,0)$ and $(1,1,1)$ are solutions, hence the line determined by these points. – Michael Hoppe Feb 14 at 14:57