coordinate change on two hypersurfaces

Question

If M is a hypersurface (embedded) on $\mathbb R^n$, then we know that we can use coordinates such that $M=\{(x_1,...,x_n) | x_n=0\}$, at least locally. If we have two hypersurfaces M, N can we do anything similar? Can we achieve coordinates such that $M: x_n=0$ and $N: x_i=0$ (where $i=n$ if $M=N$ locally, or $i\neq n$ if not). I suppose this is a question of linear algebra at the most part. What I am thinking is make a coordinate change such that $M=\{(x_1,...,x_n)| x_n=0\}$ and then work on coordinate change for N by stabilizing $x_n$ but I am not sure if this would work. Any hint would be welcome.

Answer 1

There are cases in which you can't. For example, consider the zero sets of $y-x^2=0$ and $y=0$. The tangent spaces of them at the origin don't generate the whole plane, but if we had local coordinates as desired then they would generate the whole plane.