Problem from "Differential topology" by Guillemin

Question

I am strugling one of the problems of "Differential Topology" by Guillemin:
Suppose that $Z$ is an $l$-dimensional submanifold of $X$ and that $z\in Z$. Show that there exsists a local coordinate system $\left \{ x_{1},...,x_{k} \right \}$ defined in a neighbourhood $U$ of $z$ such that $Z \cap U$ is defined by the equations $x_{l+1}=0,...,x_{k}=0$.
I assume that the solution should be based on the Local Immersion theorem, which states that "If $f:X\rightarrow Y $ is an immersion at $x$, then there exist a local coordinates around $x$ and $y=f(x)$ such that $f(x_{1},...,x_{k})=(x_{1},...,x_{k},0,...,0)$".
I would really appreciate any pointers on how to attack this problem.