system of partial differential equation and boundary condition. Let $\Omega$ be a regular domain, for example be a rectangle.
Is it true that solve system of PDE's like this:
$$u+\Delta w=0, ~~~~~~~~~~~~~~~~~~~~~~w=b1,~\frac{\partial w}{\partial n}=b2,~~~\text{on}~
 \partial \Omega $$
$$\Delta u=0,~~~~~~~~~~~~~~~~~~~~~~~~~\text{with out boundary condition.} $$

To answer the question as clarified in the comments:
Yes, your formulation is fine. (Though numerical stability I know nothing about.)
The idea is the following: just take the Laplacian of the first equation, since we know that $\Delta u = 0$ you get
$$ \Delta\Delta w = 0 $$
which is the biharmonic equation. It is well-known that as a fourth order elliptic operator, this problem on a domain is well-posed when you prescribe two boundary conditions. And as you have provided $w|_{\partial\Omega}$ and $\partial_n w|_{\partial\Omega}$ you can convert the problem to a variational formulation and solve. (You would be minimizing the square norm of either the Hessian or Laplacian of $w$ under the boundary constraints.) 
After you solve the biharmonic equation for $w$, you can easily recover $u$ as minus the Laplacian of $w$ using your original equation. 
