How to find the dimension of linear system of curves of degree $d$

Consider two curves $C_1$ and $C_2$ in $\mathbb P^2 (\mathbb C)$ . How can i find the expected and real dimension of the linear system of cuves of degree $d$ passing through points lying on the both curves .

Expected dimension can be found using the multiplicity of the intersection point and a relation . But how do i find the real dimension .