First of all, how you 10 variables are dependent?. I f they are linearly dependent than You only select one from them and after thsi you measure the regeression of your independent variable on the selected one. The impact on the others will be only gained.
You can use Structural Equation Modeling (SEM) for this kind of analysis. But I wonder if your 10 dependent variables are highly inter-rated in a way that would form one or more scales.
Simultaneous equation models work well for this type of analysis. There is a good Sage "Little Green Book" on this entitled Nonrecursive Models: Endogeneity, Reciprocal Relationships, and Feedback Loops by John Hipp and some others. You can do the analysis in most software that does Structural Equation Modeling such as Amos, Stata, R, etc. but Amos is probably the easiest I have encountered.