# $R^2$ and p-value for multivariate linear regression with confounders

I am working on a project where I am interested in the following variables:

Dependent variables: $$y_1,\, y_2,\, y_3,\, y_4,\, y_5$$ (continuous)
Independent variables: $$x_1,\, x_2, \ldots,\, x_{18}$$ (some continuous, some binary)

The first thing I did was to perform a series of simple linear regressions between each dependent and independent variable.

I found that $$x_1$$ (continuous variable) was strongly correlated with $$y_1, \, y_2,\, y_3,$$ and $$y_4$$, and that no dependent variables were correlated with any other independent variables.

Based on this, I would like to ask the following question:
"Are $$\{y_1,\ldots, y_5\}$$ related to $$x_1$$, when including $$\{x_2,\, x_3,\ldots , x_{18}\}$$ as confounders?"

I can calculate $$R^2$$ and p-values by performing a partial correlation on each dependent variable, but then:
1) Do I need to adjust the p-values based on multiple comparisons?
2) Does it matter that the dependent variables may be correlated?
3) Is there a more appropriate statistical test to answer my question (like multivariate linear regression)?

I really appreciate the help, I don't have any statistics background.

For each $$Y$$, fit a linear model with all of the covariates and their interactions that you think they are important, to see what happens (if $$Y$$ still related to $$x_1$$.)