Regression Model for explained model(Details inside)
AFAIK if you want to predict the value of one variable, you need to have one or more variables as predictors; i.e.: you assume the behaviour of one variable can be explained by the behaviour of other variables.
In your case you have three independent variables whose value you want to predict, and since you don't mention any other variables, I assume that each variable depends on the others. In that case you could fit three models (for instance, regression models), each of which would predict the value of one variable, based on the others. As an example, to predict x:

x_prediction=int+cy*y_test+cz*z_test


, where int is the intercept and cy, cz, the coefficients of the linear regression. 
Likewise, in order to predict y and z:

y_prediction=int+cx*x_test+cx*z_test
z_prediction=int+cx*x_test+cy*y_test