What values of the independent variable do I use to compute the predicted data when calculating RMSE? I have a data set with
x = [10 16.25 16.25 16.25 16.25 20 22.5 22.6875 24.57 24.57 41.86 47 47 53.8 66.43 77.9 91.201 96.2 97.2]
and
y = [1.28 4.15 3.42 1.53 3.44 4.89 2.91 8.51 9.03 14.91 9.73 8.07 18.56 20.68 42.02 65.72 65.00 83.95 82.17]
A linear regression analysis gives me y = 0.919x - 15.396 and I am trying to calculate the RMSE for this model, for which I need the observed and predicted y values.
What values of x do I use to calculate the predicted y values? Should I use values the original x values? Or do I come up with random x values within the range of the original ones?

You have to have the $y$-values to which the $x$-values correspond. Otherwise, you cannot calculate the differences between predicted $y$ and true $y$. Thus, randomly generating $x$-values does not help, even though there are no issues when it comes to making predictions for those $x$-values.
With that in mind, it is acceptable to calculate the RMSE on the data used to fit the model, and there are many techniques that hold out some observations from the model-fitting process in order to see what kind of performance the model achieves on data it has not seen before. This is related to the “test” or “validation” performance that is discussed in machine learning circles, and the “cross validation” for which this Stack is named relies on this idea.
