What can be said about usefulness of two regression lines?

Question

How one can say that the regression equation $y$ on $x$ is more useful than the regression equation $x$ on $y$ as a predicting formula?

Answer 1

before randomly running a regression of $y$ on $x$ or vice versa. You should ask the question if there is any sense if using $x$ to predict $y$. Should there exist a relationship that should grant the predictability?

If you can answer this question then you can also know the usefulness of the regression. Other checks would be to see the following:

1. t-stats/p-values of coefficients, are they significant
2. do the computed coefficient values make sense or are the values absurdly high/low?
3. Are the residuals white noise? You can do a Q-Q plot to check for this.

Good Q-Q plot

Bad Q-Q plot

Last thing to check is a good R-square value, subject to the fact that all the conditions above have "good" results.