Regression Lines and Their Point of Intersection

Each regression coefficient of the previous section is the slope of a straight line — the regression line — used for prediction. Since a regression line's whole purpose is to estimate one variable from the other for data centred on the sample, both regression lines are built to pass through the same fixed point: the pair of means $(\bar{x}, \bar{y})$.

Both equations are satisfied by $x = \bar{x}, y = \bar{y}$ — substitute those values into either equation and both sides become zero. This means the two regression lines, however different their slopes, always intersect at exactly the point $(\bar{x}, \bar{y})$. This single fact is useful in both directions:

• Given the means $\bar{x}, \bar{y}$ and the two regression coefficients, both regression-line equations can be written down directly, and their point of intersection can be verified to be $(\bar{x}, \bar{y})$ as a check.
• Conversely, if only the two regression-line equations are given (without being told the means directly), solving them simultaneously, exactly as one would solve any pair of linear equations in two unknowns, recovers $\bar{x}$ and $\bar{y}$ — because their unique common solution point is, by the fact just shown, precisely the pair of means. …