Regression Lines and Their Point of Intersection

Each regression coefficient of the previous section is the slope of a regression line, used for prediction. Since a regression line is built from data centred on the sample, both regression lines are constructed to pass through the same fixed point — the pair of means $(\bar x,\bar y)$.

Both equations are satisfied exactly by $x=\bar x, y=\bar y$ (substituting these values makes both sides zero) — so the two regression lines, however different their slopes, always intersect at exactly $(\bar x,\bar y)$. This single fact is useful in both directions:

• Given the means and the two regression coefficients, both line equations can be written directly, and their intersection verified to be $(\bar x,\bar y)$ as a check.
• Given only the two regression-line equations (without the means stated separately), solving them simultaneously — exactly as with any pair of linear equations — recovers $\bar x$ and $\bar y$, since their unique common solution point is precisely the pair of means.

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