Q.The regression line of $y$ on $x$ for a set of bivariate data is $3x-2y=1$, and the regression line of $x$ on $y$ is $4x-y=5$. Find the mean values $\bar{x}$ and $\bar{y}$.

Both regression lines pass through $(\bar x,\bar y)$, so solving the two given equations simultaneously gives the means directly.

Since both regression lines are always built to pass through the point of means $(\bar x,\bar y)$, that point is exactly the common solution of the two given equations, found by solving them simultaneously.

Step 1 — Write the two equations.

Step 2 — Express $y$ from equation (ii). $y = 4x-5$.

Step 3 — Substitute into equation (i).

Step 4 — Find $y$. $y = 4(1.8)-5 = 7.2-5=2.2$.

Step 5 — State the means. $\bar x = x = 1.8$, $\bar y = y = 2.2$ — the point where the two regression lines intersect. …

Equally, equation (i) could have been solved for $x$ first ($x=(1+2y)/3$) and substituted into (ii) — either variable can b …

Mixing up which given equation is the line of $y$ on $x$ and which is $x$ on $y$ does not actually affect this particular question (both equations are simply solved together regardless), but it matters a great deal in a question that also asks for $b_{yx}$ or $b_{xy}$ individually from the same equations — always keep track of which l …