Linear Regression

The two lines meet at the point $(13,17)$ , and hence we have $\overline{x} =13$ and $\overline{y} = 17$ . The two equations then become $y - \overline{y} \;=\; \tfrac45(x - \overline{x}) \hspace{2cm} x - \overline{x} \; = \; \tfrac{9}{20}(y - \overline{y})$ The standard theory of (least squares) linear regression tells us that $\frac{\mathrm{Cov}[x,y]}{\sigma_x^2} \; = \; \tfrac45 \hspace{2cm} \frac{\mathrm{Cov}[x,y]}{\sigma_y^2} \; = \; \tfrac{9}{20}$ where $\mathrm{Cov}[x,y]$ is the covariance of $x$ and $y$ , and hence $\frac{\sigma_y^2}{\sigma_x^2} \; = \; \tfrac{16}{9}$ so that (since $\sigma_x^2 = 9$ ) the variance of $y$ is $\sigma_y^2 = 16$ , making the answer $13 + 17 + \sqrt{16} = \boxed{34}$