Question

Identify the regression equations of X on Y and Y on X from the following equations:
$2x+3y=6$ and $5x+7y-12=0$

Answer 1

For regression equations, both coefficients both $bxy$ and $byx$ need to be of the same sign.

If $2x+3y=6$ is Y on X and $5x+7y-12=0$ is X on Y

then $Y=\frac{6}{3}-\frac{2}{3}x$ then $X=\frac{12-7y}{5}$

$by{x}_1=-\frac{2}{3}$ and $bx{y}_1=-\frac{7}{5}$

If $2x+3y=6$ is X on Y and $5x+7y=12$ is Y on X

Then $x=\frac{6-3y}{2}$ and $y=\frac{12-5x}{7}$

$bx{y}_2=-\frac{3}{2}$ and $by{x}_1=-\frac{5}{7}$

$bx{y}_1\times by{x}_1=\frac{14}{15}<1$

$by{x}_2\times bx{y}_2=\frac{15}{4}<1$

We know $bxy\times byx=r^2$ which is always $<1$

So first assumption is correct

Regression equation of Y on X is $2x+3y=6$

Regression equation of X on Y is $5x+7y-12=0$