Question

Two lines of regression are given as 4x + 3y + 7 = 0 and 3x + 4y + 8 = 0. Identify the line of regression of x on y.

Answer 1

Line l₁: 4x + 3y + 7 = 0   ...(i)

Line l₂: 3x + 4y + 8 = 0   ...(ii)

Equation (i) can be written as

For x:

4x = – 3y – 7

⇒ `x = (-3y)/4 - 7/4`

For y:

3y = – 4x – 7

⇒ `y = (-4x)/3 - 7/3`

Equation (ii) can be written as

For x:

3x = – 4y – 8

⇒ `x = (-4y)/3 - 8/3`

For y:

4y = – 3x – 8

⇒ `y = (-3x)/4 - 8/4`

Now assume `x = (-3y)/4 - 7/4` is the regression line of x on y.

And `y = (-3x)/4 - 2` is the regression line of y on x.

Slopes are `b = (-3)/4` and `d = (-3)/4`

Then, `r^2 = bd = ((-3)/4)((-3)/4)`

`r^2 = 9/16 < 1`

This is a valid solution.

The regression line of x on y is

`x = (-3)/4y - 7/4`

4x = – 3y – 7

⇒ 4x + 3y + 7 = 0.