Regression Line of X on Y:

To find the regression line of x on y, we need to first calculate the correlation coefficient between x and y. Once we have the correlation coefficient, we can use it to calculate the slope and intercept of the regression line.

Calculating Correlation Coefficient:

We can calculate the correlation coefficient using the formula:

r = (nΣxy - ΣxΣy) / √[(nΣx^2 - (Σx)^2) (nΣy^2 - (Σy)^2)]

where,
n = number of observations
Σxy = sum of the product of x and y
Σx = sum of x
Σy = sum of y
Σx^2 = sum of x squared
Σy^2 = sum of y squared

Calculating Regression Line:

Once we have the correlation coefficient, we can use the formula for the slope of the regression line:

b = r (Sy / Sx)

where,
Sy = standard deviation of y
Sx = standard deviation of x

And the formula for the intercept:

a = y̅ - b x̅

where,
y̅ = mean of y
x̅ = mean of x

Applying the Formulas:

x 2y = 4 can be written as y = 0.5x
2x 3y - 5 = 0 can be written as y = (2/3)x - (5/3)

Using these equations, we can calculate the values required to find the regression line of x on y:

n = 2
Σxy = (1*0.5) + (2*(2/3)) = 1.833
Σx = 3
Σy = 1.1667
Σx^2 = 5
Σy^2 = 0.4167

Using the formula for correlation coefficient, we get:

r = (2*1.833 - 3*1.1667) / √[(2*5 - 3^2) (2*0.4167 - 1.1667^2)]
r = 0.866

Using the formula for slope of the regression line, we get:

b = 0.866 (0.4082 / 1.2472)
b = 0.283

Using the formula for intercept, we get:

a = 0.583 - (0.283 * 1.5)
a = 0.124

Therefore, the regression line of x on y is:

ŷ = 0.124 + 0.283x

X +2y -5 = 0 is the equn.of the reg. line of x on y & 2x +3y - 8 = 0 is the equn. of the reg.
line of y on x , then the 2 equns can be written as x = -2y +5 & y = -2/3 x+ 8/3
Hence bxy = -2/3 & bxy = -2 Now r2 = 4/3 > 1 This is impossible.
Hence our assumption is wrong ∴ 2 x + 3y -8 = 0 is the equn. of reg.line of x on y.