Indicate the line of regression of on y from between the lines `x+2y=4 and 2x+3y-5=0` …………….

Answer

To find the line of regression of \( x \) on \( y \) between the lines \( x + 2y = 4 \) and \( 2x + 3y - 5 = 0 \), we will follow these steps:

### Step 1: Identify the equations of the lines
The given equations are:
1. \( x + 2y = 4 \)
2. \( 2x + 3y - 5 = 0 \)

### Step 2: Rearrange the equations to identify coefficients
We can rearrange both equations to express \( y \) in terms of \( x \):
1. From \( x + 2y = 4 \):
\[ 2y = 4 - x \implies y = 2 - \frac{x}{2} \]
Here, the coefficient of \( y \) (denoted as \( b_{xy} \)) is \( \frac{1}{2} \).

2. From \( 2x + 3y - 5 = 0 \):
\[ 3y = 5 - 2x \implies y = \frac{5}{3} - \frac{2}{3}x \]
Here, the coefficient of \( y \) (denoted as \( b_{yx} \)) is \( -\frac{2}{3} \).

### Step 3: Determine the regression coefficients
We need to check the conditions for regression coefficients:
- If \( b_{xy} \) is greater than 1, then \( b_{yx} \) should be less than 1, and vice versa.
- The product of the coefficients should not exceed 1.

From our calculations:
- \( b_{xy} = \frac{1}{2} \) (less than 1)
- \( b_{yx} = -\frac{2}{3} \) (also less than 1)

### Step 4: Check the product of coefficients
Now, we calculate the product:
\[ \text{Product} = b_{xy} \times b_{yx} = \left(\frac{1}{2}\right) \times \left(-\frac{2}{3}\right) = -\frac{1}{3} \]
This product is less than 1, which satisfies our condition.

### Step 5: Identify the regression line
Since \( b_{xy} \) is less than 1 and the product of the coefficients is also less than 1, we can conclude that the line of regression of \( x \) on \( y \) is represented by the second equation:
\[ 2x + 3y - 5 = 0 \]

### Final Answer
The line of regression of \( x \) on \( y \) is:
\[ 2x + 3y - 5 = 0 \]