Two regression lines are represented by `2x+3y -10=0` and `4x + y-5=0`
Find the line of regression of y on x.

Answer

To find the line of regression of \( y \) on \( x \) given the two regression lines \( 2x + 3y - 10 = 0 \) and \( 4x + y - 5 = 0 \), we will follow these steps:

### Step 1: Rewrite the equations of the regression lines
We have:
1. \( 2x + 3y - 10 = 0 \)
2. \( 4x + y - 5 = 0 \)

### Step 2: Express \( y \) in terms of \( x \) from the first line
From the first line:
\[ 3y = -2x + 10 \]
Dividing by 3:
\[ y = -\frac{2}{3}x + \frac{10}{3} \]

### Step 3: Express \( x \) in terms of \( y \) from the second line
From the second line:
\[ 4x = -y + 5 \]
Dividing by 4:
\[ x = -\frac{1}{4}y + \frac{5}{4} \]

### Step 4: Identify the coefficients \( b_{yx} \) and \( b_{xy} \)
From the equations derived:
- The slope of the regression line of \( y \) on \( x \) (denoted as \( b_{yx} \)) is \( -\frac{2}{3} \).
- The slope of the regression line of \( x \) on \( y \) (denoted as \( b_{xy} \)) is \( -\frac{1}{4} \).

### Step 5: Calculate the product of the slopes
Now, we calculate the product of the slopes:
\[ b_{yx} \cdot b_{xy} = \left(-\frac{2}{3}\right) \cdot \left(-\frac{1}{4}\right) = \frac{2}{12} = \frac{1}{6} \]

### Step 6: Verify the condition
Since \( \frac{1}{6} < 1 \), our assumption about the regression lines is correct.

### Step 7: Write the line of regression of \( y \) on \( x \)
Thus, the line of regression of \( y \) on \( x \) is:
\[ 2x + 3y - 10 = 0 \]

### Final Answer
The line of regression of \( y \) on \( x \) is \( 2x + 3y - 10 = 0 \).

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