The two lines of regression for a bivariate distribution (X,Y) are 3 x + y = 7 and 3x + 5y = 11 . Find the regression coefficient `b_(yx)`

Answer

To find the regression coefficient \( b_{yx} \) from the given lines of regression, we will follow these steps:

### Step 1: Write down the equations of the lines of regression.
The two lines of regression given are:
1. \( 3x + y = 7 \)
2. \( 3x + 5y = 11 \)

### Step 2: Rearrange the first equation to express \( y \) in terms of \( x \).
From the first equation:
\[ y = 7 - 3x \]

### Step 3: Rearrange the second equation to express \( y \) in terms of \( x \).
From the second equation:
\[ 3x + 5y = 11 \implies 5y = 11 - 3x \implies y = \frac{11 - 3x}{5} \]
This can be simplified to:
\[ y = -\frac{3}{5}x + \frac{11}{5} \]

### Step 4: Identify the slope of the second regression line.
The slope of the line \( y = -\frac{3}{5}x + \frac{11}{5} \) is \( -\frac{3}{5} \).

### Step 5: Determine the regression coefficient \( b_{yx} \).
The regression coefficient \( b_{yx} \) is equal to the slope of the regression line of \( y \) on \( x \). Therefore:
\[ b_{yx} = -\frac{3}{5} \]

### Final Answer:
The regression coefficient \( b_{yx} \) is \( -\frac{3}{5} \).

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