The equations of the two lines of regression are `6x + y− 31 = 0` and `3x + 2y− 26=0`.
Calculate the mean values of `x` and `y`.

Answer

To find the mean values of \( x \) and \( y \) given the equations of the two lines of regression, we can follow these steps:

### Step 1: Write down the equations of the regression lines
The equations given are:
1. \( 6x + y - 31 = 0 \)
2. \( 3x + 2y - 26 = 0 \)

### Step 2: Express \( y \) in terms of \( x \) from the first equation
From the first equation, we can express \( y \) as:
\[ y = 31 - 6x \]

### Step 3: Substitute \( y \) into the second regression equation
Now, substitute \( y \) from Step 2 into the second equation:
\[ 3x + 2(31 - 6x) - 26 = 0 \]

### Step 4: Simplify the equation
Expanding the equation gives:
\[ 3x + 62 - 12x - 26 = 0 \]
Combine like terms:
\[ -9x + 36 = 0 \]

### Step 5: Solve for \( x \)
Rearranging gives:
\[ 9x = 36 \implies x = \frac{36}{9} = 4 \]

### Step 6: Substitute \( x \) back to find \( y \)
Now substitute \( x = 4 \) back into the equation for \( y \):
\[ y = 31 - 6(4) = 31 - 24 = 7 \]

### Final Result
The mean values are:
\[ \bar{x} = 4, \quad \bar{y} = 7 \]