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

Answer

To identify the regression lines from the given equations, we will follow these steps:

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

We will rewrite these equations in the form \( y = mx + c \) where \( m \) is the slope.

**For the first equation:**
\[ y = -6x + 31 \]
Here, the slope \( b_{yx} = -6 \).

**For the second equation:**
\[ 2y = -3x + 26 \implies y = -\frac{3}{2}x + 13 \]
Here, the slope \( b_{xy} = -\frac{3}{2} \).

### Step 2: Identify the regression lines
From the rewritten equations, we can identify:
- The regression line of \( y \) on \( x \) is given by the first equation:
\[ y = -6x + 31 \]
- The regression line of \( x \) on \( y \) is given by the second equation:
\[ x = -\frac{1}{6}y + \frac{31}{6} \]

### Step 3: Calculate the correlation coefficient
The correlation coefficient \( r \) can be calculated using the formula:
\[ r^2 = b_{xy} \times b_{yx} \]
Substituting the values we found:
\[ r^2 = \left(-\frac{3}{2}\right) \times (-6) = \frac{3 \times 6}{2} = 9 \]
Thus, \( r^2 = \frac{1}{4} \), which implies \( r = -\frac{1}{2} \).

### Step 4: Verify the range of \( r \)
The value of \( r = -\frac{1}{2} \) is within the range of -1 to 1, confirming our calculations are correct.

### Conclusion
The regression lines are:
- Regression line of \( y \) on \( x \): \( 6x + y - 31 = 0 \)
- Regression line of \( x \) on \( y \): \( 3x + 2y - 26 = 0 \)