State True or False: The equations of two regression lines are `10x-4y=80` and `10y- 9x=40` . then `b_(xy) = 0.9`

Answer

To determine whether the statement "b_(xy) = 0.9" is true or false given the regression equations \(10x - 4y = 80\) and \(10y - 9x = 40\), we will follow these steps:

### Step-by-Step Solution:

1. **Rearranging the First Equation**:
Start with the first regression line equation:
\[ 10x - 4y = 80 \]
Rearranging gives:
\[ 10x = 4y + 80 \quad \Rightarrow \quad x = \frac{4}{10}y + \frac{80}{10} \quad \Rightarrow \quad x = \frac{2}{5}y + 8 \]
This indicates that the regression line of \(x\) on \(y\) has a slope \(b_{xy} = \frac{2}{5}\).

2. **Rearranging the Second Equation**:
Now, consider the second regression line equation:
\[ 10y - 9x = 40 \]
Rearranging gives:
\[ 10y = 9x + 40 \quad \Rightarrow \quad y = \frac{9}{10}x + 4 \]
This indicates that the regression line of \(y\) on \(x\) has a slope \(b_{yx} = \frac{9}{10}\).

3. **Calculating the Correlation Coefficient**:
The correlation coefficient \(r\) can be calculated using the relationship:
\[ r^2 = b_{xy} \cdot b_{yx} \]
Substituting the values we found:
\[ r^2 = \left(\frac{2}{5}\right) \cdot \left(\frac{9}{10}\right) = \frac{2 \cdot 9}{5 \cdot 10} = \frac{18}{50} = \frac{9}{25} \]

4. **Finding the Value of \(r\)**:
To find \(r\), we take the square root of \(r^2\):
\[ r = \sqrt{\frac{9}{25}} = \frac{3}{5} = 0.6 \]

5. **Conclusion**:
Since we found \(b_{xy} = \frac{2}{5} = 0.4\), and the statement claims \(b_{xy} = 0.9\), we conclude that the statement is **False**.