Two lines of regressions are represented by `4x + 10y = 9` and `6x + 3y = 4`. Find the line of regression y on x.

Answer

To find the line of regression \( y \) on \( x \) given the two regression lines represented by the equations \( 4x + 10y = 9 \) and \( 6x + 3y = 4 \), we can follow these steps:

### Step 1: Rearranging the first equation
We start with the first equation:
\[ 4x + 10y = 9 \]
To express \( y \) in terms of \( x \), we rearrange the equation:
\[ 10y = 9 - 4x \]
\[ y = \frac{9 - 4x}{10} \]
\[ y = -\frac{4}{10}x + \frac{9}{10} \]
\[ y = -\frac{2}{5}x + \frac{9}{10} \]

### Step 2: Rearranging the second equation
Next, we take the second equation:
\[ 6x + 3y = 4 \]
We also rearrange this equation to express \( x \) in terms of \( y \):
\[ 3y = 4 - 6x \]
\[ y = \frac{4 - 6x}{3} \]
\[ y = -2x + \frac{4}{3} \]

### Step 3: Finding the slopes of the regression lines
From the equations we derived, we can identify the slopes:
- The slope of the regression line of \( y \) on \( x \) (from the first equation) is \( b_{yx} = -\frac{2}{5} \).
- The slope of the regression line of \( x \) on \( y \) (from the second equation) is \( b_{xy} = -2 \).

### Step 4: Using the relationship between the slopes
The relationship between the slopes of the regression lines is given by:
\[ b_{yx} \cdot b_{xy} = r^2 \]
Where \( r \) is the correlation coefficient. We can calculate \( r^2 \):
\[ b_{yx} \cdot b_{xy} = \left(-\frac{2}{5}\right) \cdot (-2) = \frac{4}{5} \]

### Step 5: Finding the line of regression \( y \) on \( x \)
Now, we can conclude that the line of regression \( y \) on \( x \) is:
\[ y = -\frac{2}{5}x + \frac{9}{10} \]

### Final Answer
Thus, the line of regression \( y \) on \( x \) is:
\[ y = -\frac{2}{5}x + \frac{9}{10} \]