Out of the following two regression lines, find the line of regression of Y on `X:3x+12y=7, 3y+9x=46`.

Answer

To find the line of regression of Y on X from the given regression lines, we will follow these steps:

### Step 1: Write down the equations of the regression lines
The two regression lines given are:
1. \(3x + 12y = 7\) (Equation 1)
2. \(3y + 9x = 46\) (Equation 2)

### Step 2: Rearrange Equation 1 to find the regression of Y on X
We will rearrange Equation 1 to express Y in terms of X.

Starting with:
\[ 3x + 12y = 7 \]

Subtract \(3x\) from both sides:
\[ 12y = 7 - 3x \]

Now, divide by 12:
\[ y = -\frac{3}{12}x + \frac{7}{12} \]

This simplifies to:
\[ y = -\frac{1}{4}x + \frac{7}{12} \]

### Step 3: Identify the regression line of Y on X
The regression line of Y on X can be expressed in the form:
\[ y = b_{yx}x + a \]
where \(b_{yx}\) is the slope of the regression line.

From our rearranged equation, we can see that:
\[ b_{yx} = -\frac{1}{4} \]

### Step 4: Write the final answer
Thus, the line of regression of Y on X is:
\[ y = -\frac{1}{4}x + \frac{7}{12} \]