In a study on the relationship between in vesestment (X) and profit (Y), the following two regression equations were obtained based on the data on X and Y
`3X+Y-12=0`
`X+2Y-14=0`
What is the mean `barX` ?

Options

A. 6
B. 5
C. 4
D. 2

Explanation

To find the mean \( \bar{X} \) from the given regression equations, we will follow these steps:

### Step 1: Write down the regression equations
The two regression equations provided are:
1. \( 3X + Y - 12 = 0 \)
2. \( X + 2Y - 14 = 0 \)

### Step 2: Substitute \( \bar{X} \) and \( \bar{Y} \)
Since the regression equations pass through the point \( (\bar{X}, \bar{Y}) \), we can substitute \( \bar{X} \) and \( \bar{Y} \) into both equations:
1. \( 3\bar{X} + \bar{Y} - 12 = 0 \) (Equation 1)
2. \( \bar{X} + 2\bar{Y} - 14 = 0 \) (Equation 2)

### Step 3: Rearranging the equations
From Equation 1:
\[ 3\bar{X} + \bar{Y} = 12 \quad \text{(1)} \]
From Equation 2:
\[ \bar{X} + 2\bar{Y} = 14 \quad \text{(2)} \]

### Step 4: Solve for \( \bar{Y} \) in terms of \( \bar{X} \)
From Equation (1):
\[ \bar{Y} = 12 - 3\bar{X} \quad \text{(3)} \]

### Step 5: Substitute \( \bar{Y} \) in Equation (2)
Now substitute Equation (3) into Equation (2):
\[ \bar{X} + 2(12 - 3\bar{X}) = 14 \]
Expanding this gives:
\[ \bar{X} + 24 - 6\bar{X} = 14 \]
Combining like terms:
\[ -5\bar{X} + 24 = 14 \]

### Step 6: Solve for \( \bar{X} \)
Now, isolate \( \bar{X} \):
\[ -5\bar{X} = 14 - 24 \]
\[ -5\bar{X} = -10 \]
\[ \bar{X} = \frac{-10}{-5} = 2 \]

### Conclusion
Thus, the mean \( \bar{X} \) is:
\[ \bar{X} = 2 \]