For the lines of regression `4x-2y=4and2x-3y+6=0`, find the mean of 'x' and the mean of 'y'.

Answer

To find the mean of 'x' and the mean of 'y' from the given lines of regression, we can follow these steps:

### Step-by-Step Solution:

1. **Write the equations of the regression lines:**
The given lines of regression are:
\[ 4x - 2y = 4 \quad \text{(Equation 1)} \]
\[ 2x - 3y + 6 = 0 \quad \text{(Equation 2)} \]
We can rewrite Equation 2 as:
\[ 2x - 3y = -6 \quad \text{(Equation 2)} \]

2. **Multiply Equation 2 by 2:**
To eliminate 'x', we can multiply Equation 2 by 2:
\[ 2(2x - 3y) = 2(-6) \]
This gives us:
\[ 4x - 6y = -12 \quad \text{(Equation 3)} \]

3. **Subtract Equation 3 from Equation 1:**
Now, we will subtract Equation 3 from Equation 1:
\[ (4x - 2y) - (4x - 6y) = 4 - (-12) \]
Simplifying this, we get:
\[ -2y + 6y = 4 + 12 \]
\[ 4y = 16 \]
Dividing both sides by 4:
\[ y = 4 \]

4. **Substitute the value of 'y' back into Equation 1:**
Now, we can substitute \( y = 4 \) back into Equation 1 to find 'x':
\[ 4x - 2(4) = 4 \]
Simplifying this, we have:
\[ 4x - 8 = 4 \]
Adding 8 to both sides:
\[ 4x = 12 \]
Dividing both sides by 4:
\[ x = 3 \]

5. **Final Mean Values:**
Therefore, the mean values are:
\[ \bar{x} = 3 \quad \text{and} \quad \bar{y} = 4 \]

### Conclusion:
The mean of 'x' is 3 and the mean of 'y' is 4.