if 4x-5y+33=0 and 20x-9y=107 are two lines of regression, then what are the values of `barx` and `bary` respectively

Options

A. 12and18
B. 18and12
C. 13and17
D. 17and13

Answer

To find the values of \( \bar{x} \) and \( \bar{y} \) from the given regression lines, we will follow these steps:

### Step 1: Write down the equations
We have two regression equations:
1. \( 4x - 5y + 33 = 0 \) (Equation 1)
2. \( 20x - 9y - 107 = 0 \) (Equation 2)

### Step 2: Rearrange Equation 1
Rearranging Equation 1 gives us:
\[ 4x - 5y = -33 \]
Let’s denote this as Equation 1.

### Step 3: Rearrange Equation 2
Rearranging Equation 2 gives us:
\[ 20x - 9y = 107 \]
Let’s denote this as Equation 2.

### Step 4: Multiply Equation 1
To eliminate \( x \), we can multiply Equation 1 by 5:
\[ 5(4x - 5y) = 5(-33) \]
This simplifies to:
\[ 20x - 25y = -165 \]
Let’s denote this as Equation 3.

### Step 5: Add Equation 2 and Equation 3
Now we will add Equation 2 and Equation 3:
\[ (20x - 9y) + (20x - 25y) = 107 - 165 \]
This simplifies to:
\[ 20x - 9y - 20x + 25y = -58 \]
The \( 20x \) terms cancel out:
\[ 16y = -58 \]

### Step 6: Solve for \( y \)
Now, we can solve for \( y \):
\[ y = \frac{-58}{16} = -3.625 \]

### Step 7: Substitute \( y \) back into Equation 1
Now we will substitute \( y \) back into Equation 1 to find \( x \):
\[ 4x - 5(-3.625) + 33 = 0 \]
This simplifies to:
\[ 4x + 18.125 + 33 = 0 \]
\[ 4x + 51.125 = 0 \]
Now, solving for \( x \):
\[ 4x = -51.125 \]
\[ x = \frac{-51.125}{4} = -12.78125 \]

### Final Values
Thus, the values of \( \bar{x} \) and \( \bar{y} \) are:
- \( \bar{x} = -12.78125 \)
- \( \bar{y} = -3.625 \)