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

Options

A. 12 and 18
B. 18 and 12
C. 13 and 17
D. 17 and 13

Explanation

To solve the problem, we need to find the values of \( \bar{x} \) and \( \bar{y} \) given the two lines of regression:

1. \( 4x - 5y + 33 = 0 \)
2. \( 20x - 9y - 107 = 0 \)

### Step 1: Rewrite the equations
We can rearrange both equations to express \( y \) in terms of \( x \).

From the first equation:
\[ 4x - 5y + 33 = 0 \implies 5y = 4x + 33 \implies y = \frac{4}{5}x + \frac{33}{5} \]

From the second equation:
\[ 20x - 9y - 107 = 0 \implies 9y = 20x - 107 \implies y = \frac{20}{9}x - \frac{107}{9} \]

### Step 2: Set the equations equal to find \( \bar{x} \)
Since both equations represent the same \( y \) for the same \( x \), we can set them equal to each other:
\[ \frac{4}{5}x + \frac{33}{5} = \frac{20}{9}x - \frac{107}{9} \]

### Step 3: Clear the fractions
To eliminate the fractions, we can multiply through by the least common multiple of the denominators, which is 45:
\[ 45 \left( \frac{4}{5}x + \frac{33}{5} \right) = 45 \left( \frac{20}{9}x - \frac{107}{9} \right) \]
This simplifies to:
\[ 36x + 297 = 100x - 535 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ 36x + 297 + 535 = 100x \]
\[ 831 = 100x - 36x \]
\[ 831 = 64x \]

### Step 5: Solve for \( \bar{x} \)
Now, divide both sides by 64:
\[ x = \frac{831}{64} \implies \bar{x} = 12.984375 \approx 13 \]

### Step 6: Substitute \( \bar{x} \) back to find \( \bar{y} \)
Now we can substitute \( \bar{x} \) back into either of the original equations to find \( \bar{y} \). We will use the first equation:
\[ 4\bar{x} - 5\bar{y} + 33 = 0 \]
Substituting \( \bar{x} = 13 \):
\[ 4(13) - 5\bar{y} + 33 = 0 \]
\[ 52 - 5\bar{y} + 33 = 0 \]
\[ 85 - 5\bar{y} = 0 \implies 5\bar{y} = 85 \]
\[ \bar{y} = \frac{85}{5} = 17 \]

### Final Answer
Thus, the values of \( \bar{x} \) and \( \bar{y} \) are:
\[ \bar{x} = 13, \quad \bar{y} = 17 \]