Find the sum of `barx` and `bary` for the two regression lines: `40x-18y-214=0` and `8x-10y +66=0`.

Answer

To find the sum of \( \bar{x} \) and \( \bar{y} \) for the two regression lines given by the equations \( 40x - 18y - 214 = 0 \) and \( 8x - 10y + 66 = 0 \), we will follow these steps:

### Step 1: Rewrite the equations in standard form
The two regression lines can be rewritten as:
1. \( 40x - 18y = 214 \) (Equation 1)
2. \( 8x - 10y = -66 \) (Equation 2)

### Step 2: Solve for the point of intersection
To find the point of intersection of these two lines, we can solve the equations simultaneously. We will eliminate one variable by manipulating the equations.

### Step 3: Eliminate one variable
Let's eliminate \( x \) by multiplying Equation 2 by 5:
\[ 5 \times (8x - 10y) = 5 \times (-66) \]
This gives us:
\[ 40x - 50y = -330 \quad \text{(Equation 3)} \]

### Step 4: Subtract Equation 3 from Equation 1
Now, we subtract Equation 3 from Equation 1:
\[ (40x - 18y) - (40x - 50y) = 214 - (-330) \]
This simplifies to:
\[ -18y + 50y = 214 + 330 \]
\[ 32y = 544 \]

### Step 5: Solve for \( y \)
Now, divide both sides by 32:
\[ y = \frac{544}{32} = 17 \]

### Step 6: Substitute \( y \) back to find \( x \)
Now that we have \( y \), we can substitute it back into either Equation 1 or Equation 2. Let's use Equation 2:
\[ 8x - 10(17) = -66 \]
\[ 8x - 170 = -66 \]
Now, add 170 to both sides:
\[ 8x = 104 \]
Now, divide by 8:
\[ x = \frac{104}{8} = 13 \]

### Step 7: Find \( \bar{x} \) and \( \bar{y} \)
The point of intersection is \( (13, 17) \). Therefore, we have:
\[ \bar{x} = 13 \quad \text{and} \quad \bar{y} = 17 \]

### Step 8: Calculate the sum \( \bar{x} + \bar{y} \)
Now, we can find the sum:
\[ \bar{x} + \bar{y} = 13 + 17 = 30 \]

### Final Answer
Therefore, the sum of \( \bar{x} \) and \( \bar{y} \) is:
\[ \boxed{30} \]