If the regression equations are `8x – 10y + 66 = 0` and `40x – 18y = 214`, the mean value of y is…..

Answer

To find the mean value of \( y \) given the regression equations \( 8x - 10y + 66 = 0 \) and \( 40x - 18y = 214 \), we can follow these steps:

### Step 1: Rewrite the regression equations
The first regression equation can be rewritten as:
\[ 8x - 10y + 66 = 0 \implies 8x - 10y = -66 \]
Dividing the entire equation by 2 gives:
\[ 4x - 5y = -33 \quad \text{(Equation 1)} \]

The second regression equation can be rewritten as:
\[ 40x - 18y = 214 \]
Dividing the entire equation by 2 gives:
\[ 20x - 9y = 107 \quad \text{(Equation 2)} \]

### Step 2: Substitute \( x \) in terms of \( y \)
From Equation 1, we can express \( 4x \) in terms of \( y \):
\[ 4x = 5y - 33 \]

### Step 3: Substitute into Equation 2
Now substitute \( 4x \) into Equation 2:
\[ 20x - 9y = 107 \]
Since \( 20x = 5 \times 4x \), we can write:
\[ 20x = 5(5y - 33) = 25y - 165 \]
Substituting this into Equation 2 gives:
\[ 25y - 165 - 9y = 107 \]

### Step 4: Simplify the equation
Now simplify the equation:
\[ 25y - 9y - 165 = 107 \]
\[ 16y - 165 = 107 \]

### Step 5: Solve for \( y \)
Now, add 165 to both sides:
\[ 16y = 107 + 165 \]
\[ 16y = 272 \]
Now divide by 16:
\[ y = \frac{272}{16} = 17 \]

### Conclusion
The mean value of \( y \) is:
\[ \boxed{17} \]

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