If the two means of regression are 4x+3y+7=0 and 3x+4y+8=0, then find the means of x and y.

Answer

To find the means of \( x \) and \( y \) from the given regression equations, we will solve the two equations step by step.

### Step 1: Write down the given equations
The two means of regression are given as:
1. \( 4x + 3y + 7 = 0 \) (Equation 1)
2. \( 3x + 4y + 8 = 0 \) (Equation 2)

### Step 2: Rearrange the equations
We can rearrange both equations to express \( y \) in terms of \( x \).

From Equation 1:
\[ 3y = -4x - 7 \implies y = -\frac{4}{3}x - \frac{7}{3} \]

From Equation 2:
\[ 4y = -3x - 8 \implies y = -\frac{3}{4}x - 2 \]

### Step 3: Set the equations for \( y \) equal to each other
Since both expressions represent \( y \), we can set them equal to find \( x \):
\[ -\frac{4}{3}x - \frac{7}{3} = -\frac{3}{4}x - 2 \]

### Step 4: Clear the fractions
To eliminate the fractions, we can multiply through by 12 (the least common multiple of 3 and 4):
\[ 12 \left(-\frac{4}{3}x\right) - 12 \left(\frac{7}{3}\right) = 12 \left(-\frac{3}{4}x\right) - 12(2) \]
This simplifies to:
\[ -16x - 28 = -9x - 24 \]

### Step 5: Solve for \( x \)
Now, we can bring all \( x \) terms to one side and constant terms to the other side:
\[ -16x + 9x = -24 + 28 \]
\[ -7x = 4 \implies x = -\frac{4}{7} \]

### Step 6: Substitute \( x \) back to find \( y \)
Now that we have \( x \), we can substitute it back into either equation to find \( y \). We will use Equation 1:
\[ 4\left(-\frac{4}{7}\right) + 3y + 7 = 0 \]
\[ -\frac{16}{7} + 3y + 7 = 0 \]
Convert 7 to a fraction:
\[ -\frac{16}{7} + 3y + \frac{49}{7} = 0 \]
Combine the constants:
\[ 3y + \frac{33}{7} = 0 \]
\[ 3y = -\frac{33}{7} \implies y = -\frac{11}{7} \]

### Final Answer
Thus, the means of \( x \) and \( y \) are:
\[ \text{Mean of } x = -\frac{4}{7}, \quad \text{Mean of } y = -\frac{11}{7} \]