If ` a, b, c ` and d satisfy the equations
`{:( a + 7b + 3c + 5d = 0 ),( 8 a + 4b + 6 c + 2 d = - 4 ),( 2 a + 6 b + 4 c + 8 d = 4 ),(5a + 3b + 7c + d = - 4 ),("then " (a + d)//( b + c) = ? ):}`

Options

A. 0
B. 1
C. `-1`
D. `-4`

Answer

To solve the problem, we have the following equations:

1. \( a + 7b + 3c + 5d = 0 \) (Equation 1)
2. \( 8a + 4b + 6c + 2d = -4 \) (Equation 2)
3. \( 2a + 6b + 4c + 8d = 4 \) (Equation 3)
4. \( 5a + 3b + 7c + d = -4 \) (Equation 4)

We need to find the value of \( \frac{a + d}{b + c} \).

### Step 1: Add Equation 1 and Equation 4
We start by adding Equation 1 and Equation 4:

\[ (a + 7b + 3c + 5d) + (5a + 3b + 7c + d) = 0 + (-4) \]

This simplifies to:

\[ 6a + 10b + 10c + 6d = -4 \]

### Step 2: Factor out common terms
We can factor out the common terms from the left-hand side:

\[ 6(a + d) + 10(b + c) = -4 \]

### Step 3: Rearranging
Now, we can rearrange this equation:

\[ 6(a + d) = -4 - 10(b + c) \]

### Step 4: Add Equation 2 and Equation 3
Next, we add Equation 2 and Equation 3:

\[ (8a + 4b + 6c + 2d) + (2a + 6b + 4c + 8d) = -4 + 4 \]

This simplifies to:

\[ 10a + 10b + 10c + 10d = 0 \]

### Step 5: Factor out common terms
We can factor out the common terms from the left-hand side:

\[ 10(a + b + c + d) = 0 \]

### Step 6: Rearranging
This implies:

\[ a + b + c + d = 0 \]

### Step 7: Express \( d \)
From this equation, we can express \( d \):

\[ d = - (a + b + c) \]

### Step 8: Substitute \( d \) back into the equation
Now, we substitute \( d \) back into the equation we derived from Step 2:

\[ 6(a + (- (a + b + c))) + 10(b + c) = -4 \]

This simplifies to:

\[ 6(-b - c) + 10(b + c) = -4 \]

### Step 9: Combine like terms
Combining the terms gives us:

\[ (-6b - 6c + 10b + 10c) = -4 \]

This simplifies to:

\[ 4b + 4c = -4 \]

### Step 10: Divide by 4
Dividing the entire equation by 4 gives us:

\[ b + c = -1 \]

### Step 11: Substitute \( b + c \) back
Now we substitute \( b + c = -1 \) back into the equation we derived in Step 2:

\[ 6(a + d) + 10(-1) = -4 \]

This simplifies to:

\[ 6(a + d) - 10 = -4 \]

### Step 12: Solve for \( a + d \)
Rearranging gives us:

\[ 6(a + d) = 6 \]

Dividing by 6 gives:

\[ a + d = 1 \]

### Step 13: Calculate \( \frac{a + d}{b + c} \)
Now we can find \( \frac{a + d}{b + c} \):

\[ \frac{a + d}{b + c} = \frac{1}{-1} = -1 \]

Thus, the final answer is:

\[ \boxed{-1} \]