Calculate the value of (a + b),
if   y = `a+b/x` where a, b are real numbers and y = 1 when x = -1, y = 5 when x = -5.

Options

A. 9
B. 11
C. 15
D. 7

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) using the given equations based on the values of \( y \) and \( x \).

The equation we have is:
\[ y = a + \frac{b}{x} \]

We know:
1. When \( x = -1 \), \( y = 1 \)
2. When \( x = -5 \), \( y = 5 \)

### Step 1: Substitute the first set of values into the equation
Using the first condition \( y = 1 \) when \( x = -1 \):
\[ 1 = a + \frac{b}{-1} \]
This simplifies to:
\[ 1 = a - b \quad \text{(Equation 1)} \]

### Step 2: Substitute the second set of values into the equation
Using the second condition \( y = 5 \) when \( x = -5 \):
\[ 5 = a + \frac{b}{-5} \]
This simplifies to:
\[ 5 = a - \frac{b}{5} \quad \text{(Equation 2)} \]

### Step 3: Solve the two equations simultaneously
Now we have two equations:
1. \( a - b = 1 \)
2. \( a - \frac{b}{5} = 5 \)

We can solve these equations to find \( a \) and \( b \).

### Step 4: Express \( a \) from Equation 1
From Equation 1:
\[ a = b + 1 \quad \text{(Substituting this into Equation 2)} \]

### Step 5: Substitute \( a \) in Equation 2
Substituting \( a \) in Equation 2:
\[ (b + 1) - \frac{b}{5} = 5 \]
Multiply through by 5 to eliminate the fraction:
\[ 5(b + 1) - b = 25 \]
This simplifies to:
\[ 5b + 5 - b = 25 \]
Combine like terms:
\[ 4b + 5 = 25 \]
Subtract 5 from both sides:
\[ 4b = 20 \]
Divide by 4:
\[ b = 5 \]

### Step 6: Substitute \( b \) back to find \( a \)
Now substitute \( b = 5 \) back into Equation 1:
\[ a - 5 = 1 \]
Add 5 to both sides:
\[ a = 6 \]

### Step 7: Calculate \( a + b \)
Now we can find \( a + b \):
\[ a + b = 6 + 5 = 11 \]

### Final Answer
Thus, the value of \( a + b \) is:
\[ \boxed{11} \]