If `aneb and (a^(2)-b^(2))/(9)=a+b`, then what is the value of `a-b`?

Options

A. `(1)/(3)`
B. `3`
C. `9`
D. `12`

Answer

To solve the equation \( \frac{a^2 - b^2}{9} = a + b \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ \frac{a^2 - b^2}{9} = a + b \]

### Step 2: Multiply both sides by 9 to eliminate the fraction
To eliminate the fraction, we multiply both sides by 9:
\[ a^2 - b^2 = 9(a + b) \]

### Step 3: Use the difference of squares formula
Recall that \( a^2 - b^2 \) can be factored as \( (a - b)(a + b) \):
\[ (a - b)(a + b) = 9(a + b) \]

### Step 4: Simplify the equation
Since \( a + b \) is a common term on both sides, we can divide both sides by \( a + b \) (noting that \( a \neq b \) implies \( a + b \neq 0 \)):
\[ a - b = 9 \]

### Conclusion
Thus, the value of \( a - b \) is:
\[ \boxed{9} \]

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